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stringlengths 3
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|---|---|---|
keyboard that voices out digits inputted. <p>I input lots of digits during my daily work, mostly with keyboard's keypad on the right side. To improve input accuracy, i'd like to hear the digits I just inputted, so I do not need to check again what's inputted actually, as I input wrong digit or leave out digit for my reason or keyboard's reason once in a while.</p>
<p>So I am wondering is there any keyboard that can read out what's just inputted? Or is there an app to do this kind of work?</p>
<p>I am using an MacOS system.</p>
<p>Any comment is welcome.</p>
<p>Thanks!</p>
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| Stackexchange |
How do I run a test in Django using production data (Read only)?. <p>I would like to run a test in Django but in order to run the test I need access to data from the production database (I don't want to setup an entirely new database for testing).</p>
<p>How can use this data from the production database without allowing write access in my script? I don't want this test to effect anything, but I need access to this data in order to run this test. It seems ridiculous to have to launch (and pay for) an entirely new database just to grab a few rows of data, any advice? </p>
| 0non-cybersec
| Stackexchange |
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application.
I need to scale my app adding some spot instances that EB do not support.</p>
<p>So I create a second autoscaling from a launch configuration with spot instances.
The autoscaling use the same load balancer created by beanstalk.</p>
<p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p>
<p>This work fine, but:</p>
<ol>
<li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p>
</li>
<li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p>
</li>
</ol>
<p><strong>UPDATE</strong></p>
<p>Elastic Beanstalk add support to spot instance since 2019... see:
<a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>
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| Stackexchange |
I usually travel internationally but the states can provide some coastal beauty as well. Central Coast, California.. | 0non-cybersec
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VAIO laptop's wireless indicator blinking when connected to access point. <p>I have a VAIO laptop model VGN-C22GH. It has a built in wireless card with a Intel Centrino processor on his motherboard.</p>
<p>When using Ubuntu 10.04, the wireless indicator blinks violently. This symptom does not appear with earlier versions of Ubuntu like 9.10 or 8.10.</p>
<p>This leads me to believe that it is a driver issue. Is it something that I should worry about?</p>
<p>I have no idea where to look for, since this is a very limited scope problem. Should I report to Cannonical about this problem of mine??</p>
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[Video] Each. Fresh. Day.. | 0non-cybersec
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Atheros AR9285 bluetooth dual mode not working on lubuntu 16.04. <p>My problem is that I have a AR9285 bluetooth + wifi card in a HP probook 4330s. I tried several methods from the internet, still not working the bluetooth dual mode. </p>
<ul>
<li>Enabling the btcoex=1</li>
</ul>
<p>Main symptomps: first after boot the bluetooth.service is in halt state. After restart I can start the bluetooth manager. But When I open, I can't connect to any devices and it drops there is no bluetooth adapters.</p>
<p>here is the outputs.</p>
<p><code>uname -a</code>:</p>
<p><code>Linux nolex-dev 4.10.0-35-generic #39~16.04.1-Ubuntu SMP Wed Sep 13 09:02:42 UTC 2017 x86_64 x86_64 x86_64 GNU/Linux</code></p>
<p><code>lsusb -t</code>:</p>
<pre><code>/: Bus 04.Port 1: Dev 1, Class=root_hub, Driver=xhci_hcd/2p, 5000M
/: Bus 03.Port 1: Dev 1, Class=root_hub, Driver=xhci_hcd/2p, 480M
/: Bus 02.Port 1: Dev 1, Class=root_hub, Driver=ehci-pci/2p, 480M
|__ Port 1: Dev 2, If 0, Class=Hub, Driver=hub/6p, 480M
|__ Port 2: Dev 3, If 0, Class=Human Interface Device, Driver=usbhid, 12M
|__ Port 2: Dev 3, If 1, Class=Human Interface Device, Driver=usbhid, 12M
|__ Port 2: Dev 3, If 2, Class=Human Interface Device, Driver=usbhid, 12M
|__ Port 4: Dev 4, If 1, Class=Video, Driver=uvcvideo, 480M
|__ Port 4: Dev 4, If 0, Class=Video, Driver=uvcvideo, 480M
|__ Port 6: Dev 5, If 0, Class=Wireless, Driver=, 12M
|__ Port 6: Dev 5, If 1, Class=Wireless, Driver=, 12M
/: Bus 01.Port 1: Dev 1, Class=root_hub, Driver=ehci-pci/2p, 480M
|__ Port 1: Dev 2, If 0, Class=Hub, Driver=hub/6p, 480M
|__ Port 3: Dev 5, If 0, Class=Wireless, Driver=btusb, 12M
|__ Port 3: Dev 5, If 1, Class=Wireless, Driver=btusb, 12M
</code></pre>
<p><code>lsmod</code>:</p>
<pre><code>Module Size Used by
rfcomm 77824 0
bnep 20480 2
intel_rapl 20480 0
x86_pkg_temp_thermal 16384 0
intel_powerclamp 16384 0
coretemp 16384 0
kvm_intel 200704 0
ppdev 20480 0
hp_wmi 16384 0
sparse_keymap 16384 1 hp_wmi
kvm 593920 1 kvm_intel
irqbypass 16384 1 kvm
crct10dif_pclmul 16384 0
crc32_pclmul 16384 0
ghash_clmulni_intel 16384 0
pcbc 16384 0
aesni_intel 167936 0
aes_x86_64 20480 1 aesni_intel
crypto_simd 16384 1 aesni_intel
glue_helper 16384 1 aesni_intel
cryptd 24576 3 crypto_simd,ghash_clmulni_intel,aesni_intel
arc4 16384 2
snd_hda_codec_hdmi 49152 1
intel_cstate 20480 0
uvcvideo 90112 0
ath9k 147456 0
ath9k_common 36864 1 ath9k
videobuf2_vmalloc 16384 1 uvcvideo
ath9k_hw 466944 2 ath9k,ath9k_common
videobuf2_memops 16384 1 videobuf2_vmalloc
videobuf2_v4l2 24576 1 uvcvideo
snd_hda_codec_idt 57344 1
snd_hda_codec_generic 73728 1 snd_hda_codec_idt
intel_rapl_perf 16384 0
ath 28672 3 ath9k_hw,ath9k,ath9k_common
snd_hda_intel 36864 3
videobuf2_core 40960 2 uvcvideo,videobuf2_v4l2
snd_hda_codec 126976 4 snd_hda_intel,snd_hda_codec_idt,snd_hda_codec_hdmi,snd_hda_codec_generic
mac80211 782336 1 ath9k
joydev 20480 0
snd_hda_core 81920 5 snd_hda_intel,snd_hda_codec,snd_hda_codec_idt,snd_hda_codec_hdmi,snd_hda_codec_generic
videodev 172032 3 uvcvideo,videobuf2_core,videobuf2_v4l2
media 40960 2 uvcvideo,videodev
input_leds 16384 0
snd_hwdep 16384 1 snd_hda_codec
btusb 45056 0
btrtl 16384 1 btusb
serio_raw 16384 0
btbcm 16384 1 btusb
btintel 16384 1 btusb
snd_pcm 102400 4 snd_hda_intel,snd_hda_codec,snd_hda_core,snd_hda_codec_hdmi
bluetooth 557056 14 btrtl,btintel,bnep,btbcm,rfcomm,btusb
lpc_ich 24576 0
cfg80211 602112 4 mac80211,ath9k,ath,ath9k_common
jmb38x_ms 20480 0
snd_timer 32768 1 snd_pcm
memstick 16384 1 jmb38x_ms
snd 77824 14 snd_hda_intel,snd_hwdep,snd_hda_codec,snd_hda_codec_idt,snd_timer,snd_hda_codec_hdmi,snd_hda_codec_generic,snd_pcm
soundcore 16384 1 snd
shpchp 36864 0
mei_me 40960 0
mei 102400 1 mei_me
hp_accel 28672 0
lis3lv02d 20480 1 hp_accel
parport_pc 32768 0
mac_hid 16384 0
parport 49152 2 parport_pc,ppdev
input_polldev 16384 1 lis3lv02d
autofs4 40960 2
hid_logitech_hidpp 28672 0
hid_logitech_dj 20480 0
usbhid 53248 0
hid 118784 4 usbhid,hid_logitech_dj,hid_logitech_hidpp
amdkfd 139264 1
amd_iommu_v2 20480 1 amdkfd
i915 1449984 3
radeon 1507328 1
psmouse 139264 0
ahci 36864 1
libahci 32768 1 ahci
r8169 81920 0
ttm 98304 1 radeon
i2c_algo_bit 16384 2 radeon,i915
mii 16384 1 r8169
drm_kms_helper 151552 2 radeon,i915
syscopyarea 16384 1 drm_kms_helper
sysfillrect 16384 1 drm_kms_helper
sysimgblt 16384 1 drm_kms_helper
fb_sys_fops 16384 1 drm_kms_helper
sdhci_pci 28672 0
drm 352256 7 radeon,i915,ttm,drm_kms_helper
sdhci 45056 1 sdhci_pci
wmi 16384 1 hp_wmi
video 40960 1 i915
fjes 77824 0
</code></pre>
<p><code>cat /etc/modprobe.d/ath9k.conf</code>:</p>
<pre><code>options ath9k nohwcrypt=1
options ath9k ps_enable=1
options ath9k blink=1
options ath9k btcoex_enable=1
</code></pre>
<p><code>rfkill list</code>:</p>
<pre><code>1: phy0: Wireless LAN
Soft blocked: no
Hard blocked: no
2: hp-wifi: Wireless LAN
Soft blocked: no
Hard blocked: no
3: hp-bluetooth: Bluetooth
Soft blocked: no
Hard blocked: no
</code></pre>
<p><code>lspci -knn | grep Net -A3; lsusb; dmesg | grep -E 'Blue|ath'</code>:</p>
<pre><code>25:00.0 Network controller [0280]: Qualcomm Atheros AR9285 Wireless Network Adapter (PCI-Express) [168c:002b] (rev 01)
DeviceName: WLAN
Subsystem: Hewlett-Packard Company AR9285 Wireless Network Adapter (PCI-Express) [103c:1461]
Kernel driver in use: ath9k
Kernel modules: ath9k
26:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller [10ec:8168] (rev 06)
Bus 002 Device 005: ID 03f0:311d Hewlett-Packard Atheros AR9285 Malbec Bluetooth Adapter
Bus 002 Device 004: ID 04f2:b230 Chicony Electronics Co., Ltd Integrated HP HD Webcam
Bus 002 Device 003: ID 046d:c52b Logitech, Inc. Unifying Receiver
Bus 002 Device 002: ID 8087:0024 Intel Corp. Integrated Rate Matching Hub
Bus 002 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
Bus 004 Device 001: ID 1d6b:0003 Linux Foundation 3.0 root hub
Bus 003 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
Bus 001 Device 002: ID 8087:0024 Intel Corp. Integrated Rate Matching Hub
Bus 001 Device 001: ID 1d6b:0002 Linux Foundation 2.0 root hub
[ 9.025384] Bluetooth: Core ver 2.22
[ 9.025402] Bluetooth: HCI device and connection manager initialized
[ 9.025405] Bluetooth: HCI socket layer initialized
[ 9.025408] Bluetooth: L2CAP socket layer initialized
[ 9.025413] Bluetooth: SCO socket layer initialized
[ 9.213639] ath: phy0: Disabling ASPM since BTCOEX is enabled
[ 9.213642] ath: EEPROM regdomain: 0x60
[ 9.213643] ath: EEPROM indicates we should expect a direct regpair map
[ 9.213645] ath: Country alpha2 being used: 00
[ 9.213646] ath: Regpair used: 0x60
[ 9.265926] ath9k 0000:25:00.0 wlo1: renamed from wlan0
[ 14.703648] Bluetooth: BNEP (Ethernet Emulation) ver 1.3
[ 14.703650] Bluetooth: BNEP filters: protocol multicast
[ 14.703654] Bluetooth: BNEP socket layer initialized
</code></pre>
<p><code>hcitool dev</code> doesn't drop any kind of adapters. When I plug a custom BT adapter, drops back a device handler for this.</p>
<p>In my opinion the correct device descriptor is missing from the ath9k driver.</p>
<p><strong>EDIT1:</strong></p>
<p>Removed ath9k.conf</p>
<p>I tried:</p>
<pre><code>modprobe -rfv ath9k
modprobe ath9k btcoex_enable=1
[ 300.117260] ath: phy0: Disabling ASPM since BTCOEX is enabled
[ 300.117262] ath: EEPROM regdomain: 0x60
[ 300.117263] ath: EEPROM indicates we should expect a direct regpair map
[ 300.117264] ath: Country alpha2 being used: 00
[ 300.117264] ath: Regpair used: 0x60
[ 300.118861] ieee80211 phy0: Selected rate control algorithm 'minstrel_ht'
[ 300.120210] ieee80211 phy0: Atheros AR9285 Rev:2 mem=0xffffade802450000, irq=19
[ 300.120827] ath9k 0000:25:00.0 wlo1: renamed from wlan0
</code></pre>
<p>Adapter not found, same results.</p>
| 0non-cybersec
| Stackexchange |
Zen is not a word? Really?. | 0non-cybersec
| Reddit |
SHAXX MOTIVATION 80s Workout Mixtape RADIO. I made dis :D #MOTW
https://youtu.be/F1HZ8bjL8oU | 0non-cybersec
| Reddit |
Always flexing.. | 0non-cybersec
| Reddit |
A bro in need is a bro indeed. | 0non-cybersec
| Reddit |
Node.native: C++11 (aka C++0x) port for node.js. | 0non-cybersec
| Reddit |
Omit the target dir from find results. <p>How can I prevent <code>find</code> from returning the directory I use as the root to start searching from in the results? e.g.:</p>
<pre><code>$ find targetDir -name 'target*'
targetDir/target1
targetDir/target2
targetDir/subDir/target3
</code></pre>
<p>instead of:</p>
<pre><code>$ find targetDir -name 'target*'
targetDir
targetDir/target1
targetDir/target2
targetDir/subDir/target3
</code></pre>
| 0non-cybersec
| Stackexchange |
Saving Dataset in Hive as avro format merge column. <p>I try to put in hive in avro format, data from dataset. but each column in my dataset is merged into one in the hive table.</p>
<pre><code>Dataset<obj1> = ....
Dataset<obj1>.printSchema();
root
|-- a: double (nullable = true)
|-- b: string (nullable = true)
|-- c: string (nullable = true)
|-- d: string (nullable = true)
|-- e: string (nullable = true)
</code></pre>
<p>Save the dataset in Hive : </p>
<pre><code>Dataset<obj1>.write()
.mode(SaveMode.Overwrite)
.partitionBy("a")
.format("com.databricks.spark.avro")
.option("recordName", "recordName_custom")
.option("recordNamespace", "recordNamespace_custom")
.saveAsTable("DB.TABLE");
</code></pre>
<p>result of the table created in hive : </p>
<pre><code>show create table DB.TABLE;
</code></pre>
<p>result : </p>
<pre><code> CREATE TABLE `DB.TABLE`(
`col` array<string> COMMENT 'from deserializer')
PARTITIONED BY (
`a` string)
...
</code></pre>
<p>At the moment, one of the solution is to put dataset as avro file into HDFS :</p>
<pre><code> Dataset<obj1>
.write()
.mode(SaveMode.Overwrite)
.partitionBy("a")
.format("com.databricks.spark.avro")
.option("recordName", "recordName_custom")
.option("recordNamespace", "recordNamespace_custom")
.save("path");
</code></pre>
<p>then create a external table to it.
at the end of the batch, we do a MSCK REPAIR to detect new partition if needed.</p>
<p>any solution, best practice advise ? </p>
<p>Version used: </p>
<p>com.databricks, spark-avro_2.11: 4.0.0<br>
Spark: 2.3.2<br>
Hadoop: 2.3.2<br>
HDFS: 3.1.1.3.1<br>
Hive: 3.1.0</p>
| 0non-cybersec
| Stackexchange |
He has legitimate reason. | 0non-cybersec
| Reddit |
My boyfriend and I have the same birth mark on the same finger in the same location. | 0non-cybersec
| Reddit |
The future of the book—interesting essay on book technology and what it might mean for literature as we know it. | 0non-cybersec
| Reddit |
Remainder of $15^{81}$ divided by $13$ without using Fermat's Little theorem.. <p>I was requested to find the congruence of <span class="math-container">$15^{81}\mod{13}$</span> without using Fermat's theorem (since that is covered in the chapter that follows this exercise). Of course I know that by property <span class="math-container">$15^{81} \equiv 2^{81} \pmod{13}$</span>, but how could I find what is the congruence of <span class="math-container">$2^{81}$</span> without using Fermat? Needless it is to say that an exhaustive iterative method would be extremely long.</p>
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5 Delicious July 4th Recipes!. | 0non-cybersec
| Reddit |
How to use the stdin as a file. <p>I have a command that wants a file as input, but I get the input from the <code>stdin</code> and I don't want to create a temporary file.</p>
<p>I would like to do the following:</p>
<pre><code>#!/bin/bash
osadecompile /dev/stdin < input_file.scpt
</code></pre>
<p>but that causes an I/O error. The followingworks, but is it possible <em>without</em> using a temporary intermediate file_?</p>
<pre><code>#!/bin/bash
AS_FILE=$(mktemp)
cat - > "$AS_FILE"
osadecompile "$AS_FILE"
rm "$AS_FILE"
</code></pre>
<p>As you might have guessed, I'm on macOS with <strong>bash 3.2</strong>. However, a <strong>zsh 5.3</strong> solution would also be welcomed.</p>
| 0non-cybersec
| Stackexchange |
Is it possible to use custom type definitions in an ES6 project?. <p>My team works on a relatively large NodeJS project, written in ES6, transpiled by babel, and then deployed as AWS lambdas with Serverless. This project is focused around consuming, mapping/transforming, and outputting one specific object type, which we have defined. </p>
<p>Our problem is, ECMA/JavaScript is not strongly typed, so if we make a mistake like treating a field as an array somewhere and a string somewhere else, there's nothing to catch that except runtime errors. We have also poorly documented the structure of this object, so sometimes consumers send us instances of the object with data in slightly misnamed fields that we say we process, but don't actually use. </p>
<p>I am looking for a way to create some kind of schema or type definition for this specific object in our project, so we can use that to correct our code, make our processing more robust, and create much better documentation for it. Now, I know VSCode offers some <a href="https://github.com/Microsoft/TypeScript/wiki/Type-Checking-JavaScript-Files" rel="noreferrer">basic type checking</a> in JavaScript, but I don't think it's feasible to try to JSDoc a really big object and then put that doc in every file that uses the object. I have found that VSCode can also, somehow, drive that checking <a href="https://code.visualstudio.com/docs/languages/javascript#_type-checking-and-quick-fixes-for-javascript-files" rel="noreferrer">with .d.ts files</a> but I don't understand if or how I can leverage that for a specific, custom object that we've designed. Most of what I have found seems to be specifically related to pulling .d.ts files for external libraries. </p>
<p>So, TL:DR, Is it possible, in a NodeJS/ES6 project, to make one object, widely used throughout that project, strongly typed? Error checking in VSCode would be acceptable, but some kind of command-line linting that we could trigger before transpiling would be great too. </p>
| 0non-cybersec
| Stackexchange |
How can I paste text into Confluence without losing formatting?. <p>I have some verbiage in a plain-text email that I would like to paste into a Confluence doc.</p>
<p>How can I do this without losing the formatting? In particular, there are some columns of numbers aligned with spaces. Pasting strips out all the spaces, so even if I specify a "preformatted" paragraph I've lost all alignment.</p>
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Do what you can, with what you have, where you are.. | 0non-cybersec
| Reddit |
ar
X
iv
:0
71
1.
21
81
v2
[
m
at
h.
K
T
]
1
4
Ja
n
20
09
The KH-Isomorphism Conjecture and Algebraic
KK-theory
Paul D. Mitchener
University of Sheffield
e-mail: [email protected]
Web-site: http://www.mitchener.staff.shef.ac.uk
November 14, 2018
Abstract
In this article we prove that the KH-asembly map, as defined by Bartels and
Lück, can be described in terms of the algebraic KK-theory of Cortinas and
Thom. The KK-theory description of the KH-assembly map is similar to that
of the Baum-Connes assembly map. In some elementary cases, methods used
to prove the Baum-Connes conjecture also apply to the KH-isomorphism con-
jecture.
Contents
1 Introduction 2
2 Algebroids 3
3 Tensor Products 5
4 Algebraic Homotopy and Simplicial Enrichment 8
5 Path Extensions 11
6 The KK-theory spectrum 12
7 KK-theory groups 14
8 Modules over Algebroids 18
9 Equivariant KK-theory 22
10 Assembly 28
1
http://arxiv.org/abs/0711.2181v2
http://www.mitchener.staff.shef.ac.uk
11 Algebraic KK-theory and homology 31
12 The Index Map 32
1 Introduction
Various assembly maps, such as the Baum-Connes assembly map (see [2, 26])
and the Farrell-Jones assembly map (see [7]) can be described using an ab-
stract homotopy-theoretic framework, developed by Davis and Lück in [6]; the
machinery is based upon spectra and equivariant homology theories.
Recently, in [1], Bartels and Lück introduced the KH-assembly map. The
KH-assembly map is constructed using the abstract machinery, and resem-
bles the Farrell-Jones assembly map in algebraic K-theory. However, the KH-
assembly map uses homotopy algebraic K-theory, as introduced in [27], rather
than ordinary algebraic K-theory.
Kasparov’s bivariant K-theory, orKK-theory (see for example [3, 12, 24, 26]
for accounts) is in essence a way of organizing certain maps between the K-
theory groups of C∗-algebras. The Baum-Connes assembly map is such a map,
and is usually described using KK-theory; some work is needed (see [10, 22])
to show that the KK-theory assembly map can be constructed using the Davis-
Lück machinery.
Recently, in [4], Cortinas and Thom developed a version of bivariant algebraic
K-theory. This version of KK-theory induces maps between the homotopy
algebraic K-theory groups of discrete algebras. The construction is broadly
speaking similar to the construction of KK-theory for locally convex algebras
in [5], but replaces smooth homotopies with algebraic homotopies.
The purpose of this article is essentially an algebraic version of the converse
of [22]; we show that the abstract KH-assembly map can be described in terms
of algebraicKK-theory in terms similar to that of the Baum-Connes conjecture.
We also look at some elementary consequences of this description, namely some
elementary cases where one can easily show that the KH-assembly map is an
isomorphism.
It is fairly easy to write down an algebraic KK-theory version of the Baum-
Connes assembly map. However, to associate this map to the Davis-Lück ma-
chinery, we need to write this map at the level of spectra, and moreover to
generalize algebraic KK-theory from algebras to algebroids, and to equivariant
algebras based on groupoids.
Thus, before we look at assembly maps, we describe KK-theory spectra for
algebroids and groupoid algebras, before looking at some basic properties. These
definitions and properties are mainly, but not entirely, easy generalisations of
those in [4].
2
2 Algebroids
The following definition is a slight generalization of the notion of an algebroid
in [20].
Definition 2.1 Let R be a commutative unital ring. A R-algebroid, A, consists
of a set of objects, Ob(A), along with a left R-module, Hom(a, b)A for each pair
of objects a, b ∈ Ob(A), such that:
• We have an associative R-bilinear composition law
Hom(b, c)A ×Hom(a, b)A → Hom(a, c)A
• Given an element r ∈ R and morphisms x ∈ Hom(a, b)A, y ∈ Hom(b, c)A,
the equation r(xy) = (rx)y = x(ry) holds.
We call an R-algebroid A unital if it is a category. Thus, in a unital R-
algebroid we have an identity element 1a ∈ Hom(a, a)A for each object a.
An R-algebra can be considered to be the same thing as an R-algebroid with
one object.
Definition 2.2 Let A and B be R-algebroids. Then a homomorphism α : A →
B consists of maps α : Ob(A) → Ob(B) and R-linear maps α : Hom(a, b)A →
Hom(α(a), α(b))B that are compatible with the composition law.
Given R-algebroids A and B, we write Hom(A,B) to denote the set of all
homomorphisms.
Definition 2.3 Let A be an R-algebroid. Then we define the additive comple-
tion, A⊕, to be the R-algebroid in which the objects are formal sequences of
the form
a1 ⊕ · · · ⊕ an ai ∈ Ob(A)
where n ∈ N. Repetitions are allowed in such formal sequences. The empty
sequence is also allowed, and labelled 0.
The R-module Hom(a1 ⊕ · · · ⊕ am, b1 ⊕ · · · ⊕ bn)A⊕ is defined to be the set
of matrices of the form
x11 · · · x1m
...
. . .
...
xn1 · · · xnm
xij ∈ Hom(aj , bi)A
with element-wise addition and multiplication by elements of the ring R.
The composition law is defined by matrix multiplication.
3
Given an algebroid homomorphism α : A → B, there is an induced homo-
morphism α⊕ : A⊕ → B⊕ defined by writing
α⊕(a1 ⊕ · · · an) = α(a1)⊕ · · · ⊕ α(an) ai ∈ Ob(A)
and
α⊕
x1,1 · · · x1,m
...
. . .
...
xn,1 · · · xn,m
=
α(x1,1) · · · α(x1,m)
...
. . .
...
α(xn,1) · · · α(xn,m)
xi,j ∈ Hom(aj , bi)
With such induced homomorphisms, the process of additive completion de-
fines a functor from the category of R-algebroids and homomorphisms to itself.
We define the direct sum of two objects a = a1⊕· · ·⊕am and b = b1⊕· · ·⊕bn
in the additive completion A⊕ by writing
a⊕ b = a1 ⊕ · · · ⊕ am ⊕ b1 ⊕ · · · ⊕ bn
In the unital case, the additive completion of a unital R-algebroid is an
additive category;1 direct sums of objects can be defined as above. The induced
functor, α⊕, is an additive functor. Analogous properties hold in the non-unital
case.
Finally, given an R-algebroid A, we would like to define an associated R-
algebra that carries the same information as the additive completion A⊕. The
naive way to do this is to simply form the direct limit
⋃
ai∈Ob(A)
Hom(a1 ⊕ · · · ⊕ an, a1 ⊕ · · · ⊕ an)A⊕
with respect to the inclusions
Hom(a⊕ c, a⊕ c)A⊕ → Hom(a⊕ b⊕ c, a⊕ b⊕ c)A⊕
(
w x
y z
)
7→
w 0 x
0 0 0
y 0 z
Unfortunately, the above construction is not functorial. We can, however,
replace it by an equivalent functorial construction. This construction is essen-
tially the same as that used to define the K-theory of C∗-categories in [14] or
[21].
Let A be an R-algeboid. Then we define OA be the category in which the
set of objects consists of all compositions of inclusions Hom(a⊕ c, a⊕ c)A⊕ →
Hom(a⊕ b⊕ c, a⊕ b⊕ c)A⊕ of the form
(
w x
y z
)
7→
w 0 x
0 0 0
y 0 z
1See [18] or [28] for relevant definitions.
4
A morphism set between two inclusions has precisely one element if the
inclusions are composable; otherwise, it is empty.
We can define a functor, HA, from the category OA to the category of R-
algebras by associating the R-algebra Hom(a ⊕ c, a ⊕ c)A⊕ to the inclusion
Hom(a⊕ c, a⊕ c)A⊕ → Hom(a⊕ b⊕ c, a⊕ b⊕ c)A⊕ . If i and j are composable
inclusions, then the one morphism in the set Hom(i, j)OA is mapped to the
inclusion i itself.
Definition 2.4 Let A be an R-algebroid. Then we define the R-algebra AH to
be the colimit of the functor HA.
The following result is obvious from our constructions.
Proposition 2.5 The assignment A⊕ 7→ AH is a covariant functor, and we
have a natural transformation J : A⊕ → AH . The natural transformation F is
surjective on each morphism set.
Further, given a homomorphism α : A → BH , we have a factorisation
A
α′
→ B⊕
J
→ BH .
✷
3 Tensor Products
Definition 3.1 Let A and B be R-algebroids. Then we define the tensor prod-
uct A ⊗R B to be the R-algebroid with the set of objects Ob(A) × Ob(B). We
write the object (a, b) in the form a⊗ b. The R-module Hom(a⊗ b, a′⊗ b′)A⊗RB
is the tensor product of R-modules Hom(a, a′)A ⊗R Hom(b, b
′)B.
The composition law is defined in the obvious way.
If we view an R-algebra as an R-algebroid with just one object, we can form
the tensor product, A ⊗R B, of an R-algebroid A with an R-algebra B. The
objects of the tensor product A ⊗R B are identified with the objects of the
algebroid A.
Definition 3.2 Let R be a commutative ring with an identity element. An
R-moduloid, E , consists of a collection of objects Ob(E), along with a left R-
module, Hom(a, b)E defined for each pair of objects a, b ∈ Ob(E).
A homomorphism, φ : E → F , between R-moduloids consists of a map
φ : Ob(E) → Ob(F) and a collection of R-linear maps φ : Hom(a, b)E →
Hom(φ(a), φ(b))F .
The difference between an R-moduloid and an R-algebroid is that there is
no composition law between the various R-bimodules Hom(a, b)E . There is of
course a forgetful functor, F , from the category of R-algebroids and homomor-
phisms to the category of R-moduloids and homomorphisms.
5
Definition 3.3 Let A be an R-moduloid. Given objects a, b ∈ Ob(A), let us
define
Hom(a, b)
⊗(k+1)
A =
⊕
ci∈Ob(A)
Hom(a, c1)⊗R Hom(c1, c2)⊗R · · · ⊗Hom(ck, b)
The tensor algebroid, TA, is the R-algebroid with the same set of objects as
the R-moduloid A where the morphism set Hom(a, b)TA is the direct sum
∞
⊕
k=1
Hom(a, b)⊗kA
Here the the R-moduleHom(a, b)⊗1A is simply the morphism set Hom(a, b)A.
Composition of morphisms in the tensor category is defined by concatenation
of tensors.
Formation of the tensor category defines a functor, T , from the category ofR-
moduloids to the category of R-algebroids. We will abuse notation slightly, and
also write TA to denote the tensor algebroid when R is already an R-algebroid.
This definition of course ignores the multiplicative structure.
Proposition 3.4 The functor T is naturally left-adjoint to the forgetful functor
F .
Proof: We need a natural R-linear bijection between the morphism sets
Hom(TA,B) and Hom(A, FB) when A is an R-moduloid and B is an R-
algebroid.
Let α : TA → B be a homomorphism of R-algebroids. The morphism set
Hom(a, b)TA is the sum
∞
⊕
k=1
ci∈Ob(A)
Hom(a, c1)⊗R Hom(c1, c2)⊗R · · · ⊗Hom(ck, b)
We have an induced homomorphism of R-moduloids, G(α) : A → FB, de-
fined to be α on the set of objects, and by the restriction G(α) = α|Hom(a,b)A
on morphism sets.
Conversely, given an R-moduloid homomorphism β : A → FB, we have an
R-algebroid homomorphism H(β) : TA → B, defined to be β on the set of
objects, and by the formula
H(β)(x1 ⊗ · · · ⊗ xk) = β(x1) · · ·β(xk)
for each morphism of the form x1 ⊗ · · · ⊗ xk in the tensor category.
It is easy to check that the mapsG andH areR-linear, natural, and mutually
inverse. ✷
Now, let A be an R-algebroid. Then there is a canonical homomorphism of
R-moduloids σ : A → TA defined by mapping each morphism set of the category
A onto the first summand.
6
Proposition 3.5 Let α : A → B be a homomorphism of R-algebroids. Then
there is a unique R-algebroid homomorphism ϕ : TA → B such that α = ϕ ◦ σ.
Proof: We can define the required homomorphism ϕ : TA → B by writing
ϕ(a) = α(a) for each object a ∈ Ob(A), and
ϕ(x1 ⊗ · · · ⊗ xn) = α(xn) . . . α(x1)
for morphisms xi ∈ Hom(ci, ci+1)A. It is easy to see that ϕ is the unique
R-algebroid homomorphism with the property that α = ϕ ◦ σ. ✷
We have a natural homomorphism π : TA→ A defined to be the identity on
the set of objects, and by the formula
ϕ(x1 ⊗ · · · ⊗ xn) = xn . . . x1
for morphisms xi ∈ Hom(ci, ci+1). It follows that there is an R-algebroid JA
with the same objects as the category A, and morphism sets
Hom(a, b)JA = ker(π : Hom(a, b)TA → Hom(a, b)A)
Definition 3.6 A sequence of R-algebroids and homomorphisms
0→ I
i
→ E
j
→ B → 0
is called a short exact sequence if the R-algeboids I, E , and B all have the same
sets of objects, and for all such object a and b our sequence restricts to a short
exact sequence of abelian groups:
0→ Hom(a, b)I → Hom(a, b)A → Hom(a, b)E → 0
We call the above short exact sequence F -split if there is a homomorphism of
R-moduloids (but not necessarily R-algebroids), s : B → E such that j ◦ s = 1C .
We call the homomorphism s an F -splitting of the short exact sequence.
Our definitions make it clear that, for an R-algeboid A, the tensor algebroid
fits into a natural short exact sequence
0→ JA →֒ TA
π
→ A→ 0
with natural F -splitting σ : A → TA.
Definition 3.7 The above short exact sequence is called the universal extension
of A.
The following result is easy to check.
Proposition 3.8 Let A and C be R-algebroids. Then we have a natural F -split
short exact sequence
0→ (JA)⊗R C → (TA)⊗R C → A⊗R C → 0
✷
7
Theorem 3.9 Let
0→ I
i
→ E
j
→ A → 0
be an F -split short exact sequence. Then we have a natural homomorphism
γ : JA → I fitting into a commutative diagram
JA → TA → A
↓ ↓ ‖
I → E → A
If our F -splitting is a homomorphism of R-algebroids, then the homomor-
phism γ is the zero map.
Proof: Since the exact sequence we are looking at is F -split, there is an R-
moduloid homomorphism s : A → E such that j ◦ s = 1A. By proposition 3.4,
there is a natural R-algebroid homomorphism H(s) : TA → E fitting into the
above diagram.
The homomorphism γ is defined by restriction of the homomorphism H(s).
Now, suppose that the above R-moduloid homomorphism s is actually an
algebroid homomorphism. Then, by exactness, we have a commutative diagram
JA → TA
π
→ A
↓ ↓ ‖
I → E → A
where the vertical homomorphism on the left is zero, and central vertical homo-
morphism is the composition s ◦ π. ✷
Definition 3.10 We call the above homomorphism γ the classifying map of
the diagram
JA → TA → A
↓ ↓ ‖
I → E → A
4 Algebraic Homotopy and Simplicial Enrich-
ment
For each natural number, n ∈ N, write
Z
∆n =
Z[t0, . . . , tn]
〈1 −
∑n
i=0 ti〉
In particular, we have Z∆
0
= Z and Z∆
1
= Z[t]. The sequence of rings (Z∆
n
)
combines with the obvious face and degeneracy maps to form a simplicial ring
Z
∆. We refer to [4] for details.
8
Given an R-algebroid A, we can also consider A to be a Z-algebroid, and so
form tensor products A⊗Z Z
∆n , and the simplicial R-algebroid A∆.
In the case n = 1, there are obvious homomorphisms ei : A
∆1 → A defined
by the evaluation of a polynomial at a point i ∈ Z.
Definition 4.1 Let f0, f1 : A → B be homomorphisms of R-algebroids. An
elementary homotopy between f0 and f1 is a homomorphism h : A → B
∆1 such
that e0 ◦ h = f0 and e1 ◦ h = f1.
We call f0 and f1 algebraically homotopic if they are linked by a chain of
elementary homotopies. We write [A,B] to denote the set of algebraic homotopy
classes of homomorphisms from A to B.
Let us view the geometric n-simplex ∆n as a simplicial set. Given a simplicial
set X , recall (see for example [8]) that a simplex of X is a simplicial map
f : ∆n → X . We define the simplex category of X , ∆ ↓ X , to be the category in
which the objects are the simplices of X , and the morphisms are commutative
diagrams
∆n → X
↓ ‖
∆m → X
The simplex category has the feature that the simplicial set X is naturally
isomorphic to the direct limit
lim
∆n→X
∆n
taken over the simplex category.
Definition 4.2 Let X be a simplicial set. Let A be an R-algebroid. Then we
define AX to be the direct limit
AX = lim
∆n→X
A∆
n
taken over the simplex category.
The assignment A− : X 7→ AX is a contravariant functor. Let X be a
simplicial set with a basepoint, + (in the sense of simplicial sets; see again [8]).
Then there is a functorially induced map AX → A = A+ arising from the
inclusion + →֒ X . We define
AX,+ = ker(AX → A)
Given pointed simplicial sets X and Y , it is easy to check that the R-
algebroids (AX)Y and AX∧Y are naturally isomorphic. The product X ∧ Y
is of course the smash product of pointed simplicial sets. In particular, if Sm
and Sn are simplicial spheres, then the R-algebroids (AS
m
)S
n
and AS
m+n
are
naturally isomorphic.
Recall from proposition 2.5 that there is a functor A 7→ AH from the cate-
gory of R-algebroids to the category of R-algebras. Naturality of the relevant
constructions immediately gives us the following result.
9
Proposition 4.3 Let A be an R-algebroid, and let X be a simplicial set. Then
the R-algebras (AH)
X and (AX)H are naturally isomorphic. ✷
Because of the above proposition, we do not distinguish between the R-
algebras (AH)
X and (AX)H , and simplify our notation by writing A
X
H in both
cases.
Now, let Simp denote the category of simplicial sets. Let AlgR denote the
category of R-algebroids. Then the contravariant functor A− can be written as
a covariant functor A− : Simpop → AlgR.
Let us call a directed object in the category of R-algebroids and homomor-
phisms a directed R-algebroid. If we write AlgindR to denote the category of
directed R-algebroids, then the above functor has an extension
A− : Simpop → AlgindR
Given a simplicial set X , we can form its subdivision sd(X); there is a
natural simplicial map h : sd(X) → X . The process of repeated subdivision
yields a sequence of simplicial sets, sd•X :
sd
0
X
hX
← sd
1
X
hsd(X)
← sd
2
X ← · · ·
Definition 4.4 Let A and B be R-algebroids. Then we define HOM(A,B) to
be the simplicial set defined by writing
HOM(A,B)[n] = lim
→
k
Hom(A,Bsd
k∆n)
The face and degeneracy maps are those inherited from the simplicial R-
algebroid B∆.
The following result will by used later on in this article to formulate algebraic
KK-theory in terms of spectra. It is proven in exactly the same way as theorem
3.3.2 in [4].
Theorem 4.5 Let A be a simplicial R-algebroid, and let B be a directed R-
algebroid. Then
[A,BS
n,+] = πnHOM(A,B)
✷
Of course, in the above theorem, we use Sn to denote the simplicial n-sphere,
and we are using simplicial homotopy groups; see for instance [8] or some other
standard reference on simplicial homotopy theory for further details.
10
5 Path Extensions
Let A be an R-algebroid. Let A⊕A be the R-algebroid with the same objects
as A, with morphism sets
Hom(a, b)A⊕A = Hom(a, b)A ⊕Hom(a, b)A
Write ΩA = AS
1,+. Then we obtain a short exact sequence
0→ ΩA → A∆
1 (e0,e1)
→ A⊕A → 0
with an F -splitting s : A⊕A → A∆
1
given by the formula
s(x, y) = (1− t)x+ ty
If we write PA = A∆
1,+, we have a commutative diagram
ΩA → PA
e1
→ A
‖ ↓ ↓
ΩA → A∆
1 (e0,e1)
→ A⊕A
↓ ↓
A = A
where the homomorphism A → A⊕A is the inclusion of the second factor, the
homomorphism A ⊕ A → A is projection onto the first factor, and the map
A∆
1
→ A is the evaluation map e0. Further, the complete rows and columns
are short exact sequences.
The column on the right has a natural splitting (1, 0): A → A ⊕ A. The
second row is the path extension, which has a natural F -splitting as we observed
above. By a diagram chase, it follows that the top row and middle column have
natural F -splittings.
Hence, by theorem 3.9, there is a natural map ρ : JA → ΩA.
Definition 5.1 Let f : A → B be a homomorphism of R-algebroids. Then the
path algebroid of f is the unique R-algebra PB⊕BA fitting into a commutative
diagram
ΩB → PB ⊕B A → A
‖ ↓ ↓
ΩB → PB → B
where the bottom row is the path extension, and the upper row is a short exact
sequence.
It is straightforward to check that the path algebroid is well-defined, and
that the upper row in the above diagram has a natural F -splitting. There is
therefore a natural map η(f) : JA → ΩB.
We can compose with the map i : ΩA → AS
1
to obtain a map η(f) : JA →
BS
1
.
Definition 5.2 We call the above homomorphism η(f) the classifying map of
the homomorphism f .
11
6 The KK-theory spectrum
Consider a homomorphism α : A → BH . Then we have a classifying map
η(α) : JA → BS
1
H . Given a a natural number m ∈ N, we define the R-algebroid
JmA iteratively, by writing
J0A = A Jk+1A = J(JkA)
Given a homomorphism α : J2nA → (BS
n
H )
sd•∆n , we have a structure map
η(α) : J2n+1 → (BS
n+1
H )
sd•∆n
Applying the classifying map construction again to the above, we see that
we have a simplicial map
ǫ : HOM(J2nA,BS
n
H )→ HOM(J
2n+2A,BS
n+2
H )
∼= ΩHOM(J
2n+2A,BS
n+1
H )
Definition 6.1 We defineKK(A,B) to be the spectrum with sequence of spaces
HOM(J2nA,BS
n
H ), with structure maps defined as above.
Note that, by proposition 2.5, elements of the space HOM(J2nA,BS
n
H ) are
defined by homomorphisms α : J2nA → (BS
n
⊕ )
sdk∆n .
Recall from [13] that a symmetric spectrum is a spectrum, E, equipped with
actions of symmetric groups Σn × En → En that commute with the relevant
structure maps. The extra structure means that the smash product, E ∧ F of
symmetric spectra E and F can be defined.
Proposition 6.2 The spectrum KK(A,B) is a symmetric spectrum.
Proof: There is a canonical action of the permutation group Σn on the
simplicial sphere Sn ∼= S1∧· · ·∧S1 defined by permuting the order of the smash
product of simplicial circles, and therefore on the space HOM(J2nA,BS
n
⊕ ).
By construction, the iterated structure map ǫk : HOM(J2nA,BS
n
⊕ ) →
ΩHOM(J2n+2kA,BS
n+k
⊕ ) is Σn × Σk-equivariant, and so we have a symmetric
spectrum as required. ✷
Proposition 6.3 Let B be an R-algebroid, and let k and l be natural numbers.
Then there is a natural homomorphism s : Jk(BS
l
)→ (J lB)S
k
.
Proof: The classifying map of the F -split exact sequence
0→ (JA)S
1
→ (TA)S
1
→ AS
1
→ 0
is a natural homomorphism J(AS
1
)→ (JA)S
1
. The homomorphism s is defined
by iterating the above construction. ✷
12
Definition 6.4 Let A, B, and C be R-algebroids. Consider homomorphisms
α : J2mA → BS
m
⊕ and β : J
2nB → CS
n
⊕ . Then we define the product α♯β to be
the composition
J2m+2nA
J2nα
→ J2n(BS
m
⊕ )
s
→ (J2nB⊕)
Sm β⊕→ CS
m+n
⊕
We can of course also define the above in the case of directed R-algebroids.
This extension of the above definition is needed in the following theorem.
Theorem 6.5 Let A, B, and C be R-algebroids. Then there is a natural map
of spectra
KK(A,B) ∧KK(B, C)→ KK(A, C)
defined by the formula
α : J2mA → (BS
m
⊕ )
sd•∆k α : J2nB → (CS
n
⊕ )
sd•∆l
Further, the above product is associative in the obvious sense. Given homo-
morphisms α : A → B and β : B → C, we have the formula α♯β = β ◦ α.
Proof: For convenience, we will simply consider homomorphisms of the form
α : J2mA → BS
m
⊕ and β : J
2nB → CS
n
⊕ .
Our construction gives us a natural continuous Sm × Sn-equivariant map
HOM(JmA,BS
m
H ) ∧ HOM(J
nB, CS
n
H ) → HOM(J
m+nA, CS
m+n
⊕ ). Compatibility
with the structure maps follows since naturality of the classifying map construc-
tion gives us a commutative diagram
J2m+2n+2A
J2n+2(α)
→ J2n+2(BS
m
⊕ )
s
→ (J2n+2B)S
n
⊕
η2(β⊕)
→ Σm+n+2CS
m+n+2
⊕
‖ ↓ ↓ ‖
J2m+2n+2A
J2nη2(α)
→ J2n(BS
m+2
⊕ )
s
→ (J2nB)S
m+2
⊕
β⊕
→ CS
m+n+2
⊕
We now need to check the statement concerning associativity. Consider
homomorphisms
α : J2mA → BS
m
⊕ β : J
2nB → CS
n
⊕ γ : J
2pC → DS
p
⊕
Then we have a commutative diagram
J2m+2n+2pA = J2m+2n+2pA
↓ ↓
J2n+2p(BS
m
⊕ ) = J
2n+2p(BS
m
⊕ )
↓ ↓
J2p(J2nBS
m
⊕ )
s
→ (J2n+2pB)S
m
)⊕
↓ ↓
J2p(CS
m+n
⊕ )
s
→ J2p((CS
m
)S
n
⊕ )
↓ ↓
(J2pC)S
m+n
⊕ = J
2p(CS
m+n
⊕ )
↓ ↓
DS
m+n+p
⊕ = Σ
m+n+pDS
m+n+p
⊕
13
But the column on the left is the product (α♯β)♯γ and the column on the
right is the product α♯(β♯γ) so associativity of the product follows. ✷
Proposition 6.6 Let A, B, and C be R-algebroids. Then there is a map
∆: KK(A,B)→ KK(A⊗R C,B⊗R C). This map is compatible with the product
in the sense that we have a commutative diagram
KK(A,B) ∧KK(B, C) → KK(A, C)
↓ ↓
KK(A⊗R D,B ⊗R D) ∧KK(B ⊗R D, C ⊗R D) → KK(A⊗R D, C ⊗R D)
where the horizontal maps are defined by the product, and the vertical maps are
copies of the map ∆.
Proof: Let α : J2nA → BS
n
⊕ be a homomorphism. Then we have a naturally
induced homomorphism α⊗ 1: (J2nA)⊗R C → B
Sn
⊕ ⊗R C).
There is an obvious natural homomorphism β : BS
n
⊕ ⊗R C → (B⊗R C)
Sn
⊕ . By
proposition 3.8, we have an F -split short exact sequence
0→ (JA)⊗R C → (TA)⊗R C → A⊗R C → 0
We thus obtain a natural homomorphism γ : J(A⊗R C)→ (JA)⊗R C as the
classifying map of the diagram
A⊗ C
‖
0 → (JA)⊗ C → (TA)⊗ C → A⊗ C → 0
We now define the map ∆ by writing ∆(α) = β ◦ (α⊗ 1) ◦ γn. The relevant
naturality properties are easy to check. ✷
The following result follows directly from the relevant definitions in the cat-
egory of symmetric spectra (see [13]) along with the above proposition and
theorem.
Theorem 6.7 Let A be an R-algebroid. Then the spectrum KK(A,A) is a
symmetric ring spectrum.
Let B be another R-algebroid. Then the spectrum KK(A,B) is a symmetric
KK(R,R)-module spectrum. ✷
7 KK-theory groups
Note that, by theorem 4.5,
π0KK(A,B) = lim
→
n
[J2nA,BS
2n
H ]
14
and when B is anR-algebra, we have a natural isomorphismBH ∼= B⊗RM∞(R),
where M∞(R) denotes the R-algebra of infinite matrices, indexed by N, with
entries in the ring R.
Thus, if we define
KK(A,B) = π0KK(A,B)
then in the case where A and B are R-algebras, we recover the definition of
bivariant algebraic KK-theory in [4].
Definition 7.1 Let A and B be R-algebroids. Then we can define the KK-
theory groups
KKp(A,B) = πpKK(A,B)
By theorem 4.5, we have natural isomorphisms
[A,BS
n,+] ∼= πnHOM(A,B)
Hence, by definition of the KK-theory spectrum, we know that
KKp(A,B) ∼= lim
→
n
[JnA,BS
n+p
⊕ ]
In section 5, that we defined an R-algebroid ΩB = BS
1,+. Iterating this
construction, we see that ΩnB = BS
n,+. Hence, when p ≥ 0 we can write
KKp(A,B) = KK0(A,Ω
pB)
Similarly, we can write
KK−p(A,B) = KK0(J
pA,B)
By definition, elements of the group KK0(A,B) arise from homomorphisms
α : J2nA → BS
n
⊕ . Given two such homomorphisms α, β : J
2nA → BS
n
⊕ , we can
define the sum, α⊕ β : J2nA → BS
n
⊕ , by writing
(α⊕ β)(a) = α(a) ⊕ β(a)
for each object a ∈ Ob(JnA), and
(α⊕ β)(x) =
(
α(x) 0
0 β(x)
)
for each morphism x.
The following result is obvious from the construction of KK-theory.
Proposition 7.2 Let α : J2nA → BS
n
⊕ be a homomorphism. Let [α] be the
equivalence class defined by the following conditions.
• Let α, β : J2nA → BS
n
⊕ be algebraically homotopic. Then [α] = [β].
15
• Let α : J2nA → BS
n
⊕ be a homomorphism, and let 0: A → B
Sn
⊕ be the
homomorphism that is zero for each morphism in the category JnA. Then
[α] = [α⊕ 0].
• Let α : J2nA → BS
n
⊕ , and let η(α) : J
2n+2A → BS
n+1
⊕ be the corresponding
classifying map. Then [η(α)] = [α].
Then the group KK0(A,B) is the set of equivalence classes of homomor-
phisms α : JnA → BS
n
⊕ . The group operation is defined by the formula
[α⊕ β] = [α] + [β]. ✷
At the level of groups, the product is an associative map
KKp(A,B)⊗KKq(B, C)→ KKp+q(B, C)
Given homomorphisms α : A → B and β : B → C, the product [α] · [β] is the
equivalence class of the composition, [β ◦ α].
An essentially abstract argument, as described in section 6.34 of [4] for R-
algebras rather than R-algebroids, yields the following result.
Theorem 7.3 Let
0→ A→ B → C → 0
be an F -split short exact sequence of R-algebroids. Let D be an R-algebroid.
Then we have natural maps ∂ : KKp(A,D) → KKp+1(C,D) inducing a long
exact sequence of KK-theory groups
→ KKp(C,D)→ KKp(B,D)→ KKp(A,D)
∂
→ KKp+1(C,D)→
✷
A similar result holds in the other variable; we do not need it here.
Definition 7.4 We call a homomorphism α : J2nA → BS
n
⊕ a KK-equivalence
if there is an element [α]−1 ∈ KK0(B,A) such that [α]
−1 · [α] = [1A] and
[α] · [α]−1 = [1B].
We call two R-algebroids A and B KK-equivalent if there is a KK-
equivalence α : J2nA → BS
n
⊕ .
Proposition 7.5 Let α : J2nA → BS
n
⊕ be a KK-equivalence. Then the product
with α induces equivalences of spectra
KK(B, C)→ KK(A, C) KK(C,A)→ KK(C,B)
for every R-algebroid C.
16
Proof: Let us just look at the map
α♯ : KK(B, C)→ KK(A, C)
since the other case is almost identical. The map α♯ induces an isomorphism
π0KK(B, C) = KK0(B, C)→ KK0(A, C) = π0KK(A, C)
by definition of the KK-theory group and the term KK-equivalence.
Let p ≥ 0. Replacing the R-algebroid C by the R-algebroid ΩpC, we see that
the map α♯ also induces an isomorphism
πpKK(A, C) = KKp(B, C)→ KKp(A, C) = πpKK(A, C)
Thus the map α♯ is an equivalence of spectra as desired, and we are done.
✷
Observe that the category-theoretic concept of natural isomorphism makes
sense when we are talking about homomorphisms of unital R-algebroids.
Lemma 7.6 Let α, β : A → B be naturally isomorphic homomorphisms of uni-
tal R-algebroids. Then the maps α and β are simplicially homotopic at the level
of KK-theory spectra.
Proof: By definition of the KK-theory spectrum, the space KK(A,B)0 is the
simplicial set HOM(A,BH). In this space, the homomorphisms α and β are the
same as the homomorphisms α′ : A → B⊕ and β
′ : A → B⊕ defined by writing
α′(a) = α(a)⊕ β(a), β′(a) = α(a) ⊕ β(a) a ∈ Ob(A)
and
α′(x) =
(
α(x) 0
0 1
)
β′(x) =
(
1 0
0 β(x)
)
where x ∈ Hom(a, b)A.
Since the homomorphisms α and β are naturally equivalent, we can find
invertible morphisms ga ∈ Hom(α(a), β(a))B for each object a ∈ Ob(A) such
that β(x)ga = g
−1
a α(x) for all x ∈ Hom(a, b)A.
Let
W =
(
1− t2 (t3 − 2t)g−1a
tga 1− t
2
)
Then the matrix W is an invertible morphism in the R-algebroid (A⊗Z[t])⊕;
the inverse is the matrix
W−1 =
(
1− t2 (2t− t3)g−1a
−tga 1− t
2
)
Further,
e0(W ) =
(
1 0
0 1
)
= e0(W
−1) ev1(W ) =
(
0 −g−1a
ga 0
)
= −e1(W
−1)
17
Hence
e0(W
(
α(x) 0
0 1
)
W−1) = α′(x)
and
e1(W
(
α(x) 0
0 1
)
W−1) = β′(x)
Therefore the homomorphisms α′ and β′ are algebraically homotopic. The
result now follows by theorem 4.5. ✷
The following result is immediate from the above lemma and proposition
7.5.
Theorem 7.7 Let A and A′ be equivalent unital R-algebroids. Let B be another
R-algebroid. Then the spectra KK(A,B) and KK(A′,B) are stably equivalent,
and the spectra KK(B,A) and KK(B,A′) are homotopy-equivalent. ✷
The following result is proved similarly.
Proposition 7.8 The symmetric spectra KK(A,B) and KK(A,B⊕) are stably
equivalent. ✷
Let R be a ring where every R-algebra is central. It is shouwn in [4] that for
an R-algebra A there is a natural isomorphism KKn(R,A) ∼= KHn(A). The
proof depends on certain universal properties of KK-theory and homotopy K-
theory, and a universal characterisation of algebraic KK-theory of the type first
considered for C∗-algebras in [11, 25]. The characterisation involves triangulated
categories; see [19, 23].
The following result is now immediate.
Corollary 7.9 Let R be a ring where every R-algebra is central. Let A be
an R-algebroid that is equivalent to an R-algebra. Then we have a natural
isomorphism
KKn(R,A) ∼= KHn(A)
✷
Definition 7.10 Let A be an R-algebroid. Then we define the homotopy alge-
braic K-theory spectrum
KH(A) = KK(R,A)
8 Modules over Algebroids
The modules we consider in this section were introduced for algebras in [15].
18
Definition 8.1 Let A be an R-algebroid. Then a right A-module is an R-linear
contravariant functor from A to the category of R-modules.
A natural transformation, T : E → F , between two A-modules is called a
homomorphism.
We write L(A) to denote the category of all right A-modules and homomor-
phisms. The category L(A) is clearly a unital R-algebroid.2
Definition 8.2 Given an object c ∈ Ob(A), we define Hom(−, c)A to be the
right A-module with spaces Hom(−, c)A(a) = Hom(a, c)A; the action of the
R-algebroid A is defined by multiplication.
Definition 8.3 Let E and F be right A-modules. Then we define the direct
sum, E ⊕ F , to be the right A-module with spaces
(E ⊕ F)(a) = E(a)⊕F(a) a ∈ Ob(A)
We call a right A-module E finitely generated and projective if there is a right
A-module F such that the direct sum E ⊕ F is isomorphic to a direct sum of
the form
Hom(−, c1)A ⊕ · · · ⊕Hom(−, cn)A
Let us write L(Afgp) to denote the category of all finitely-generated projec-
tive A-modules and homomorphisms. The following result follows directly from
theorem 7.7 and proposition 7.8.
Proposition 8.4 Let A and B be R-algebroids. Then there is a natural stable
equivalence of spectra
KK(A,B)→ KK(A,L(Bfgp))
✷
Let f : A → B be a homomorphism of R-algebroids, and let E be a right
A-module. Then we can define a right B-module, f∗E = B⊗A E using a similar
method to that used for Hilbert modules over C∗-algebras (see for example [17]).
Actually, the procedure is slightly easier since we do not need to worry about
the analytic issues that are present in the C∗-algebra case.
Definition 8.5 Let A and B be R-algebroids. Then an (A,B)-bimodule, F , is
an R-algebroid homomorphism F : A → L(B). An (A,B)-bimodule F is termed
finitely generated and projective if it is a homomorphism F : A → L(Bfgp).
2Assuming we take all of our right A-modules in a given universe, so that the category
L(A) is small.
19
Let F be an (A,B)-bimodule. For each object a ∈ Ob(A), let us write
F(−, a) to denote the associated right B-module. Then for each object b ∈
Ob(B) we have an R-module F(b, a). For each morphism x ∈ Hom(a, a′)A, we
have an induced homomorphism x : F(−, a)→ F(−, a′).
Given an R-algebroid homomorphism f : A → B, there is an associated
(A,B)-bimodule which we will simply label B. The right B-module associated
to the object a ∈ Ob(A) is Hom(−, f(a))B. The homomorphisms associated to
morphisms in the category A are defined in the obvious way through the functor
f .
Definition 8.6 Let E be a A-module, and let F be a (A,B)-bimodule. Let
b ∈ Ob(B). Then we define the algebraic tensor product E ⊗A F(b,−) to be the
R-module consisting of all formal linear combinations
λ1(η1, ξ1) + · · ·+ λn(ηn, ξn)
where λi ∈ R, ηi ∈ E(a), ξi ∈ F(b, a), and a ∈ Ob(A), modulo the equivalence
relation ∼ defined by writing
• (η1 + η2, ξ) ∼ (η1, ξ) + (η2, ξ)
• (η, ξ1 + ξ2) ∼ (η, ξ1) + (η, ξ2)
• (η, xξ) ∼ (ηx, ξ) for each morphism x ∈ Hom(a, a′)A.
Let us write η ⊗ ξ to denote the equivalence class of the pair (η, ξ).
Definition 8.7 We write E ⊗A F to denote the right B-module defined by
associating the R-module E ⊗A F(b,−) to an object b ∈ Ob(B).
Given a homomorphism of right A-modules, T : E → E ′, there is an induced
homomorphism T ⊗ 1: E ⊗A F → E
′ ⊗A F , defined in the obvious way.
Recall that a homomorphism f : A → B turns the R-algebroid B into an
(A,B)-bimodule. Thus, given an A-module E , we can form the tensor product
E ⊗A B.
We can write f∗E = E ⊗A B. A homomorphism of A-modules, T : E → E
′
yields a functorially induced map f∗T = T ⊗ 1: E ⊗A B → E
′ ⊗A B.
Theorem 8.8 Let E be a finitely generated projective right A-module. Then we
have a canonical morphism of spectra
E∧ : KK(A,B)→ KK(R,B)
This morphism is natural in the variable B in the obvious sense. Given an
R-algebroid homomorphism f : A → A′, we have a commutative diagram
E∧ : KK(A,B) → KK(R,B)
↑ ‖
E ⊗A A
′∧ : KK(A′,B) → KK(R,B)
20
Proof: The right A-module E defines a homomorphism R → L(Afgp) by
mapping the one object of R, considered as an R-algebroid, to the Hilbert
module E , and the unit 1 to the identity homomorphism on E . Thus the Hilbert
module E defines an element in the 0-th space of the spectrum KK(R,L(Afgp)).
By proposition 8.4, we have a natural equivalence of symmetric spectra
KK(R,L(Afgp))→ KK(R,A)
By composition of the formal inverse of the above equivalence with the prod-
uct, we obtain a canonical map
E∧ : KK(A,B)→ KK(R,B)
Naturality in the variable B follows by associativity of the product. To prove
that the stated diagram is commutative, we need to prove that the diagram
E∧ : KK(L(Afgp),B) → KK(R,B)
↑ ‖
E ⊗A A
′∧ : KK(L(A′fgp),B) → KK(R,B)
is commutative.
The homomorphism f∗ : L(Afgp)→ L(A
′
fgp) is defined by mapping the right
A-module E to the right A-module E ⊗AA
′, and the homomorphism T : E → F
to the homomorphism T ⊗ 1: E ⊗A A
′ → F ⊗A A
′.
Now, the map at the bottom of the diagram is defined by the product with
the homomorphism R → L(A′fgp) defined by the right A
′-module E ⊗A A
′.
The composition of the vertical map on the left and the map at the top of the
diagram is defined by the product with the composition of the homomorphism
f∗ and the homomorphism C→ L(A) defined by the right A
′-module E .
By construction, these two homomorphisms are the same, and we are done.
✷
We are actually going to use a slight generalisation of the above theorem;
the proof is essentially the same as the above.
Definition 8.9 Let A be an R-algebroid. Let a ∈ Ob(A). Then we define the
path-component of a, A|Or(a), to be the full subcategory of A containing all
objects b ∈ Ob(A) such that Hom(a, b)A 6= 0.
We call a right A-module E almost finitely generated and projective if the
restriction to each path-component is finitely generated and projective.
Theorem 8.10 Let E be an almost finitely generated projective A-module.
Then we have a canonical morphism of spectra
E∧ : KK(A,B)→ KK(R,B)
21
This morphism is natural in the variable B in the obvious sense. Given an
R-algebroid homomorphism f : A → A′, we have a commutative diagram
E∧ : KK(A,B) → KK(R,B)
↑ ‖
E ⊗A A
′∧ : KK(A′,B) → KK(R,B)
✷
9 Equivariant KK-theory
Let G be a discrete groupoid, and let R be a ring. We can regard G as a small
category in which every morphism is invertible.
Definition 9.1 A G-algebra over R is a functor from the category G to the
category of R-algebras and homomorphisms.
Thus, if A is a G-algebra, then for each object a ∈ Ob(G) we have an algebra
A(a). A morphism g ∈ Hom(a, b)G induces a homomorphism g : A(a)→ A(b).
We can regard an ordinary algebra C as a G-algebra by writing C(a) = C
for each object a ∈ Ob(G) and saying that each morphism in the groupoid G
acts as the identity map.
Definition 9.2 A G-module over R is a functor from the groupoid G to the
category of R-bimodules and R-linear maps.
The notation used for G-modules is the same as that used for G-algebras.
There is a forgetful functor, F , from the category of G-algebras to the category
of G-modules.
An equivariant map between G-algebras or G-modules is the same thing as
a natural transformation.
Definition 9.3 Let A and B be G-algebras. Then we define the tensor product
A⊗RB to be the G-algebra where (A⊗RB)(a) = A(a)⊗R B(a) for each object
a ∈ Ob(G), and the G-action is defined by writing g(x⊗y) = g(x)⊗g(y) whenever
g ∈ Hom(a, b)G , x ∈ A(a), and y ∈ B(a).
We define the direct sum A⊕B to be the G-algebra where (A⊕B)(a) is the
direct sum A(a) ⊕ B(a) for each object a ∈ Ob(G) and the G-action is defined
by writing g(x ⊕ y) = g(x) ⊕ g(y) whenever g ∈ Hom(a, b)G , x ∈ A(a), and
y ∈ B(a).
We similarly define tensor products and direct sums of G-modules.
Recall from section 4 that we have a sequence of rings, (Z∆
n
); these rings
combine to form a simplicial ring Z∆. If we equip each such ring with the trivial
G-action, and A is a G-algebra, we can form the tensor product A⊗Z ∆
n.
Just as in section 4, given a G-algebra A and a simplicial set X , we can form
the G-algebra. AX . If the simplicial set X has a basepoint +, we also form the
G-algebra A(X,+) = ker(AX → A).
22
Definition 9.4 Let f0, f1 : A → B be equivariant maps of G-algebras. An
equivariant map h : A → B∆
1
such that e0 ◦ h = f0 and e1 ◦ h = f1 is called an
elementary homotopy between the maps f0 and f1.
We call f0 and f1 algebraically homotopic if they can be linked by a chain
of elementary homotopies. We write [A,B]G to denote the set of algebraic
homotopy classes.
Let us call a directed object in the category of G-algebras and equivariant
maps a directed G-algebra. As in the non-equivariant case, simplicial subdivision
gives us a directed G-algebra Asd
•X .
Definition 9.5 Let A and B be G-algebras. Then we define HOMG(A,B) to
be the simplicial set defined by writing
HOMG(A,B)[n] = lim
→
k
Hom(A,Bsd
k∆n)
The face and degeneracy maps are those inherited from the simplicial G-
algebra B∆.
The following result is proved in the same way as theorem 3.3.2 in [4].
Theorem 9.6 Let A be a G-algebra, and let B be a directed G-algebra. Then
[A,BS
n
]G = πnHOMG(A,B)
✷
A short exact sequence of G-algebras is a sequence of G-algebras and equiv-
ariant maps
0→ A
i
→ B
j
→ C → 0
such that the sequence
0→ A(a)
i
→ B(a)
j
→ C(a)→ 0
is exact for each object a ∈ Ob(G). A splitting of a short exact sequence is
defined in the obvious way.
We call a short exact sequence
0→ A→ B
j
→ C→ 0
F -split if there is an equivariant map of G-modules s : C → B such that j ◦ s =
1C . Such a map s is called an F -splitting.
Definition 9.7 Let A be a G-module. Then we define the tensor G-algebra
A⊗k to be the tensor product of A with itself k times. We define the equivariant
tensor algebra, TA, to be the iterated direct sum
TA = ⊕∞k=1A
⊗k
23
Composition of elements in the tensor G-algebra is defined by concatenation
of tensors. Formation of the tensor G-algebra defines a functor, T from the
category of G-modules to the category of G-algebras. The following result is
proved in the same way as proposition 3.4.
Proposition 9.8 The functor T is naturally adjoint to the forgetful functor F .
✷
Given a G-algebra A, there is a canonical equivariant map σ : A → TA of
G-modules defined by mapping each morphism set of the G-algebra A onto the
first summand. As we should by now expect, there is an associated universal
property.
Proposition 9.9 Let α : A → B be an equivariant map between G-algebras.
Then there is a unique homomorphism ϕ : TA→ B such that α = ϕ ◦ σ. ✷
We can thus define a G-algebra JA by writing
JA(a) = kerπ : TA(a)→ A(a)
for each object a ∈ Ob(G). The G-action is inherited from the tensor G-algebra.
There is a natural short exact sequence
0→ JA →֒ TA
π
→ A→ 0
with F -splitting σ : A→ TA. The following result is proved in the same way as
theorem 3.9.
Theorem 9.10 Let
0→ I
i
→ E
j
→ A→ 0
be an F -split short exact sequence. Then we have a natural homomorphism
γ : JA→ I fitting into a commutative diagram
0 → JA → TA → A → 0
↓ ↓ ‖
0 → I → E → A → 0
✷
As before, the homomorphism γ is called the classifying map of the short
exact sequence.
The last construction we need to define an equivariant KK-theory spectrum
is an equivariant version of the path extension. Analogously to the algebroid
case, we can write ΩA = AS
1,+ and obtain an F -split short exact sequence
0→ ΩA→ A∆
1 (e0,e1)
→ A⊕A→ 0
As before, a diagram chase enables us to construct a natural classifying map
ρ : JA→ ΩA.
24
Definition 9.11 Let f : A→ B be an equivariant map of G-algebras. Then the
path algebra of f is the unique G-algebra PB ⊕B A fitting into a commutative
diagram
ΩB → PB ⊕B A → A
‖ ↓ ↓
ΩB → PB → B
where the bottom row is the path extension, and the upper row is a short exact
sequence.
It is straightforward to check that the path algebra is well-defined, and that
the upper row in the above diagram has a natural F -splitting. There is therefore
a naturalmap η(f) : JA→ ΩB.
We can compose with the map i : ΩA → AS
1
to obtain a map η(f) : JA →
BS
1
.
Definition 9.12 We call the above equivariant map η(f) the classifying map
of the equivariant map f .
Given a G-algebra A, let us write M∞(A) to denote the direct limit of the
G-algebras Mn(A) = A⊗R Mn(R) under the inclusions
x 7→
(
x 0
0 0
)
Consider an equivariant map α : A → M∞(B
Sn), where A and B are G-
algebras. Then by the above construction, there is an induced classifying map
η(α) : JA→M∞(B
Sn+1).
Definition 9.13 We define KKG(A,B) to be the symmetric spectrum with
sequence of spaces HOMG(J
2nA,M∞(B
Sn)). The structure map
ǫ : HOMG(J
2nA,M∞(B
Sn))→ ΩHOMG(J
2n+2A,M∞(B
Sn+1)) ∼= HOMG(J
2n+2A,M∞(B
Sn+2))
is defined by applying the above classifying map construction twice, that is to
say writing ǫ(α) = η(η(α)) whenever α ∈ HOM(J2nA,M∞(B
Sn+1)).
The action of the permutation group Sn is induced by its canonical action
on the simplicial sphere Sn.
The product is constructed as for R-algebroids, in the non-equivariant case.
Definition 9.14 Let A, B, and C be G-algebras. Let α ∈
HOMG(J
2mA,M∞(B
Sm)) and β ∈ HOMG(J
2nB,M∞(C
Sn)). Then we
define the product α♯β to be the composition
J2m+2nA
J2nα
→ J2n(M∞(B
Sm)→ (J2nM∞(B))
Sm β→M∞(C)
Sm+n
25
Theorem 9.15 Let A, B, and C be G-algebras. Then there is a natural map
of spectra
KKG(A,B) ∧KKG(B,C)→ KKG(A,C)
defined by the formula
α∧β 7→ α♯β α ∈ HOMG(J
2mA,M∞(B
Sm)), β ∈ HOMG(J
2nB,M∞(C
Sn))
Further, the above product is associative in the usual sense. Given equiv-
ariant maps α : A → B and β : B → C, we have the formula α♯β = β ◦ α.
✷
Just as in the non-equivariant case, the following result follows from our
constructions.
Corollary 9.16 Let A be a G-algebra. Then the spectrum KKG(A,A) is a
symmetric ring spectrum.
Let B be another G-algebra. Then the spectrum KKG(A,B) is a symmetric
KKG(R,R)-module spectrum. ✷
Let θ : G → H be a functor between groupoids, and let A be an H-algebra.
Abusing notation, we can also regard A as a G-algebra; we write A(a) = A(θ(a))
for each object a ∈ Ob(G), and define a homomorphism g = θ(g) : A(θ(a)) →
A(θ(b)) for each morphism g ∈ Hom(a, b)G .
There is an induced map θ∗ : KKH(A,B) → KKG(A,B) defined by the ob-
servation that any H-equivariant map is also G-equivariant.
Definition 9.17 We call the map θ∗ : KKH(A,B)→ KKG(A,B) the restriction
map.
Proposition 9.18 Let θ : G → H be a functor between groupoids, and let A
and B be H-algebras. Then the restriction map θ∗ : KKH(A,B) → KKG(A,B)
is compatible with the product in the sense that we have a commutative diagram
KKH(A,B) ∧KKH(B,C) → KKH(A,C)
↓ ↓
KKG(A,B) ∧KKG(B,C) → KKG(A,C)
where the horizontal map is defined by the product and the vertical maps are
restriction maps.
Proof: Abusing notion, we can also regard the H-algebra A as a G-algebra; we
write A(a) = A(θ(a)) for each object a ∈ Ob(G), and define a homomorphism
g = θ(g) : A(θ(a))→ A(θ(b)) for each morphism g ∈ Hom(a, b)G .
We similarly regard the H-algebra B as a G-algebra. An H-equivariant map
α : JnA → M∞(B
Sn) is also G-equivariant. We use this construction to define
our map θ∗ : KKH(A,B)→ KKG(A,B).
The result is now straightforward to check. ✷
26
Given a G-algebra A, we define the convolution algebroid, AG, to be the
algebroid with the same set of objects as the groupoid G, and morphism sets
Hom(a, b)AG = {
m
∑
i=1
xigi | xi ∈ A(b), gi ∈ Hom(a, b)G , m ∈ N}
Composition of morphisms is defined by the formula
(
m
∑
i=1
xigi
)
n
∑
j=1
yjhi
=
m,n
∑
i,j=1
xigi(yj)gihj
Theorem 9.19 Let G be a groupoid, and let A and B be G-algebras. Then there
is a map
D : KKG(A,B)→ KK(AG, BG)
which is compatible with the product in the sense that we have a commutative
diagram
KKG(A,B) ∧KKG(B,C) → KK(A,C)
↓ ↓
KK(AG, BG) ∧KK(BG, CG) → KK(AG, CG)
where the horizontal maps are defined by the product.
Proof: Let α : J2nA → M∞(B
Sn) be an equivariant map. Then we have
a functorially induced homomorphism α∗ : (J
2nA)G → M∞(B
Sn)G defined by
writing
α∗(
n
∑
i=1
xigi) =
n
∑
i=1
α(xi)gi
We have a natural homomorphism γ : J(AG) → (JA)G defined as the clas-
sifying map of the diagram
AG
‖
0 → (JA)G → (TA)G → AG → 0
For any G-algebra C, we can regard the G-algebra M∞(C) as the tensor
product C ⊗R M∞(R). It follows that there is an obvious homomorphism
β : M∞(B
Sn)G →M∞(BG)
Sn .
We thus have a map D : KKG(A,B) → KK(AG, BG) defined by writing
D(α) = β ◦ α∗ ◦ γ
2n. The relevant naturality properties are easy to check. ✷
Corollary 9.20 Let G be a discrete groupoid, and let A and B be G-algebras.
Then the spectrum KK(AG, BG) is a symmetric KKG(R,R)-module spectrum.
✷
27
Let G be a discrete groupoid, and let A and B be G-algebras. Then we can
define groups
KKGp (A,B) := πpKKG(A,B)
The following result is proved in the same way as theorem 7.3.
Theorem 9.21 Let
0→ A→ B → C → 0
be an F -split short exact sequence of G-algebras. Let D be an R-algebroid. Then
we have natural maps ∂ : KKGp (A,D) → KK
G
p+1(C,D) inducing a long exact
sequence of KK-theory groups
→ KKGp (C,D)→ KK
G
p (B,D)→ KK
G
p (A,D)
∂
→ KKGp+1(C,D)→
✷
Theorem 9.22 Let θ : G → H be an equivalence of discrete groupoids. Let A
and B be H-algebras. Then the restriction map θ∗ : KKH(A,B) → KKG(A,B)
is an isomorphism of spectra.
Proof: Since the functor θ is an equivalence, there is a functor φ : H → G
along with natural isomorphisms G : φ ◦ θ → 1G and H : θ ◦ φ→ 1H.
Thus, for each object a ∈ Ob(G), there is an isomorphism Ha ∈
Hom(φθ(a), a)G . Let α : J
2nA → M∞(B
Sn) be an H-equivariant map. Then
the map α can be defined in terms of the restriction φ∗θ∗α : J2nA→M∞(B
Sn)
by the formula
α(x) = φ∗θ∗α(H−1a xHa) x ∈ A(a)
Thus the equivariant map α is determined by the restriction φ∗θ∗α.
The natural isomorphism H therefore induces a isomorphism of spectra
H∗ : KKH(A,B) → KKH(A,B) such that H∗ ◦ φ
∗ ◦ θ∗ = 1KKH(A,B). There is
similarly a isomorphism G∗ : KKG(A,B)→ KKG(A,B) such that G∗ ◦ θ
∗ ◦φ∗ =
1KKG(A,B).
It follows that the map θ∗ is a isomorphism, and we are done. ✷
10 Assembly
Given a functor, E, from the category of G-CW -complexes to the category
of spectra, we call E G-homotopy-invariant if it takes G-homotopy-equivalent
equivariant maps of G-spaces to maps of spectra that induce the same maps
between stable homotopy groups.
We call the functor E G-excisive if it is G-homotopy-invariant, and the col-
lection of functors X 7→ π∗E(X) forms a G-equivariant homology theory.
3
3See for example chapter 20 of [16] for the relevant definitions.
28
Definition 10.1 Let G be a discrete group. We define the classifying space for
proper actions, EG, to be the G-CW -complex with the following properies:
• For each point x ∈ X the isotropy group
Gx = {g ∈ G | xg = x}
is finite.4
• For a given subgroup H ≤ G the fixed point set EGH is equivariantly
contractible if H is finite, and empty otherwise.
The classifying space EG always exists, and is unique up to G-homotopy-
equivalence; see [2, 6].
The following two results from [6] are the main abstract results on assembly
maps we need in this article.
Theorem 10.2 Let G be a discrete group, and let E be a G-homotopy invariant
functor from the category of G-CW -complexes to the category of spectra. Then
there is a G-excisive functor E′ and a natural transformation α : E′ → E such
that the map
α : E′(G/H)→ E(G/H)
is a stable equivalence whenever H is a finite subgroup of G.
Further, the pair (E ′, α) is unique up to stable equivalence. ✷
Definition 10.3 Let G be a discrete group. Then we define the orbit category,
Or(G), to be the category in which the objects are the G-spaces G/H , where
H is a subgroup of G, and the morphisms are G-equivariant maps.
An Or(G)-spectrum is a functor from the category Or(G) to the category of
symmetric spectra.
Theorem 10.4 Let E be an Or(G)-spectrum. Then there is a G-excisive func-
tor, E′, from the category of G-CW-complexes to the category of spectra such
that E′(G/H) = E(G/H) whenever H is a subgroup of G.
Further, given a functor F from the category of G-CW-complexes to the
category of spectra, there is a natural transformation
β : (F|Or(G))
′ → F
such that the map
β : (F|Or(G))
′(G/H)→ F(G/H)
is a stable equivalence whenever H is a subgroup of the group G. ✷
The constant map c : EG → + induces a map c∗ : E
′(EG) → E(+). This
map is called the assembly map. The corresponding isomorphism conjecture is
the assertion that this assembly map is a stable equivalence.
4As a convention, if we mention a G-space, we assume that the group G acts on the right.
29
Definition 10.5 Consider a group G, and a G-space X . Then we define the
transport groupoid, X , to be the groupoid in which the set of objects is the space
X , considered as a discrete set, and we have morphism sets
Hom(x, y)X = {g ∈ G |xg = y}
Composition of morphisms in the transport groupoid is defined by the group
operation. There is a faithful functor i : X → G defined by the inclusion of each
morphism set in the group.
Given a ring R, a group G, and a G-algebra A over R, let X be a G-CW -
complex. Then (through the functor i), the G-algebra A can also be considered
an X-algebra, and there is a homotopy-invariant functor, E, to the category of
spectra, defined by writing
E(X) = KH(AX)
By theorem 10.2, there is an associated G-excisive functor E ′, and an assem-
bly map
α : E ′(X)→ KH(AX)
such that the map
α : E′(G/H)→ E(G/H)
is a stable equivalence whenever H is finite.
Definition 10.6 The composition of the above map β with the map
i∗ : KH(AX) → KH(AG) induced by the faithful functor i : X → G is called
the KH-assembly map for the group G over the ring R with coefficients in the
G-algebra A.
We say that the group G satisfies the KH-isomorphism conjecture over R
with coefficients in the G-algebra A if the assembly map
β : E(EG)→ K(AG)
is a stable equivalence.
The KH-assembly map is a variant of the Farrell-Jones assembly map. It
was first defined and examined in [1], where in particular its relationship to the
Farrell-Jones assembly map in algebraic K-theory is analysed.5
The following result can be deduced directly from the above definition.
Theorem 10.7 Consider the Or(G)-spectrum
E(G/H) = KH(G/H) = KH(AG/H)
5Actually, the Farrell-Jones conjecture and the corresponding KH-isomorphism conjecture
are usually formulated in terms of virtually cyclic groups rather than finite groups. But by
remark 7.4 in [1], the above formulation of the KH-isomorphism conjecture is equivalent to
the original.
30
and let E′ be the associated excisive functor.
Let X be a path-connected space, and let c : X → + be the constant map.
Then up to stable equivalence the induced map
c∗ : E
′(X)→ E′(+)
is the KH-assembly map. ✷
11 Algebraic KK-theory and homology
Let G be a discrete group, and let X be a right G-simplicial complex (or just
a G-complex for short). Let us term X G-compact if the quotient X/G is a
finite simplicial complex. Any G-complex is a direct limit of its G-compact
subcomplexes.
Given a ring R, the right action of G on the space X induces a left-action
of G on the simplicial algebroid RX .
Definition 11.1 Let A be a G-algebra, and let X be a G-complex. Then we
define the equivariant algebraic K-homology spectrum of X with coefficients in
A to be the direct limit
K
G
hom(X ;A) = lim
K⊆X
G−compact
KKG(R
K , A)
The associated equivariant algebraic K-homology groups are defined by the
formula
KGn (X ;A) = πnKhom(X ;A)
Theorem 11.2 The functor X 7→ KGhom(X ;A) is G-homotopy-invariant and
excisive. For the one-point space, +, we have KGhom(+;A) = KKG(R,A).
Proof: Suppose that X and Y are G-compact simplicial G-complexes. Let
f, g : X → Y be equivariant simplicial maps. Suppose that there is an ele-
mentary equivariant simplicial homotopy, F : X × ∆1 → Y , between f and g.
Then we have an induced equivariant homomorphism F ∗ : RY → RX⊗RR
∆1 =
RX ⊗Z Z[t] such that e0F
∗ = f∗ and e1F
∗ = g∗.
By construction of equivariant KK-theory, algebraically G-homotopic maps
α, β : B → B′ induce homotopic maps α∗, β∗ : KKG(B
′, A) → KKG(B,A). It
follows that the induced maps f∗, g∗ : KKG(X ;A)→ KKG(Y ;A) are homotopic.
More generally, two G-homotopic equivariant simplicial maps f, g : X →
Y can be linked by a finite chain of elementary homotopies, and the above
argument again shows that the induced maps f∗, g∗ : KKG(X ;A)→ KKG(Y ;A)
are homotopic. Generalising to non-compact G-complexes and taking direct
limits, it follows that the functor X 7→ KGhom(X ;A) is G-homotopy-invariant.
Let X and C be G-compact simplicial G-complexes, and suppose we have
a subcomplex B ⊆ X and an equivariant map f : B → C. Let i : B →֒ X be
the inclusion map, and let F : X → X ∪B C and I : C → X ∪B C be the maps
31
associated to the push-out X ∪B C. Then by functoriality, we have an induced
pullback diagram
RC → RB
↑ ↑
RX∪BC → RX
It is easy to check that we have a short exact sequence
0→ RX∪BC
(F∗,I∗)
→ RX ⊕RC
i∗−f∗
→ RB → 0
The map i : B → X is an inclusion of a subcomplex. It follows that we have
an induced inclusion i∗ : R
B → RX . This induced inclusion is an equivariant
map, but not an equivariant homomorphism. Hence the above short exact
sequence has an F -splitting (i∗, 0).
Therefore, by theorem 9.21, we have natural maps ∂ : KGn (X ∪B C;A) →
KGn−1(B;A) such that we have a long exact sequence
→ KGn (B;A)
(i∗,−f∗)
→ KGn (X ;A)⊕Kn(C;A)
F∗+I∗
→ KGn (X∪BC;A)
∂
→ KGn−1(A)→
Similar exact sequences can be seen to exist in the non-compact case by
taking direct limits.
To check the final axiom required of a G-homology theory, let {Xi | i ∈ I}
be a family of G-compact simplicial G-complexes. Let ji : Xi → ∐i∈IXi be the
canonical inclusion. Then by definition of equivariant K-homology as a direct
limit, the map
⊕i∈I(ji)∗ : ⊕i∈I K
G
n (Xi;A)→ K
G
n ((∐i∈IXi);A)
is an isomorphism for all n ∈ Z.
By definition, for the one-point space, +, we haveKGhom(+;A) = KKG(R,A).
✷
12 The Index Map
In [22], the author used natural constructions of analytic KK-theory spectra to
prove that the Baum-Connes assembly map fits into to the general assembly map
machinery. The KH-assembly map is defined using the general machinery; the
constructions in this section prove that the map can be described using algebraic
KK-theory. The methods are somewhat similar to those of [22].
Let G be a discrete group, and let R be a ring. Consider a G-algebra A,
over R, and a G-compact complex K. According to theorem 9.19, we have a
natural map of spectra
D : KKG(R
K , A)→ KK(RKG,AG)
32
The algebra RKG is itself a finitely generated projective RKG-module; let
us label this module EK . Then by theorem 8.8 we have an induced morphism
EK∧ : KK(R
KG,AG)→ KH(AG)
Let X = EG. Then X can be viewed as a G-simplicial complex. Combining
the above two maps and taking the direct limit over G-compact subcomplexes,
we obtain a map
β : KGhom(X ;A)→ KH(AG)
We call this map the index map, by analogy with the map appearing in the
Baum-Connes conjecture (see [2]).
Observe that by definition of the space EG, we can take the space EG to
be a single point if the group G is finite. In this case, the above map is simply
the composition
KKG(R,A)→ KK(RG,AG)→ KK(R,AG) ∼= KH(AG)
Now, for a ring R, let M∞(R) be the R-algebra of all infinite matrices
over R, indexed by N, where almost all entries are zero. Given an R-algebra
A, let κ : A → A ⊗R M∞(R) be the homomorphism defined by the formula
κ(a) = a ⊗ P , where P ∈ M∞(R) is the infinite matrix with 1 at the top left
corner, and every other entry zero. By construction of algebraic KK-theory,
given R-algebras A and B the induced map
κ∗ : KKp(A,B)→ KKp(A,B ⊗R M∞(R))
is an isomorphism.
Let G be a finite group. LetMG(R) be the R-algebra of matrices with entries
in R indexed by the group G. Then there is certainly a natural isomorphism
M∞(R) ∼= M∞(R)⊗R MG(R)
It follows that, given G-algebras A and B, we have a natural map κ : B →
B ⊗R MG(R) inducing isomorphisms
κ∗ : KK
G
p (A,B)→ KK
G
p (A,B ⊗R MG(R))
at the level of KK-theory groups.
Lemma 12.1 Let G be a finite group, and let A be a G-algebra over a ring R.
Then there is a natural injective homomorphism σ : AG→ A⊗RMG(R), where
the image is the set of all elements of the R-algebra A⊗R MG(R) that are fixed
by the group G.
Proof: Let VG be the R-module consisting of all maps from G to the ring R.
Then End(VG) = MG(R). Note that elements of the tensor product VG ⊗R A
can be viewed as maps s : G→ A. Define homomorphism
ρ : G→ End(VG ⊗R A) π : A→ End(VG ⊗R A)
33
by the formulae
ρ(g)s(g1) = s(gg1) π(a)s(g1) = π(g1(a))s(g1)
respectively, where s : G→ A is an element of the module VG ⊗R A, g, g1 ∈ G,
and a ∈ A.
Then we have an associated injective homomrphism σ : AG→ End(VG⊗RA)
defined by the formula
σ
(
m
∑
i=1
aigi
)
=
m
∑
i=1
π(ai)ρ(gi)
Now End(VG ⊗R A) = A ⊗R MG(R). It is easy to check that the image is
the fixed point set. ✷
The proof of the following result is now adapted from the corresponding
result on the Baum-Connes conjecture, where it is sometimes called the Green-
Julg theorem. See for example [9] for an elementary account.
Theorem 12.2 The index map is a stable equivalence for finite groups.
Proof: Let G be a finite group, and let A be a G-algebra over the ring
R. As we remarked above, we have a natural isomorphism κ∗ : KK
G
p (R,A) →
KKGp (R,A ⊗R MG(R)). Let σ∗ : KK
G
p (R,AG) → KK
G
p (R, (A ⊗R MG(R)) be
induced by the isomorphism in the above lemma.
We have an obvious canonical map
KHp(AG) = KKp(R,AG)→ KK
G
p (R,AG)
and we can define a homomorphism γ : KHp(AG)→ KK
G
p (R,A) fitting into a
commutative diagram
KKGp (R,AG)
σ∗
→ KKGp (R,A⊗R MG(R))
↑ ↑
KHp(AG)
γ
→ KK
p
G(R,A)
We claim that γ is an inverse to the map β : KKG(R,A) → KK(R,AG) at
the level of groups, thus proving the result. Looking at suspensions, it suffices
to prove the case when p = 0.
Note that, by construction of the algebraic KK-theory groups, an ar-
bitary KK-theory class [α] ∈ KKG0 (B,C) is represented by a homomorphism
α : J2nB → CS
n
⊗R M∞(R).
Hence, consider a homomorphism α : J2nR → AS
n
⊗R M∞(R). Then we
have a commutative diagram
J2nR → J2nRG
α∗
→ (AS
n
⊗R M∞(R))G
↓ ↓
J2nR⊗R MG(R)
α∗
→ AS
n
⊗R M∞(R)
34
where the vertical maps are versions of the map σ in the above lemma.
Now, at the level of KK-theory groups, the class β ◦ γ[α] is the composition
of the top maps in the above diagram and the vertical map on the right.
On the other hand, the composition of the first map on the top and the first
map on the left is simply the stabilisation map κ. Hence [α] = [α⊗1] = β ◦γ[α].
We have shown that β ◦ γ = 1KH(AG).
Conversely, consider an equivariant homomorphism α : J2nR→ (AG)S
n
⊗R
M∞(R). Then the composition with the map σ defines a class in the KK-theory
group KKG(R,B).
The image β[α] is defined by the composition
J2nR→ J2n(RG)
α∗
→ ((AR)S
n
G)G
We can form the diagram
J2nRG
α∗
→ ((AR)S
n
G)G
σ∗
→ (A⊗MR(G))
SnG
↑ ↑ ↑
J2nR
α
→ (AR)S
n
G = (AR)S
n
G
The left square of the diagram commutes. The right square of the diagram
does not commute. The issue is that the middle vertical map gives us a copy of
the group G on the right of the relevant expession, whereas the vertical map on
the right gives us a copy of the ring MR(G) in the centre of the expression.
If we apply the homomorphsim σ : (A ⊗R MG(R))G → A ⊗R MG(R) ⊗R
MG(R), we see that the square on the right commutes modulo the isomorphism
s : MG(R)⊗RMG(R)→MG(R)⊗RMG(R) defined by writing s(x⊗y) = y⊗x.
The map s is clearly naturally isomorphic to the identity map. It follows by
lemma 7.6 that the right square in the above diagram commutates at the level
of KK-theory groups.
Now the composite of the map on the left and the maps on the top row of
our diagram are the composition give us the class γ ◦ β[α]. Thus γ ◦ β[α] = [α].
We see that the homomorphism γ is the inverse of the homomorphism β,
and we are done. ✷
Given an equivariant map of G-complexes f : X → Y , there is an induced
functor f∗ : X → Y between the transport groupoids. There is an obvious
faithful functor i : X → G. If A is a G-algebra, it can therefore also be regarded
as an X-algebra.
Now, let K be a G-compact subcomplex. Then we have an induced restric-
tion map
i∗ : KKG(R
K , A)→ KKX(R
K , A)
By theorem 9.19, there is a natural map
D : KKX(R
K , A)→ KK(RKX,AX)
35
Definition 12.3 Let x ∈ X . Then we write EK(x) to denote the set of collec-
tions
{ηy ∈ Hom(x, y)RKX | y ∈ X}
such that the formula
ηyg = ηz
holds for all elements g ∈ G such that yg = z.
The assignment x 7→ EK(x) is an R
KX-module, which we again label EK .
The RKX-action is defined by composition of morphisms.
Since the action of the group G on the space X is not necessarily transitive,
the module RK is not necesarily finitely generated and projective. However, the
restriction, E|Or(x), to the path-component (R
KX)Or(x) is isomorphic to the
module Hom(−, x)RKX|Or(x) . Thus, the module EK is almost finitely-generated
and projective, and we have a canonical morphism
EK∧ : KK(R
KX,AX)→ KH(AX)
We can compose with the map
D : KKX(R
K , A)→ KK(RKX,AX)
to form the composite
γK : KKG(R
K , A)→ K(AX)
and take direct limits to obtain a map
γ : KGhom(X ;A)→ K(AX)
Proposition 12.4 Let f : (X,K) → (Y, L) be a map of pairs of G-simplicial
complexes, where the subcomplexes K and L are G-compact. Let
f∗ : RLX → RKX f∗ : R
LX → RLY
be the obvious induced maps. Then there is an almost finitely generated projec-
tive RLX-module, θ ,such that f∗θ = EL and f
∗θ = EK .
Proof: Let x ∈ X . Let θ(x) denote the set of collections
{ηy ∈ Hom(x, y)RLX | y ∈ X}
such that the formula
ηyg = ηz
is satisfied for all elements g ∈ G such that yg = z.
The assignment x 7→ θ(x) forms an RLX-module, θ. The C0(K)X-action is
defined by composition of morphisms.
The equations f∗(θ) = EL and f
∗θ = EK are easily checked. ✷
36
Lemma 12.5 The map γ : KGhom(X ;A) → K(AX) is natural for proper G-
complexes.
Proof: Let f : (X,K) → (Y, L) be a map of pairs of G-simplicial complexes,
where the subcomplexes K and L are G-compact. Then by the above propo-
sition, and naturality of the product and descent map in algebraic KK-theory,
we have a commutative diagram.
KKX(R
K , A) //
��
KK(RKX,AX)
�� ''OO
OO
O
OO
OO
O
O
KKG(R
K , A)
77nnnnnnnnnnnn
��
KKX(R
L, A) // KK(RLX.AX)
��
// KH(AX)
��
KKG(R
L, A)
77nnnnnnnnnnnn
// KKY (R
L, A)
OO
((QQ
QQ
QQ
QQ
QQ
QQ
KK(RLX.AY ) // KH(AY )
KK(RLY ,AY )
OO 77oooo
ooooooo
Taking direct limits, the desired result follows. ✷
Lemma 12.6 The composite map i∗γ : K
G
hom(X ;A) → KH(AG) is the index
map.
Proof: Let K be a G-compact subcomplex of X . The naturality properties
of the various descent maps and products give us a commutative diagram
KKG(R
K , A) //
��
KK(RKG,AG) //
��
KH(AG)
KK(RKX,AG)
77oooooooooooo
KKX(R
K , A) // KK(RKX,AX) //
OO
KH(AX)
OO
Here the top row is the index map, and the composite of the bottom row
and the vertical map on the left is the map γ. Taking direct limits, the desired
result follows. ✷
Lemma 12.7 Let H be a finite subgroup of G. Then the map
γ : KGhom(G/H ;A)→ KH(AG/H) is a stable equivalence of spectra.
37
Proof: Let i : H →֒ G be the inclusion isomorphism. Then the group H and
groupoidG/H are equivalent. By theorem 9.22, and the naturality of restriction
maps, the map γ is equivalent to the composition
KKG(R
G/H , A)
i∗
→ KKH(R
G/H , A)
D
→ KK(RG/HH,AH)
EG/H∧
→ KH(AH)
Let j : R → RG/H be induced by the constant map G/H → +. Then we
have a commutative diagram
KKG(R
G/H , A)
↓
KKH(R
G/H , A)
j∗
→ KKH(R,A)
↓ ↓
KKH(R
G/HH,AH)
j∗
→ KKH(RH,AH)
↓ ↓
K(AH) = K(AH)
A straightforward calculation tells us that the composite
j∗i∗ : KKG(R
G/H , A) → KKH(R,A) is a stable equivalence of spectra.
The composite map on the right, β : KKH(R,A) → K(AH) is the index map.
Since the group H is finite, the space + is a model for the classifying space
E(H,EG).
But the index map is a stable equivalence for finite groups by theorem 12.2,
so we are done. ✷
By theorem 11.2. the equivariant K-homology functor KGhom(−;A) is G-
excisive. We can therefore use the above three lemmas to apply theorem 10.2
to the study of the index map; we immediately obtain the following result.
Theorem 12.8 Let E′ be a G-excisive functor from the category of proper G-
CW -complexes to the category of symmetric spectra. Suppose we have a natural
transformation α : E′(X)→ K(AX) such that the map
α : E′(G/H)→ K(AG/H)
is a stable equivalence for every finite subgroup, H, of the group G.
Then, up to stable equivalence, the composite αi∗ : E
′(X) → K(AG) is the
map β. ✷
By definition of the KH-assembly map, the following therefore holds.
Corollary 12.9 The index map is the KH-assembly map. ✷
Now, the KH-isomorphism conjecture holds for finite groups with any coef-
ficients. It follows from more general results in [1] that the conjecture also holds
for the integers.
38
Theorem 12.10 Let G be a finitely generated abelian group. Then the KH-
isomorphism conjecture holds for G with any coefficients.
Proof: By the fundamental theorem of abelian groups, we have an isomor-
phism
G ∼= Z
q ⊕ Z/p1 ⊕ · · · ⊕ Z/pk
where the pi are prime numbers. We have classifying spaces for proper actions
EZ = R EZ/pi = +.
Thus the group G has classifying space Rq. The copies of the group Z act
by translation, and the copies of finite groups act trivially.
It follows that we have a commutative diagram
KGn (EG;A)
∼= KZn(R;A)
q ⊕K
Z/p1
n (+;A)⊕ · · · ⊕K
Z/pk
n (+;A)
↓ ↓
KHGn (AG)
∼= KHn(AZ)
q ⊕KHn(AZ/p1)⊕ · · · ⊕Kn(AZ/pk)
where the vertical arrows are copy of the assembly map at the level of groups.
We know that the KH-isomorphism conjecture holds for finite groups and
for the integers. The direct sum of assembly maps on the right is thus an
isomorphism.
It follows that the map on the left is also an isomorphism, and we are done.
✷
References
[1] A.C. Bartels and W. Lück. Isomorphism conjecture for homotopyK-theory
and groups acting on trees. Journal of Pure and Applied Algebra, 205:660–
696, 2006.
[2] P. Baum, A. Connes, and N. Higson. Classifying spaces for proper actions
andK-theory of group C∗-algebras. In S. Doran, editor, C∗-algebras: 1943–
1993, volume 167 of Contemporary Mathematics, pages 241–291. American
Mathematical Society, 1994.
[3] B. Blackadar. K-Theory for Operator Algebras, volume 5 of Mathemati-
cal Sciences Research Institute Publications. Cambridge University Press,
1998.
[4] G. Cortinas and A. Thom. Bivariant algebraic K-theory. Journal für die
Reine und Angewandte Mathematik (Crelle’s Journal), 610:267–280, 2007.
[5] J. Cuntz and A. Thom. Algebraic K-theory and locally convex algebras.
Mathematische Annalen, 334:339–371, 2006.
39
[6] J. Davis and W. Lück. Spaces over a category and assembly maps in
isomorphism conjectures in K- and L-theory. K-theory, 15:241–291, 1998.
[7] F.T. Farrell and L. Jones. Isomorphism conjectures in algebraic K-theory.
Journal of the American Mathematical Society, 6:377–392, 1993.
[8] P.G. Goerss and J.F. Jardine. Simplicial Homotopy Theory, volume 174 of
Progress in Mathematics. Birkhäuser Verlag, 1999.
[9] E. Guentner, N. Higson, and J. Trout. Equivariant E-theory for C∗-
algebras, volume 148 of Memoirs of the American Mathematical Society.
American Mathematical Society, 2000.
[10] I. Hambleton and E.K. Pedersen. Identifying assembly maps in K- and
L-theory. Mathematische Annalen, 328:27–57, 2004.
[11] N. Higson. A characterization of KK-theory. Pacific Journal of Mathe-
matics, 253–276, 1987.
[12] N. Higson. A primer on KK-theory. In Operator theory: operator alge-
bras and applications, part 1 (Durham, NH, 1988), volume 51, part 1 of
Proceedings of Symposia in Pure Mathematics, pages 239–283. American
Mathematical Society, 1990.
[13] M. Hovey, B. Shipley, and J. Smith. Symmetric spectra. Journal of the
American Mathematical Society, 13:149–208, 2000.
[14] M. Joachim. K-homology of C∗-categories and symmetric spectra repre-
senting K-homology. Mathematische Annalen, 327:641–670, 2003.
[15] C. Kassel. Charactére de Chern bivariant. K-theory, 3:367–400, 1989.
[16] M. Kreck and W. Lück. The Novikov Conjecture, volume 33 of Oberwolfach
Seminars. Birkhäuser, 2005.
[17] E. Lance. Hilbert C∗-modules. Lectures in Mathematics, ETH Zurich.
Birkhäuser, 2002.
[18] S. MacLane. Categories for the working mathematician, volume 5 of Grad-
uate Texts in Mathematics. Springer, Berlin, 1998.
[19] R. Meyer and R. Nest. The Baum-Connes conjecture via localisation of
categories. Topology, 45:209–250, 2006.
[20] B. Mitchell. Separable algebroids, volume 333 of Memoirs of the American
Mathematical Society. American Mathematical Society, 1985.
[21] P.D. Mitchener. Symmetric K-theory spectra of C∗-categories. K-theory,
24:157–201, 2001.
40
[22] P.D. Mitchener. C∗-categories, groupoid actions, equivariant KK-theory,
and the Baum-Connes conjecture. Journal of Functional Analysis, 214:1–
39, 2004.
[23] A. Neeman. Triangulated Categories, volume 148 of Annals of Mathematics.
Princeton University Press, 2001.
[24] G. Skandalis. Kasparov’s bivariantK-theory and applications. Expositiones
Mathematicae, 9:193–250, 1991.
[25] A. Thom. Connective E-theory and bivariant homology for C∗-algbras.
Available at http://www.uni-math.gwdg.de/thom.
[26] A. Valette. Introduction to the Baum-Connes Conjecture. Lectures in Math-
ematics, ETH Zurich. Birkhäuser, 2002.
[27] C.A. Weibel. Homotopy algebraic K-theory. In Algebraic K-theory and
Algebraic Number Theory (East-West Center, 1987), volume 83 of Con-
temporary Mathematics, pages 461–488. American Mathematical Society,
1989.
[28] C.A. Weibel. An introduction to homological algebra, volume 38 of Cam-
bridge Studies in Advanced Mathematics. Cambridge University Press,
1994.
41
http://www.uni-math.gwdg.de/thom
Introduction
Algebroids
Tensor Products
Algebraic Homotopy and Simplicial Enrichment
Path Extensions
The KK-theory spectrum
KK-theory groups
Modules over Algebroids
Equivariant KK-theory
Assembly
Algebraic KK-theory and homology
The Index Map
| 0non-cybersec
| arXiv |
Cursed_Baby. | 0non-cybersec
| Reddit |
Remastered Edition of Sidereal Confluence coming in May 2020, with stunning new artwork from Kwanchai Moriya. | 0non-cybersec
| Reddit |
Should I keep using the same frameworks for projects if they get the job done?. Some more context:
I'm currently working on the two following projects:
* https://github.com/purrcat259/charitybot2
* https://github.com/purrcat259/peek
The second one arose out of the need to see how nginx will handle itself during CharityBot2's public beta test during [Gameblast17](https://www.gameblast17.com/).
I was about to reach for Flask to write an API to get data out of peek's SQLite database and exposes it on an API to be viewable on a browser, which is exactly what I did with CharityBot2, so been there, done that.
My question lies here: is it better to challenge myself and write the API using some other framework (such as Django) or even another language (such as Ruby, using Sinatra), or stick to what I know how to use? My GitHub profile is also my portfolio that I show to prospective employers - I'm not sure whether experience in a broad number of frameworks or deep knowledge of a few is more attractive.
Thanks in advance. | 0non-cybersec
| Reddit |
Nobody uploaded Greenscreen Videos so here's a late We are Number One Greenscreen. | 0non-cybersec
| Reddit |
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application.
I need to scale my app adding some spot instances that EB do not support.</p>
<p>So I create a second autoscaling from a launch configuration with spot instances.
The autoscaling use the same load balancer created by beanstalk.</p>
<p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p>
<p>This work fine, but:</p>
<ol>
<li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p>
</li>
<li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p>
</li>
</ol>
<p><strong>UPDATE</strong></p>
<p>Elastic Beanstalk add support to spot instance since 2019... see:
<a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>
| 0non-cybersec
| Stackexchange |
wagtail pathoverflow on adding new child page. <p>I have developed a wagtail site and that works just fine. It's just a simple blog. I was able to add a blog index page and blog posts under them. Hover recently when try to add a new page it gives the error
<code>PathOverflow at /admin/pages/add/blog/blogpage/8/</code></p>
<p>Bellow is the complete traceback.</p>
<pre><code>Environment:
Request Method: POST
Request URL: https://vikatakavi.info/admin/pages/add/blog/blogpage/8/
Django Version: 2.1.5
Python Version: 3.5.2
Installed Applications:
['home',
'search',
'wagtail.contrib.forms',
'wagtail.contrib.redirects',
'wagtail.embeds',
'wagtail.sites',
'wagtail.users',
'wagtail.snippets',
'wagtail.documents',
'wagtail.images',
'wagtail.search',
'wagtail.admin',
'wagtail.core',
'modelcluster',
'taggit',
'django.contrib.admin',
'django.contrib.auth',
'django.contrib.contenttypes',
'django.contrib.sessions',
'django.contrib.messages',
'django.contrib.staticfiles',
'django.contrib.sitemaps',
'blog']
Installed Middleware:
['django.contrib.sessions.middleware.SessionMiddleware',
'django.middleware.common.CommonMiddleware',
'django.middleware.csrf.CsrfViewMiddleware',
'django.contrib.auth.middleware.AuthenticationMiddleware',
'django.contrib.messages.middleware.MessageMiddleware',
'django.middleware.clickjacking.XFrameOptionsMiddleware',
'django.middleware.security.SecurityMiddleware',
'wagtail.core.middleware.SiteMiddleware',
'wagtail.contrib.redirects.middleware.RedirectMiddleware']
Traceback:
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/django/core/handlers/exception.py" in inner
34. response = get_response(request)
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/django/core/handlers/base.py" in _get_response
126. response = self.process_exception_by_middleware(e, request)
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/django/core/handlers/base.py" in _get_response
124. response = wrapped_callback(request, *callback_args, **callback_kwargs)
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/django/views/decorators/cache.py" in _wrapped_view_func
44. response = view_func(request, *args, **kwargs)
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/wagtail/admin/urls/__init__.py" in wrapper
102. return view_func(request, *args, **kwargs)
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/wagtail/admin/decorators.py" in decorated_view
34. return view_func(request, *args, **kwargs)
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/wagtail/admin/views/pages.py" in create
224. parent_page.add_child(instance=page)
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/treebeard/mp_tree.py" in add_child
1013. return MP_AddChildHandler(self, **kwargs).process()
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/treebeard/mp_tree.py" in process
387. newobj.path = self.node.get_last_child()._inc_path()
File "/home/ubuntu/projects/blogger/env/lib/python3.5/site-packages/treebeard/mp_tree.py" in _inc_path
1114. raise PathOverflow(_("Path Overflow from: '%s'" % (self.path, )))
Exception Type: PathOverflow at /admin/pages/add/blog/blogpage/8/
Exception Value: Path Overflow from: '000100010005ZZZZ'
</code></pre>
<p>I'm absolutely clueless on this.</p>
| 0non-cybersec
| Stackexchange |
Can't get past the unetbootin option screen. <p>Guys I'm losing my mind here... For the last 2 days I'm trying to install either Elementary OS l, Lubuntu or Linux mint and I haven't had luck with any of them.</p>
<p>I tried with dd and unetbootin... With dd I didn't get anywhere, with unetbootin I get to the splash screen and when I choose whatever option it gets stuck.</p>
<p>I have previously installed elementary os using this usb without any problems...</p>
<p>What the hell am I doing wrong?</p>
<p>I tried downloading iso, I tried with the built in unetbootin option to let it download it for me, same thing every time... I get to the splash screen, choose whatever option and blinking underscore appears and thats it.</p>
<p>I've searched everywhere for a solution and can't find one.</p>
<p>Highly appreciated!</p>
<p>Eidt: I chose the "run in compatibility mode" with Mint on the usb and it ran! I isntalled it all fine, but it doesn't boot... When it goes past the bios checks it gives me a black screen and that's it... Nothing else.</p>
| 0non-cybersec
| Stackexchange |
On the first page of Google images when I searched for "Lesbian hair cut".. | 0non-cybersec
| Reddit |
17 Brilliant Short Novels You Can Read in a Sitting. | 0non-cybersec
| Reddit |
7 days difference! It may not be much to y'all, but it's my first veggie ever and I'm proud.. | 0non-cybersec
| Reddit |
determine whether or not a subset is closed or open. <p>determine whether or not a subset is closed or open:</p>
<blockquote>
<p>(a) For <span class="math-container">$X=\Bbb R^2$</span> and <span class="math-container">$d$</span> the Euclidean metric on <span class="math-container">$\Bbb R^2$</span>:</p>
<ol>
<li><p><span class="math-container">$A_1=$</span>{<span class="math-container">$(x,y): x^2+y^2 <1$</span>} <span class="math-container">$\cup $</span> {<span class="math-container">$(1,0)$</span>}.</p>
</li>
<li><p><span class="math-container">$A_2=$</span>{<span class="math-container">$(x,0): 0 < x < 1$</span>}.</p>
</li>
</ol>
<p>(b) For <span class="math-container">$X=$</span>{all continuous functions <span class="math-container">$f: [0,1]\to [0,1]$</span> } with the metric <span class="math-container">$d(f,g) = \sup_{x\in [0,1]}|f(x) - g(x)|$</span>:</p>
<ul>
<li><span class="math-container">$A_3=$</span>{<span class="math-container">$f\in X : f(0)=f(1)$</span>}.</li>
</ul>
</blockquote>
| 0non-cybersec
| Stackexchange |
Study of 79 countries: Religiosity linked to anti-gay attitudes -- but effect smallest among Buddhists. | 0non-cybersec
| Reddit |
Intersection of sets in modulo. <p>For a range of numbers in modulo 7 {1,2,3,4,5,6,7}
let A be a continuous set/range of m elements
and B be a continuous set/range of n elements</p>
<p>How to calculate the number of elements (x) in the intersection A∩B?</p>
<p>examples:</p>
<ol>
<li>A={6,7}={6-7}, B={1,2,3}={1-3} -> A∩B={}, i.e.x=0</li>
<li>A={6,1}={6-1}, B={3,4,5,6}={3-6} -> A∩B={6}, i.e. x=1</li>
<li>A={6,7}={6-7}, B={5,6,7,1}={5-1} -> A∩B={6,7}, i.e. x=2</li>
</ol>
<p>So we know the ranges min & max values and we want to calculate the number of common elements (using simple arithmetics).
This particular case is about weekdays but the solution should work in any modulo N scenario, I guess.</p>
| 0non-cybersec
| Stackexchange |
.htaccess is not working. <p>I have a .htaccess file in /var/www, the contents of which is: <code>ErrorDocument 404 /404.html</code>.</p>
<p>In apache2.conf <code>AccessFileName</code> is <code>.htaccess
</code>, and <code>AllowOverride</code> does not exist. For some reason, visiting myurl.com/anything does not show 404.html.</p>
<p>Help greatly appreciated.
~JJ56.</p>
| 0non-cybersec
| Stackexchange |
Where can I get a gameboy in good condition?. Hi reddit! I own a gameboy advance, I've had it since I was a kid, and because of that, its in poor condition. I've wanted to get a gameboy with a backlight so I can play my favorite games. I am willing to buy a knockoff brand, like how you see the consoles that play NES to N64. Anything is nice! | 0non-cybersec
| Reddit |
Missing cmssbx28 font (and other sizes). <p>This is trying to build ods (<a href="https://github.com/patmorin/ods.git" rel="nofollow noreferrer">https://github.com/patmorin/ods.git</a>) (the "Open Data Structures in *" by Pat Morin set of books), and it complains it can't build cmssbx28. Running <code>mktexmf cmssbx28</code> (and the other missing sizes) gives "!I can't find file 'cmssbx28'."</p>
<p>This is texlive 2019 on Fedora 31. Am I missing some package? I can't find anything relevant with my limited font-fu.</p>
| 0non-cybersec
| Stackexchange |
I think I’ve misread signs. I’ve got a huge crush on this girl who I thought liked me back as she was always laughing at whatever I said and constantly looking at me and we had easy flowing conversations but this is only during class I’m with her. On Friday I was sorta making it obvious that I’d have a free car and that I didn’t want to drive back alone. She didn’t ask if she could have the lift. Today it just felt we didn’t speak as much and have fun conversations. Today I actually offered her a lift and she said her mum was picking her up and I don’t even live in the same direction. What do I do? I can ask her friend if that’s a good idea? I need help badly | 0non-cybersec
| Reddit |
[WP] You die and go to heaven but unlike most arrivals you get a special role, 'Gods Jester'. You have never considered yourself funny but god says "You are the only mortal that could ever make me laugh, here let me show you".. | 0non-cybersec
| Reddit |
update-grub ignores GRUB_CMDLINE_LINUX in /etc/default/grub when generating menu.lst. <p>I created <code>/etc/default/grub</code> with the contents <code>GRUB_CMDLINE_LINUX="cgroup_enable=memory swapaccount=1"</code>. When I run <code>update-grub</code>, I expect these additional arguments to be appended to the kernel lines in <code>/boot/grub/menu.lst</code>. But it never works.</p>
<p>I tried several variants (<code>GRUB_CMDLINE_{XEN,LINUX}{_DEFAULT}</code>) but no luck. The file <code>/etc/default/grub</code> <em>does</em> get executed, tried it with an test echo output. When (un)installing kernels, <code>menu.lst</code> always gets updated. But the above arguments are always ignored.</p>
<p>Do you have any ideas to debug this? I'm out of ideas.</p>
<p>System info:</p>
<ul>
<li>Ubuntu 14.04 x64</li>
<li>grub 0.97-29ubuntu66</li>
<li>It's an Xen DomU booted via pvgrub64</li>
</ul>
| 0non-cybersec
| Stackexchange |
Nigeria bomb rips through marketplace. | 0non-cybersec
| Reddit |
Any quizPack format spec?. <p>I'm a Windows software developer interested in reading the <code>.quizPack</code> file format. It is designed for Apple iQuiz software, and contains text quizzes for iPod. The text file format, unpacked from quizPack, "trivia.txt" is fully documented <a href="http://macdevcenter.com/pub/a/mac/2007/04/30/building-custom-iquiz-data.html" rel="nofollow">here</a>. But how to unpack these files?</p>
<p>I cannot find any specification of this format on Apple website or elsewhere. </p>
<p>Here is <a href="http://files.aspyr.com/iquiz_maker/quizzes/CatsAndDogs.quizPack" rel="nofollow">a downloadable example of quizPack file</a>.</p>
<p>This format can be created, for example, by <a href="http://www.iquizmaker.com" rel="nofollow">iQuizMaker software</a>. iTunes can read these files and export them to iPod as "unpacked" quizzes.</p>
| 0non-cybersec
| Stackexchange |
All other things equal, what would you consider the most/least sexy job a man could have?. Let's assume you meet a guy that is handsome, charming, etc. What occupations would you be the most and least excited to hear that he has? | 0non-cybersec
| Reddit |
Communication between Windows Store app and native desktop application. <p><strong>! For the sake of simplifying things I will refer to <em>Windows Store applications</em> (also known as Metro or Modern UI) as "app" and to common <em>desktop applications</em> as "application" !</strong></p>
<p>I believe this is still one of the most unclear yet important questions concerning app-development for developers who already have established applications on the market:
How to manage communication between apps and applications on a Windows 8 system? (please let's not start a debate on principles - there're so many use cases where this is really required!)</p>
<p>I basically read hundrets of articles in the last few days but still it remains unclear how to proceed doing it right from the first time. Mainly because I found several conflicting information.
With my question here I'd like to re-approach this problem from the viewpoint of the final Windows 8 possibilities.</p>
<p><strong>Given situation:</strong></p>
<ul>
<li>App and application run on same system</li>
<li>1:1 communication</li>
<li>Application is native (written in Delphi)</li>
<li>Administrator or if required even system privileges are available for the application</li>
<li>In 90% of the use cases the app requests an action to be performed by the application and receives some textual result. The app shouldn't be left nor frozen for this!</li>
<li>In 10% the application performs an action (triggered by some event) and informs the app - the result might be: showing certain info on the tile or in the already running and active app or if possible running the app / bringing it to the foreground.</li>
</ul>
<p><strong>Now the "simple" question is, how to achieve this?</strong></p>
<ul>
<li>Is local webserver access actually allowed now? (I believe it wasn't for a long time but now is since the final release)</li>
<li>WCF? (-> apparently <a href="http://msdn.microsoft.com/en-us/library/hh556233.aspx" rel="noreferrer">MS doesn't recommend that anymore</a>)</li>
<li>HTTP requests on a local REST/SOAP server?</li>
<li><a href="http://msdn.microsoft.com/en-us/library/windows/apps/windows.web.syndication" rel="noreferrer">WinRT syndication API</a>? (another form of webservice access with RSS/atom responses)</li>
<li><a href="http://msdn.microsoft.com/en-us/library/windows/apps/br212061.aspx" rel="noreferrer">WebSockets</a> (like <a href="http://msdn.microsoft.com/en-us/library/windows/apps/windows.networking.sockets.messagewebsocket.aspx" rel="noreferrer">MessageWebSocket</a>)?</li>
<li>Some other form of TCP/IP communication?</li>
<li>Sharing a text file for in- and output (actually simply thinking of this hurts, but at least that's a possibility MS can't block...)</li>
<li>Named Pipes are not allowed, right?</li>
</ul>
<p>There are some discussions on this topic here on SO, however most of them are not up-to-date anymore as MS changed a lot before releasing the final version of Windows 8. Instead of mixing up old and new information I'd like to find a definite and current answer to this problem for me and for all the other Windows application and app developers. Thank you!</p>
| 0non-cybersec
| Stackexchange |
What's first anime you are going to (or you've already) finished in 2016?. First of all
[Happy New Year.](#idoruwinkdesu)
everyone.
So I decided to finish Owarimonogatari before the end of 2015 and failed so it's now the first show I've finished of the year.
Well at least I started with something I loved, unlike the last year when I started it with Akame ga Kill!
How about you guys? | 0non-cybersec
| Reddit |
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application.
I need to scale my app adding some spot instances that EB do not support.</p>
<p>So I create a second autoscaling from a launch configuration with spot instances.
The autoscaling use the same load balancer created by beanstalk.</p>
<p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p>
<p>This work fine, but:</p>
<ol>
<li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p>
</li>
<li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p>
</li>
</ol>
<p><strong>UPDATE</strong></p>
<p>Elastic Beanstalk add support to spot instance since 2019... see:
<a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>
| 0non-cybersec
| Stackexchange |
The dryer.... | 0non-cybersec
| Reddit |
Angular 2 - script5007: Unable to get property 'apply' of undefined or null reference. <p>I created a simple project:</p>
<pre><code>ng new my-app
cd my-app
ng serve
</code></pre>
<p>Chrome and Firefox it's OK, but when I execute in IE 11, it returns this error: SCRIPT5007</p>
<p><a href="https://i.stack.imgur.com/7qo9U.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/7qo9U.jpg" alt="enter image description here"></a>
<a href="https://i.stack.imgur.com/2DQfX.jpg" rel="noreferrer"><img src="https://i.stack.imgur.com/2DQfX.jpg" alt="enter image description here"></a></p>
<p>Can someone help me?</p>
| 0non-cybersec
| Stackexchange |
Confusion with (strict) 2-adjoints. <p>This is an elementary question on 2-categories, namely on the naturally arising notion of 2-adjunction when strict 2-functors are involved. Perhaps I don't understand some "internal" universal property of Kan extensions... (We restrict to the strict 2-category of categories and ignore size issues.)</p>
<ul>
<li><p>One eventually encounters the universal property of the Yoneda embedding as a free cocompletion of a category. This seems to have three possible meanings (of interest). There's a forgetful strict 2-functor $U:\mathsf{CocompleteCat}\to\mathsf{Cat}$ from cocomplete categories and cocontinuous functors, and it may have some notion of left adjoint.</p>
<ol>
<li>There could be strictly 2-natural <em>isomorphisms</em> of categories $$\mathsf{CocompleteCat}(\widehat{\mathsf{C}},\mathsf D)\cong\mathsf{Cat}(\mathsf C,U\mathsf D).$$</li>
<li>There could be strictly 2-natural <em>equivalences</em> of categories $$\mathsf{CocompleteCat}(\widehat{\mathsf{C}},\mathsf D)\simeq\mathsf{Cat}(\mathsf C,U\mathsf D).$$</li>
<li>There could be 2-natural (pseudonatural) <em>equivalences</em> of categories $$\mathsf{CocompleteCat}(\widehat{\mathsf{C}},\mathsf D)\simeq\mathsf{Cat}(\mathsf C,U\mathsf D).$$</li>
</ol></li>
<li><p>Given a category $\mathsf C$ one can consider its family fibration $\mathsf{Fam}(\mathsf C)\to \mathsf{Set}$. The objects of the domain are set-indexed families of objects of $\mathsf C$ while an arrow is given by a set-function between the indexing sets and a family of arrows in $\mathsf C$ in the obvious way. One may prove this assignment extends to a strict 2-functor $\mathsf{Fam}:\mathsf{Cat}\to \mathsf{Cat}$ which moreover lands in the category $\amalg$-$\mathsf{Cat}$ of categories with (small) coproducts and coproduct preserving functors. Again there's a forgetful strict 2-functor $U:\amalg$-$\mathsf{Cat}\to \mathsf{Cat}$ and again the $\mathsf{Fam}$ functor
is the free coproduct cocompletion of $\mathsf C$. In fact it's even the extensive completion/envelope of $\mathsf C$. Thus once more three options arise.</p>
<ol>
<li>There could be strictly 2-natural <em>isomorphisms</em> of categories $$\mathsf{ExtCat}(\mathsf{Fam}(\mathsf{C}),\mathsf D)\cong\mathsf{Cat}(\mathsf C,U\mathsf D).$$</li>
<li>There could be strictly 2-natural <em>equivalences</em> of categories $$\mathsf{ExtCat}(\mathsf{Fam}(\mathsf{C}),\mathsf D)\simeq\mathsf{Cat}(\mathsf C,U\mathsf D).$$</li>
<li>There could be 2-natural (pseudonatural) <em>equivalences</em> of categories $$\mathsf{ExtCat}(\mathsf{Fam}(\mathsf{C}),\mathsf D)\simeq\mathsf{Cat}(\mathsf C,U\mathsf D).$$</li>
</ol></li>
</ul>
<p>What is the correct sense in which there are 2-adjunctions $\widehat{(-)}\dashv U$ and $\mathsf{Fam}\dashv U$? I think the correct answer should be the second option, since the non-strict notion is usually the "correct" but pseudonaturality of the equivalences should somehow disappear because the 2-categories and 2-functors are strict. Also I've never seen a formulation of the free cocompletion result with isomorphisms of categories. On the other hand both adjoints are given by Kan extension whose universal property gives bijection of sets, so I don't know how to get the equivalences..</p>
| 0non-cybersec
| Stackexchange |
Possible to write protect an external disk?. Heya,
Apologies if this is the wrong place for this. Is there any software or method for specifically write protecting an external USB disk? I.e the drive would be encrypted, when you use a password to access the USB you can see and read the content but it's still read\-only, and if you wish to write to the USB from there you would need another password \(allowing a specific pc would be ideal but unlikely understanding that it's not on a domain\).
Any advice would be much appreciated! | 1cybersec
| Reddit |
[AMA Request] Someone who has written code for or been involved with implementing and maintaining a chat bot for sex site advertising.. Recently had a random person add me on Skype. Figured out quickly that it was a bot to try to entice me in to going to a sex site. I started thinking that well, someone had to code that. They waited for responses, acknowledged (but was not deterred by) me not being single when I answered as such.
How do you get a job like that? What was the description in the job posting?
What are some of the metrics? How many "sales" do they get? How often reported? Average time before bans? | 0non-cybersec
| Reddit |
Comunión en Torre La Mina { Lucía } Reportaje de comunión en exteriores. | 0non-cybersec
| Reddit |
Justin Roiland the totally normal human. | 0non-cybersec
| Reddit |
Unable to use ternary operator to conditionally assign `istream &`?. <p>I have a constructor that accepts a <code>std::istream &</code> and checks it before assigning one of the members (a <code>std::istream &</code>). For example:</p>
<pre><code>class Stream
{
public:
Stream(std::istream &is) : s_ {is.good() ? is : throw std::runtime_error {"Invalid input stream\n"}}
{}
private:
std::istream &s_;
};
</code></pre>
<p>The compiler complains that the constructor for <code>std::basic_istream(const basic_istream &)</code> is deleted (understandably, since you can't copy streams). But, I don't see where any copying is being done here? It must be within the ternary operator, because</p>
<pre><code>Stream(std::istream &is) : s {is} {}
</code></pre>
<p>with no checking works fine. Where is the <code>std::istream</code> attempting to be copied? And how can I fix this? </p>
| 0non-cybersec
| Stackexchange |
[SV][NSV] Lost 8 pounds last month; got awesome compliment at work. So, first: I lost 8 more pounds last month! Yay! I'm only weighing myself on the first Monday of each month to avoid getting caught up in day-to-day fluctuations re: Shark Week, weather conditions, .... (Yes, I retain water like a MoFo when the humidity is high.)
Also, was told by a coworker that someone told her "I've noticed that GarageQueen has lost a *ridiculous* amount of weight. She looks really good!" Hee! (And yes, knowing this person, 'ridiculous' was totally meant as a slangy complement.) Was grinning the whole rest of the day as the words "*ridiculous* amount of weight" echoed in my head.
Hee! "ridiculous" :D | 0non-cybersec
| Reddit |
Here’s one tally of the losses from WannaCry ransomware global attack. | 1cybersec
| Reddit |
For ergodic Markov chains, when does $\lim_{N\to\infty} \mathbb{E}[\sum_{n=1}^{N}f(X_n)] - N\mu(f)$ exist. <p>For an ergodic Markov chain (process would be even better) <span class="math-container">$X_{n}$</span> with stationary distribution <span class="math-container">$\mu$</span>, under which conditions does
<span class="math-container">$$
L:=\lim_{N\to\infty} \mathbb{E}[\sum_{n=1}^{N}f(X_n)] - N\mu(f)
$$</span>
exist? In the other direction, do you know of examples where this limit does not exist?</p>
<hr>
<p>Ergodicity says
<span class="math-container">$$
\mathbb{E}[\frac{1}{N}\sum_{n=1}^{N}f(X_n)] - \mu(f) \to 0.
$$</span></p>
<p>If the limit <span class="math-container">$L$</span> above exists, this could be refined to
<span class="math-container">$$
\mathbb{E}[\frac{1}{N}\sum_{n=1}^{N}f(X_n)] - \mu(f) = \frac{L}{N} + \mathcal{o}(\frac{1}{N}).
$$</span>
If we denote by <span class="math-container">$\mu_{n}$</span> the distribution of <span class="math-container">$X_n$</span>, we have
<span class="math-container">$$
\mathbb{E}[\sum_{n=1}^{N}f(X_n)] = N \mu(f) + \sum_{n=1}^{N} (\mu-\mu_n)(f)
$$</span>
and my question can be rephrased as</p>
<blockquote>
<p>Under which conditions on the Markov chain <span class="math-container">$(X_n)$</span> and the function
<span class="math-container">$f$</span> is <span class="math-container">$(\mu-\mu_n)(f)$</span> summable? In the other direction, do you know of examples where it is not summable?</p>
</blockquote>
<hr>
<p>I know that for many Markov chains we have <span class="math-container">$\mu_n =\mu+ \mathcal{O}(e^{-cn})$</span>. I'm interested in theory for cases where the convergence is not geometric but still manageable. This question is therefore maybe a reference request or maybe can just be answered by a simple class of examples with easily tunable convergence properties.</p>
| 0non-cybersec
| Stackexchange |
A summary of pewds Detroit game play. | 0non-cybersec
| Reddit |
In the SHA hash algorithm, why is the message always padded?. <p>In the SHA hash algorithm the message is always padded, even if initially the correct length without padding; the padding is of the form "1" followed by the necessary number of 0s.</p>
<p>Why is it necessary that the message always be padded?</p>
| 0non-cybersec
| Stackexchange |
My eyebrows finally grew back in after shaving the ends :') plus eyeliner !! Ccw :-). | 0non-cybersec
| Reddit |
Mission failed. | 0non-cybersec
| Reddit |
Verify gpg key from stdin. <p>I'm using these commands in a <code>Dockerfile</code> to add the <a href="https://apt.llvm.org/" rel="nofollow noreferrer">LLVM Ubuntu package repository</a>:</p>
<pre><code>RUN echo deb http://apt.llvm.org/artful/ llvm-toolchain-artful-6.0 main > \
/etc/apt/sources.list.d/llvm.list && \
wget -O - https://apt.llvm.org/llvm-snapshot.gpg.key | apt-key add -
</code></pre>
<p>This will add the repository and register the key. However, I'd also like to verify the key using the fingerprint given on the website. How can I extend this command to verify the key?</p>
| 0non-cybersec
| Stackexchange |
Whale Shark Discovers Hole in Net. | 0non-cybersec
| Reddit |
reddit's now running on Cassandra. | 0non-cybersec
| Reddit |
If there was a way to make sure that no innocent person is ever executed, would you be for or against the death penalty? Why?. | 0non-cybersec
| Reddit |
Realm: Map JSON to Realm-Objects with Alamofire. <p>I would like to use Realm and Alamofire to map JSON to my database objects. Are there good tutorials out there?</p>
| 0non-cybersec
| Stackexchange |
China in pole position for 5G era with a third of key patents. | 0non-cybersec
| Reddit |
Reclaimed Bowling Alley Table (inspired by neonDion). | 0non-cybersec
| Reddit |
Failed usb port. <p>What can I do to revive a failed usb port
I have an older pavilion desktop that only has the ports on front working on the front of the tower. None of the rear USB port function. </p>
| 0non-cybersec
| Stackexchange |
Warning : configuration file not found. <p>I just made a bootable usb drive with kali linux 2016.1 (rolling) with <code>dd</code> command. After booting in the usb drive it gives me this thing : <code>Warning : configuration file not found</code>. I can select configuration file from <code>/EFI/boot</code>.
I looked in my usb drive with gparted and i saw an unknown partition + fat16 partiton(EFI).</p>
<p>P.S I have uefi bios.</p>
| 0non-cybersec
| Stackexchange |
Wife confessed that she made a [sextape] with 4 guys in college, and I really wanna see it.. I knew she was promiscuous in college, and I'm fine with it. In fact that's how we met. I was one of her many fuck buddies in her college years, and I just happened to be the only one she kept in touch with after she graduated.
After a few glasses of wine last night, my wife confessed to me that she made a sextape with a group of frat boys during her junior year in college. She said she doesn't have a copy of it, since it was hard to copy VHS back in the day, and she doesn't keep in touch with any of those guys. I am sincerely not jealous at all. I just really want to watch it.
Would any of you watch your S.O. in a sex tape?
edit:
Thank you to everyone for your feedback. I guess I should have mentioned a few things to clarify the situation a little more. Yes, my wife has been promiscuous in her past, and she doesn't regret any of it. She was not too drunk when she participated in the group sex, and she remembers that she was very turned on during the whole thing. However, she says those days are over for her, and I believe her 100%. I don't mind that she's had many partners, because we're both disease free, and what she did with her time before committing to our marriage is okay by me. I probably would not like to watch my present day wife bang someone else. She still sexy don't get me wrong, but I don't want to share haha. However, I still want to see that video of her in the 5-some, because I think it's pretty hot, and I won't feel like our relationship will be threatened by her running off with them or something since it was so far in the past. | 0non-cybersec
| Reddit |
Not able to boot into ubuntu after reinstalling it. <p>On my system with ssd+hdd, there is windows on ssd and ubuntu on a partion in hdd. I tried installing ubuntu again as blutooth and some others issues are present. Installation is happening properly(boot loader - /dev/sda, root - /dev/sda5, home - /dev/sda2( home folder for previous ubuntu )). But while booting in grub it is showing previous ubuntu (multiple kernels - i tried installing latest kernel)only, and not able to boot.</p>
<p>Here is a link of error report from boot repair.
<a href="http://paste.ubuntu.com/p/Jq2mGRnYZJ/" rel="nofollow noreferrer">http://paste.ubuntu.com/p/Jq2mGRnYZJ/</a></p>
<p>Can you please help me to rectify this?</p>
| 0non-cybersec
| Stackexchange |
Safety and Rescue Boats in UK. | 0non-cybersec
| Reddit |
NSFetchedResultsController v.s. UILocalizedIndexedCollation. <p>I am trying to use a FRC with mixed language data and want to have a section index.</p>
<p>It seems like from the documentation you should be able to override the FRC's</p>
<pre><code>- (NSString *)sectionIndexTitleForSectionName:(NSString *)sectionName
- (NSArray *)sectionIndexTitles
</code></pre>
<p>and then use the UILocalizedIndexedCollation to have a localized index and sections. But sadly this does not work and is not what is intended to be used :(</p>
<p>Has anyone been able to use a FRC with UILocalizedIndexedCollation or are we forced to use the manual sorting method mentioned in the example UITableView + UILocalizedIndexedCollation example (example code included where I got this working).</p>
<p>Using the following properties</p>
<pre><code>@property (nonatomic, assign) UILocalizedIndexedCollation *collation;
@property (nonatomic, assign) NSMutableArray *collatedSections;
</code></pre>
<p>and the code:</p>
<pre><code>- (UILocalizedIndexedCollation *)collation
{
if(collation == nil)
{
collation = [UILocalizedIndexedCollation currentCollation];
}
return collation;
}
- (NSArray *)collatedSections
{
if(_collatedSections == nil)
{
int sectionTitlesCount = [[self.collation sectionTitles] count];
NSMutableArray *newSectionsArray = [[NSMutableArray alloc] initWithCapacity:sectionTitlesCount];
collatedSections = newSectionsArray;
NSMutableArray *sectionsCArray[sectionTitlesCount];
// Set up the sections array: elements are mutable arrays that will contain the time zones for that section.
for(int index = 0; index < sectionTitlesCount; index++)
{
NSMutableArray *array = [[NSMutableArray alloc] init];
[newSectionsArray addObject:array];
sectionsCArray[index] = array;
[array release];
}
for(NSManagedObject *call in self.fetchedResultsController.fetchedObjects)
{
int section = [collation sectionForObject:call collationStringSelector:NSSelectorFromString(name)];
[sectionsCArray[section] addObject:call];
}
NSArray *sortDescriptors = self.fetchedResultsController.fetchRequest.sortDescriptors;
for(int index = 0; index < sectionTitlesCount; index++)
{
[newSectionsArray replaceObjectAtIndex:index withObject:[sectionsCArray[index] sortedArrayUsingDescriptors:sortDescriptors]];
}
}
return [[collatedSections retain] autorelease];
}
- (NSInteger)numberOfSectionsInTableView:(UITableView *)tableView
{
// The number of sections is the same as the number of titles in the collation.
return [[self.collation sectionTitles] count];
}
- (NSInteger)tableView:(UITableView *)tableView numberOfRowsInSection:(NSInteger)section
{
// The number of time zones in the section is the count of the array associated with the section in the sections array.
return [[self.collatedSections objectAtIndex:section] count];
}
- (NSString *)tableView:(UITableView *)tableView titleForHeaderInSection:(NSInteger)section
{
if([[self.collatedSections objectAtIndex:section] count])
return [[self.collation sectionTitles] objectAtIndex:section];
return nil;
}
- (NSArray *)sectionIndexTitlesForTableView:(UITableView *)tableView {
return [self.collation sectionIndexTitles];
}
- (NSInteger)tableView:(UITableView *)tableView sectionForSectionIndexTitle:(NSString *)title atIndex:(NSInteger)index {
return [self.collation sectionForSectionIndexTitleAtIndex:index];
}
</code></pre>
<p>I would love to still be able to use the FRCDelegate protocol to be notified of updates. It seems like there is no good way making these two objects work together nicely.</p>
| 0non-cybersec
| Stackexchange |
How to reformat a USB drive with "no file system"?. <p>I have a flash drive I got as some sort of promotion and just for the heck of it I tried to erase it and use it for my purposes. However there is nothing I could do to reformer, erase, zero out, etc this drive. Diskutil info says: </p>
<pre><code> Device Identifier: disk2
Device Node: /dev/disk2
Part of Whole: disk2
Device / Media Name: SMI USB DISK Media
Volume Name: Not applicable (no file system)
Mounted: Not applicable (no file system)
File System: None
Content (IOContent): FDisk_partition_scheme
OS Can Be Installed: No
Media Type: Generic
Protocol: USB
SMART Status: Not Supported
Total Size: 1.0 GB (1018167296 Bytes) (exactly 1988608 512-Byte-Units)
Volume Free Space: Not applicable (no file system)
Device Block Size: 512 Bytes
Read-Only Media: Yes
Read-Only Volume: Not applicable (no file system)
Ejectable: Yes
Whole: Yes
Internal: No
OS 9 Drivers: No
Low Level Format: Not supported
</code></pre>
<p>I am doing this on a Mac with OSX 10.9 and it shows up like a perfectly normal file system, folders, hidden files etc. </p>
<p>Is there anything I can do with this drive except throwing it away?
Thanks.</p>
| 0non-cybersec
| Stackexchange |
Do you know why I have this warning: underfull \hbox(\badness 10000), using visual studio code with Latex extension. <pre><code>\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[]{amsthm} %lets us use \begin{proof}
\usepackage[]{amssymb} %gives us the character \varnothing
\usepackage[fleqn]{amsmath}
\usepackage{amsfonts}
\begin{document}
\maketitle
\subsection*{2. P18. Exercise 3:}
Assume we have a fair eight-sided die with the numbers 1, 2, 3, 3, 4, 5, 5, 5 on its sides.
What is the probability that each of the numbers 1 through 5 will be rolled?
If we roll two of these dice, what is the range of possible totals of the values showing
on the two dice? What is the chance that each of the numbers in this range will be rolled?\\
</code></pre>
| 0non-cybersec
| Stackexchange |
Elon Musk Predicts How the Martian Government Will Operate: “Most likely, the form of government on Mars would be something of a direct democracy […] where people vote directly on issues instead of going through representative government.”. | 0non-cybersec
| Reddit |
Storing Json Object in Mongoose String key. <p>In my Mongoose schema, I have a field which is a String, and I want to be able to store an JSON object in it. Is it possible? In Postgres, it's possible to store a dictionary in a string column. </p>
<p>I want to do that because the dictionary (actually a JSON object in JS) is just a simple read and write value and doesn't need queries, but also, because it is just one value and not an array of values.</p>
| 0non-cybersec
| Stackexchange |
How to use <> or != operator. <p>Here is my query: </p>
<pre><code>select c._id, c.cat_name
from category c
inner join deptcat dc
on c._id != dc.cat_id
inner join department d
on dc.dept_id != d._id
where d._id = 1
</code></pre>
<p>I have 3 tables <code>department</code>, <code>category</code> and <code>deptcat</code>:</p>
<pre><code>CREATE TABLE department
(_id, dept_name)
AS
VALUES
( 1, 'CSE' ),
( 2, 'E&E' ) ;
CREATE TABLE category
(_id, cat_name)
AS
VALUES
( 1, 'a' ),
( 2, 'b' ),
( 3, 'c' ),
( 4, 'd' ) ;
CREATE TABLE deptcat
(_id, dept_id, cat_id)
AS
VALUES
( 1, 1, 1 ),
( 2, 1, 2 ) ;
</code></pre>
<p>My output should be like this:</p>
<pre><code>_id | cat_name
----|-----------
3 | c
4 | d
</code></pre>
<p>I need to fetch the <code>cat_id</code>'s which is not present in the <code>deptcat</code> table and also associated with <code>dept_id 1</code>, so can some one help me here?</p>
| 0non-cybersec
| Stackexchange |
They'd rather throw their water bowl all over my kitchen and drink my shower water out of the tub... | 0non-cybersec
| Reddit |
Trolling the U.S.: Q&A on Russian Interference in the 2016 Presidential Election. | 0non-cybersec
| Reddit |
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application.
I need to scale my app adding some spot instances that EB do not support.</p>
<p>So I create a second autoscaling from a launch configuration with spot instances.
The autoscaling use the same load balancer created by beanstalk.</p>
<p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p>
<p>This work fine, but:</p>
<ol>
<li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p>
</li>
<li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p>
</li>
</ol>
<p><strong>UPDATE</strong></p>
<p>Elastic Beanstalk add support to spot instance since 2019... see:
<a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>
| 0non-cybersec
| Stackexchange |
What's so special about these $17$th deg equations?. <p>While browsing the <strong><em>Database of Number Fields</em></strong>, I came across <strong><em><a href="http://galoisdb.math.upb.de/search/display?req=p413">17T8</a></em></strong>. It only had four equations, one of which is,</p>
<p>$$\small{x^{17} - 5x^{16} + 40x^{15} - 140x^{14} + 610x^{13} - 1622x^{12} + 4870x^{11} - 10220x^{10} + 22720x^9 - 38080x^8 + 63500x^7 - 84100x^6 + 102200x^5 - 102400x^4 + 83000x^3 \color{brown}{- 55864x^2 + 24080x - 9400}=0}$$</p>
<p>It may not look much, but all four examples had the same coefficients <em>except</em> for the part in brown (the $x^2, x^1, x^0$ terms), so I assume there might be a parameterization. After some fiddling, I found the rather simple,</p>
<p>$$\frac{(x^5 - 2x^4 + 10x^3 - 10x^2 + 20x - 10)^3\,(x^2 + x + 4)}{(4x^2 - 5x + 25)} = -36m\tag1$$</p>
<p>The four were just the cases $m = -6, -2, -12, 9$. The discriminant of $(1)$ is,</p>
<p>$$D = 2^{36}\, 3^{52}\, 5^{18}\, m^{10}(16m + 81)^8$$</p>
<p><strong><em>Update:</em></strong> Note that,</p>
<p>$$\frac{(x^5 - 2x^4 + 10x^3 - 10x^2 + 20x - 10)^\color{red}3\,(x^2 + x + 4)}{4x^2 - 5x + 25}-\frac{27^2}{2^2} = \frac{(x-1)(2x^8 - 4x^7 + 32x^6 - 40x^5 + 170x^4 - 136x^3 + 362x^2 - 166x + 185)^\color{red}2}{4x^2 - 5x + 25}\tag2$$</p>
<p>Note that the <em>quintic</em> and <em>octic</em> factors (both irreducible) have <em>solvable</em> Galois groups. Also, I've seen similar factoring behavior in formulas for the <em>j-function</em> like the well-known <em>icosahedral equation</em>,</p>
<p>$$j(\tau)-1728 =-\frac{(r^{20} - 228r^{15} + 494r^{10} + 228r^5 + 1)^\color{red}3}{r^5(r^{10} + 11r^5 - 1)^5}-12^3 = -\frac{(r^{30} + 522r^{25} - 10005r^{20} - 10005r^{10} - 522r^5 + 1)^\color{red}2}{r^5(r^{10} + 11r^5 - 1)^5}\tag3$$</p>
<p>Makes me wonder if $(1)$ is a formula for something. </p>
<p><strong>Questions:</strong></p>
<ol>
<li>Does the whole family, except for special $m$, belong to <strong><em>17T8</em></strong>?</li>
<li>Can one derive it from first principles, instead of a computer search? (Its simple form seem to suggest there might be others.)</li>
</ol>
| 0non-cybersec
| Stackexchange |
"Next Song" function key (Mac) on Ubuntu. <p>I have installed Ubuntu 12.04 on my iMac 2011. I want to use the function keys on the keyboard to skip to the next song, but I get this splash when I press the key:</p>
<p><img src="https://i.stack.imgur.com/IthHx.png" alt="enter image description here"></p>
<p>Can you help me?</p>
| 0non-cybersec
| Stackexchange |
Proof Verification for Baby Rudin's Chapter 4 Exercise 4. <p>I'm trying to prove:</p>
<blockquote>
<p><span class="math-container">$f: X \to Y$</span> is continuous, <span class="math-container">$g: X \to Y$</span> is continuous, <span class="math-container">$E \subset X$</span> is dense in <span class="math-container">$X \implies$</span> <span class="math-container">$f(E)$</span> is dense in <span class="math-container">$f(X)$</span> and <span class="math-container">$\forall p \in E, g(p) = f(p) \implies g(p) = f(p) \forall p \in X$</span>.</p>
</blockquote>
<p><strong>My attempt:</strong>
First, we show that <span class="math-container">$f(E)$</span> is dense in <span class="math-container">$f(X)$</span>. Note that it might be that some members of <span class="math-container">$f(X)$</span> also belong to <span class="math-container">$f(E)$</span> which means that we only need to show that those members of <span class="math-container">$f(X)$</span> that do not additionally belong to <span class="math-container">$f(E)$</span> are limits points of <span class="math-container">$f(E)$</span>. To this end, let <span class="math-container">$y \in f(X)$</span> such that <span class="math-container">$y \notin f(E)$</span>. Then, <span class="math-container">$y = f(p)$</span> for some <span class="math-container">$p \in X \setminus E$</span>. Since <span class="math-container">$E$</span> is dense in <span class="math-container">$X$</span>, <span class="math-container">$p$</span> is a limit point of <span class="math-container">$E$</span>. Since <span class="math-container">$f$</span> is continuous at <span class="math-container">$p$</span>, for any <span class="math-container">$x \in E$</span> satisfying <span class="math-container">$x \to p$</span>, we have that <span class="math-container">$f(x) \to f(p) = y$</span>. Since <span class="math-container">$y$</span> was arbitrarily chosen, <span class="math-container">$f(E)$</span> is dense in <span class="math-container">$f(X)$</span>.</p>
<p><strong>My question</strong>: Is my proof for showing that "<span class="math-container">$f(E)$</span> is dense in <span class="math-container">$f(X)$</span>" inaccurate in any way?</p>
| 0non-cybersec
| Stackexchange |
Logging detailed network activity on Windows Server 2008. <p>One of my clients is experiencing internet delays and outages within their small network. They've verified with their cable company that the issue is not related to any cable company outages. The concern has been raised that either there's too much network activity for their one server to handle, or their website might be getting attacked. Due to this, we'd like to start logging their main server's network activity.</p>
<p>This server has the following responsibilities:</p>
<ul>
<li>DNS </li>
<li>Web Server hosting a public site</li>
<li>Hosts a WCF service that their "admin" desktop application <em>heavily</em> interacts with </li>
<li>Runs SQL Server 2008, which the WCF server and public website interact with</li>
</ul>
<p><strong>Goal:</strong>
Ideally, I'd like to log network bandwidth used by the processes, as I feel this would give a good handle on what services/software is eating too much bandwidth. I started looking at the native Windows Performance Monitor tool, but I can't figure out how to log the data to a file, and I'm wondering if there's a better [free] logging tool that will give details about how much bandwidth a given process is using.</p>
<p><strong>Questions:</strong> </p>
<ol>
<li><p>How do I log the activity seen from the Perfomance Monitor Tool (if possible)</p></li>
<li><p>Is there a better logging tool that will log how much bandwidth a given process is using?</p></li>
</ol>
<p><img src="https://i.stack.imgur.com/hbjjE.png" alt="enter image description here"></p>
| 0non-cybersec
| Stackexchange |
SQL Server in Azure VM + Azure AD + SSIS - how to authenticate?. <p>Our configuration: </p>
<ul>
<li>Windows Server 2016 VM in Azure, running SQL Server (same as on-prem version)</li>
<li>Azure Active Directory with users etc.</li>
<li>Local Windows 10 PCs are joined to the AAD domain</li>
<li>SQL Server is configured for mixed-mode auth.</li>
<li>Azure firewall allows connections to the SQL Server from the Internet (for now at least). </li>
</ul>
<p>One of our developers is trying to setup an SSIS package to run in the server, and is running into the error stating you can only deploy/run SSIS packages from SSMS if you are using Windows authentication. I therefore am now looking into setting up "proper" authentication.</p>
<p>I've looked into Azure AD Domain Services and I can set that up to allow the Windows Server VM to join the domain. </p>
<p>My questions:</p>
<ul>
<li>If I join the Windows Server VM to Azure AD using Azure AD DS, will the Windows authentication work from <em>client</em> PCs, out on the Internet, that are joined to the Azure AD via Windows 10's "Azure AD Join" feature? I noticed that in SSMS, the username is listed as "AzureAD\". </li>
<li>If I need to add more infrastructure, I'm obviously looking to minimize cost, so the options I see are as follows - which one of these makes the most sense/actually works?:
<ul>
<li>Deploy a local Windows Server on prem. This is at least a $500 cost (Server Essentials) and would restrict people from only being able to log in while on prem, unless I also setup a VPN service. It also excludes anyone from working on a home machine or basically any machine not joined to the domain - I think?</li>
<li>Not sure if this is possible, but, setup a site-to-site VPN with the Azure VNet containing Azure AD DS, and then use a Linux box on prem (free of software cost, could be a very minimal mini-box or even an ARM-based board) to connect local to the Azure VNet. Then, I should be able to theoretically just join all my machines to Azure AD DS as if I had a local server? This would also potentially enable us to restrict access to the SQL Server and just have everyone use the same VPN to connect to Azure and get to the SQL Server. Still doesn't really help people using non-joined machines...</li>
<li>Forget AD DS entirely. Not a good idea since we also have Office365 licensing already set up for everyone along with Email addresses, which integrates nicely with the Azure AD. </li>
</ul></li>
</ul>
<p>I've done a lot of work on local Windows Server infrastructure but am still new to Azure and how all of it fits together. Again, the ultimate goal here is to allow SQL Server devs and admins to use Windows Authentication against an SQL Server running in an Azure VM, with the directory being an Azure AD directory.</p>
| 0non-cybersec
| Stackexchange |
Do you know this girl? Yankee Swap. | 0non-cybersec
| Reddit |
Dealing with Grief. Last night, my 6 year old daughter and I had to put down one of our cats. This was essentially my cat (its brother is considered her cat). It was very sudden and she was hysterical on the way to the hospital and in the waiting room. My sister picked her up before I made the decision and took her home (LO got to gorge herself on ice cream and stay up late- lucky). I made the decision and went home alone.
My question is, I'm obviously distraught. Been having crying fits on and off for the last 12 hours (yeah, its only been 12 hours). LO stayed home from school and is with her Nana and Papa. Because I am distraught and grieving, I dont know how to help her grieve. Nana is helping her work through the Rainbow Bridge workbook and talking with her. (We told her that Monkey went to play with Bullet - our cat we put down 5 years ago).
How do I help her grieve while grieving myself? Do I hide my feelings or should we cry together? I dont know what to do. This is my first time going through this with her where she is at an age that she somewhat understands "forever".
Thank you. | 0non-cybersec
| Reddit |
On a non-standard approach to the classification of conics?. <p>I've been introduced to a method of classifying conics but it's too cumbersome for me. I've discovered something that seems a little more promising on Eves' <em>Elementary Matrix Theory</em>:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/URAge.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/URAge.png" alt="enter image description here"></a></p>
</blockquote>
<p>And then,taking the rank $\rho$ of the matrix:</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/z14fW.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/z14fW.png" alt="enter image description here"></a></p>
</blockquote>
<p>And then, using also the discriminant of the conic (from Aarts' <em>Plane and Solid Geometry</em>):</p>
<blockquote>
<p><a href="https://i.stack.imgur.com/SJVhV.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/SJVhV.png" alt="enter image description here"></a></p>
</blockquote>
<p>But I'm a little unsafe about using them. I mean, what could possibly go wrong against the usual approach of using rotations and translations? What would I lose by using this approach?</p>
<p>From what I know, the only problem that could happen is when the conic is an imaginary ellipse or hyperbola.</p>
| 0non-cybersec
| Stackexchange |
Can git be installed locally (including its dependencies). <p>I've developed a <a href="/questions/tagged/bash" class="post-tag" title="show questions tagged 'bash'" rel="tag">bash</a> script which needs a specific version of <a href="/questions/tagged/git" class="post-tag" title="show questions tagged 'git'" rel="tag">git</a> (2.11.1).</p>
<p>Each developer in my team has his/her own virtual machine on the same remote physic server.
My colleagues need that script, the problem is that some of them are using the shared <a href="/questions/tagged/git" class="post-tag" title="show questions tagged 'git'" rel="tag">git</a> which is outdated and there is no way to update it, and the others are using their own installation <code>/usr/bin/git</code>, which is not always up-to-date either.</p>
<p>In order for that script to work, the best solution I decided to work on is that before the script runs, it will install the wanted git LOCALLY (I mean under <code>~/custom/empty_dir</code>), but this didn't work as I'm getting many errors when running <code>make all doc info</code> as described in <a href="https://git-scm.com/book/en/v2/Getting-Started-Installing-Git#_installing_from_source" rel="nofollow noreferrer">git-scm</a></p>
<pre><code>$make all doc info
* new build flags
CC credential-store.o
In file included from credential-store.c:1:0:
cache.h:40:18: fatal error: zlib.h: No such file or directory
#include <zlib.h>
^
compilation terminated.
make: *** [credential-store.o] Error 1
</code></pre>
<p>Isn't it possible to install <a href="/questions/tagged/git" class="post-tag" title="show questions tagged 'git'" rel="tag">git</a> locally as explained above without of course impacting <code>/usr/bin/git</code> and <code>/company/shared/softwares/git/bin/git</code> ?</p>
<blockquote>
<p>ps: PRETTY_NAME="SUSE Linux Enterprise Server 11 SP4"</p>
</blockquote>
| 0non-cybersec
| Stackexchange |
Metroid Prime on the Vive and Rift using Dolphin emulator.. | 0non-cybersec
| Reddit |
Clojure: how to execute shell commands with piping?. <p>I found <code>(use '[clojure.java.shell :only [sh]])</code> for executing shell commands with clojure. Now, while <code>(sh "ls" "-a")</code> does the job, <code>(sh "ls" "-a" "| grep" "Doc")</code> doesn't. What's the trick?</p>
| 0non-cybersec
| Stackexchange |
Trump asks Saudi Arabia to increase oil production. | 0non-cybersec
| Reddit |
how to initialize a QString to null?. <p>What is the difference between <code>QString::number(0)</code> and <code>((const char*) 0)</code>?</p>
<p>I want to initialize a <code>QString</code> say <code>phoneNumber</code> to <code>null</code>. Will <code>phoneNumber(QString::number(0))</code> and <code>phoneNumber((const char*) 0)</code> both work?</p>
| 0non-cybersec
| Stackexchange |
This car covered in pennies. | 0non-cybersec
| Reddit |
which exception to throw if list is empty in java?. <p>I have the following doubt concerning which exception to throw if list is empty</p>
<pre><code>public class XYZ implements Runnable {
private List<File> contractFileList;
@Override
public void run() {
contractFileList = some method that will return the list;
//now i want to check if returned contractFile is empty or not , if yes then raise the exception
if (contractFileList.isEmpty()) {
// throw new ?????
}
}
}
</code></pre>
<p>I am runing this code inside a batch, I want to throw some exception that will stop the batch execution.</p>
| 0non-cybersec
| Stackexchange |
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