Quillen-Segal algebras and Stable homotopy theory
Hugo Bacard

Magellan Adaptive Optics Imaging of PDS 70: Measuring the Mass Accretion Rate of a Young Giant Planet within a Gapped Disk
    Kevin Wagner1,2,3,^⋆, Katherine B. Follette4, Laird M. Close1, Dániel Apai1,3,5,6, Aidan Gibbs1, Miriam Keppler6, André Müller6, Thomas Henning6, Markus Kasper7, Ya-Lin Wu8, Joseph Long1, Jared Males1, Katie Morzinski1, and Melissa McClure9
    April 14, 2020
====================================================================================================================================================================================================================================================


Let    be a monoidal model category that is also combinatorial and left proper. If Ø is a monad, operad, properad, or a PROP; following Segal's ideas we develop a theory of Quillen-Segal Ø-algebras and show that we have a Quillen equivalence between usual Ø-algebras and Quillen-Segal algebras. We use this theory to get the stable homotopy category by a similar method as Hovey. 






§ INTRODUCTION

This paper is part of a project that aims to develop a homotopy theory of  weak algebraic structures encoded by objects such as operads, properads, PROP's or monads. And we hope that this theory can be useful to understand in some way the algebra of higher categories and its applications.  The notion of symmetric operad was introduced by May <cit.> as a solution to the delooping problem in stable homotopy theory.  Segal  <cit.>  gave a solution to the same problem that is of simplicial nature with the notion of Γ-spaces.  Both solutions have inspired different directions in the development of Higher Category Theory, and we shall refer the reader to the book of Simpson <cit.> for a detailed account on the subject. 



We choose  a general formalism that we hope, will be somehow “a bridge” between the operadic and the simplicial method for the delooping problem.  Let  be a model category and let () be its category of morphisms (or arrows). For n ∈, define inductively ^n+1()= (^n()), the category of hyper-cubes in ; with ^0()=. In order to simplify the treatment of the homotopy theory in  “Segal situations” we start with the following definition. 

Let  be an arbitrary category. 


  * An -valued Quillen-Segal theory  on  is a family of functors called Segal data functors: {_i: →() }_i ∈ I,
for some small set I.

  * Say that an object c ∈ satisfies the generalized Segal conditions, if for every i ∈ I _i(c) ∈() is a weak equivalence in . 

  * Inductively, an  n-fold -valued Quillen-Segal theory  on  is a family of functors, called Segal n-data functors: {_i: →^n() }_i ∈ I,
for some small set I.



In the original paper of Segal <cit.>,  is the category of Γ-spaces,  = and the theory is given by the family functors {_n: →() }_n ≥ 1, where _n() is the nth Segal map: p_n: (n) →(1) ×_(0)⋯×_(0)(1).  The same formula defines the theory for Segal spaces  as in Rezk <cit.>, and classical Segal categories (see <cit.>, <cit.>,<cit.>, <cit.>). These theory have been extended to Segal n-categories and Segal enriched -categories (see <cit.>, <cit.>). One can iterate the process à la Simpson-Tamsamani to consider a -valued theory on a category : {_j':→()}_j∈ J  to get a 2-fold -valued theory {(_i) ∘_j':→^2()}_(i,j) ∈ I× J and so on.
 

There are many categories that are equipped with a relevant Quillen-Segal theory, and some of them will be reviewed later. But in this paper we want to extend Segal's formalism to the following situations.


  * Let = Ω-Spec be the category of Ω-prespectrum with the theory given by the functors _n: →(_∗) that takes a prespectrum X to the connecting morphism X_n →Ω (X_n+1) which is adjoint to the map S^1 ∧ X_n → X_n+1. Then a fibrant prespectrum satisfying the Segal conditions is simply an Ω-spectrum, thus a generalized cohomology theory. 

  * Let = be  the comma category associated to a (right Quillen) functor : →. We remind the reader  that an object of (↓) is a triple  = [_0, _1, π_: _0 →(_1)] ∈××(). The theory is given by the functor that projects the maps :
 
    : →(),  with    () = .



If Ø is an operad, monad, or a PROP; a Quillen-Segal Ø-algebra is an object ∈=, that satisfies the Segal condition for the forgetful functor : () →.
The first example of Quillen-Segal algebras comes from  co-Segal algebras and their generalizations to co-Segal categories <cit.>.
  

Before discussing further the theory of Quillen-Segal algebras we have the following theorem on Quillen-Segal theories in general (see Theorem <ref>)

Let  be a cofibrantly generated model category and let =(_i) be a Quillen-Segal theory on .
Assume that one the following conditions holds.



  *  has a model structure (,,) which is cellular and left proper such that every _i ∈(, _inj) is right Quillen.

  *  has a model structure (,,) which is combinatorial and left proper such that every _i ∈(, _inj) is right Quillen.

Then there is a new model structure _B()=(_B,_B,_B) on  which is a left Bousfield localization, such that:


  * ={_i: →}_i ∈ I defines an Quillen-Segal theory such that every _i is also right Quillen with respect to the model structure _B();

  * every fibrant object in this new model structure satisfies the generalized Segal conditions.



Here _inj is the injective model structure that can be found for example in Hovey <cit.>. There is also a projective model structure on  denoted by _proj. We will review them later in a more general context. 
We also have the following theorem if the cofibrations in  are generated by a set of maps between cofibrant objects. Example of such model categories are tractable model categories in the sense of Barwick <cit.> . The following theorem is valid outside tractable model categories (see Theorem <ref>).

Let  be a tractable model category and let =(_i) be a Quillen-Segal theory on .
Assume that one the following conditions holds.



  *  has a model structure (,,) which is cellular and left proper such that every _i ∈(, _proj) is right Quillen.

  *  has a model structure (,,) which is combinatorial and left proper such that every _i ∈(, _proj) is right Quillen.

Then there is a new model structure _B()=(_B,_B,_B) on  which is a left Bousfield localization, such that:


  * ={_i: →}_i ∈ I defines an Quillen-Segal theory such that every _i is also right Quillen with respect to the model structure _B();

  * every fibrant object in this new model structure satisfies the generalized Segal conditions.




These theorems require the left properness of the model structure on  . But it's possible to have a localization of a theory without left properness using a result of Beke <cit.> which is itself a consequence of a theorem of Jeff Smith. This is what we do for the comma category := and in particular for Quillen-Segal algebras for a combinatorial model category . 

First we prove that:


  * There is an injective model structure on the comma category := (Theorem <ref>)

  * There is a projective model structure on := (Theorem <ref>).

  * If = we recover the injective and projective model structure on .
 

Category theory provides an embedding ι: → with ι()=[(), , _()] (see Proposition <ref>) ; and this is how usual strict algebras are QS-algebras (_() is always a weak equivalence). We have a functor :→ that is simultaneously a left adjoint and a retraction for ι. The object _1 is a usual Ø-algebra and with the weak equivalence :_0 →(_1), one may want to lift the Ø-algebra structure to _0. This is a classical problem of homotopy invariance for algebras that goes back to Boardman and Vogt <cit.>, Dwyer, Kan and Smith <cit.> and others. There are many results in this direction that can be found for example in Berger-Moerdijk <cit.>, Johnson-Yau <cit.> and the many references therein. We will discuss it in a future work. 

 

One of the central results on the homotopy theory on  is that we can localize directly the injective and projective model structure along the functor : →. 

Let :⇆: be Quillen adjunction between combinatorial model categories where  is right adjoint.   
Then the following hold.


  * There is a model structure on the category = such that:


  * fibrant objects satisfy the Segal condition and 

  * we have a   a Quillen equivalence ι: ⇆:, where ι is right Quillen and ()=_1.


  * If the Quillen pair :⇆: is a Quillen equivalence then the functor 
    : →
 is a right Quillen equivalence and  is a composite of Quillen equivalences:

    ι




If  is tractable we can show that this localization is in fact a left Bousfield localization. The proof is just a consequence of Ken Brown's lemma. One gets the homotopy theory of Quillen-Segal Ø-algebras when  is the forgetful functor. 


 
The relation with Stable Homotopy Theory comes when = and we consider  the product ∏_n ∈ for an endofunctor : → such as the loop space functor Ω: _∗→_∗. This product is in fact a functor category (, ) where  is the set of integers regarded as a discrete category and not as posetal category.  The category , of -prespectrum, is equivalent to a category of objects in (, ) that are linked in the sense of Definition <ref>. This link condition is just a 2-pullback condition, that can be considered as a descent condition in some sense. 

Our goal here is to have a picture where the operations on spectra are seen in the diagram category (, ). We take the benefit of the existing results on the homotopy theory and the algebra of diagram categories that have been studied for decades.  For example if we index over  the strict projective model structure that goes back to Bousfield and Friedlander <cit.>, follows directly from the projective (=Reedy) model structure on (↓)= in Theorem <ref>. 



The functor P: →(, ) that forgets the link has a left adjoint (Proposition <ref>); and we use this (monadic) adjunction  to get various  model structure on the category of -prespectrum. If  is locally presentable, there are 4 model structures on  that follow from the injective and projective model structures on (, _inj) and (, _proj). All of them are combinatorial and left proper if  is left proper. In this paper we restrict to the case where  is combinatorial and left proper to keep the paper short. We will discuss in a different paper the case where  is cellular. The arguments remain essentially the same. 

We show in virtue of Theorem <ref> that:

Let  be the category of prespectrum in simplicial sets  with the  Quillen-Segal theory _n: →(_∗) given  _n(X)= [X_n →Ω(X_n+1)]. 


  * Then there is a model structure on  such that the fibrant objects are the Ω-spectrum X  that are level fibrant (Kan).

  * The homotopy category is equivalent to the stable homotopy category of Bousfield-Friedlander obtained by Hovey <cit.> and Schwede <cit.>.



Finally we set up a definition of generalized -chain complexes for future applications as linked -sequences that satisfy the chain condition.  It's not new that chain complexes and spectra behave similarly.



 §.§ Applications
 
There are various applications of the theory that is being developed here. We outline for the moment some  directions of interest. They will appear in the subsequent papers. 


  * Our interest in comma categories comes when we consider a right Quillen  : → that is a  Quillen equivalence. An important example is the homotopy coherent nerve of Cordier and Porter <cit.> from simplicial categories to quasicategories.  Joyal showed that this right Quillen functor is in fact a Quillen equivalence (see Bergner  <cit.>). Our theorem shows that the comma category that lives in between is Quillen equivalent to both simplicial categories and quasicategories. On the one hand, Lurie and Joyal gave a significant amount of work on quasicategories. And on the other hand, simplicial categories are -enriched categories, and there is a lot in the literature on the subject. A classical reference is the book of Kelly <cit.>. We would like to understand how the existing constructions for enriched categories such as weighted limits, colimits, adjunction, Cauchy completion, etc; interact with the corresponding notions introduced by Joyal and Lurie.



  * Following the ideas of Jardine <cit.>, Hirshowitz-Simpson <cit.>, Morel-Voevodsky <cit.>, Toën-Vezzozi <cit.> and others the study of presheaves on a Grothendieck site with coefficient in (, ) has application in motivic homotopy theory. Indeed we can attain presheaves of spectra this way.

  * The theory of operads, properads, PROP's and algebras over them has applications in several fields of mathematics (see for example Loday-Vallette <cit.>, Markl-Shnider-Stasheff <cit.>). Many of the existing construction for usual algebras can be transported to Quillen-Segal algebras. When we consider a general model category,  the problem of homotopy invariance for algebras doesn't always have a solution. It turns out that this problem is intimately related to a conjecture of Simpson about weak units in Higher categories (see Joyal-Kock <cit.>). We will address  this later.



  * One of the consequences  of this work is a strictification theorem for co-Segal algebras and co-Segal categories. This is the analogue of a theorem of Bergner <cit.> for the strictification of Segal categories. 




 


 §.§ Organization of the paper



  * Section 2 is about Quillen-Segal theories. The main result is the “Bousfield localization of the theory”.

  * We introduce Quillen-Segal algebras in Section 3.

  * Section 4 contains the category theory of comma constructions.

  * In Section 5, we develop the homotopy theory on the comma category . We follow closely the same method as Hovey <cit.> to get an injective and projective model structure. 

  * In Section 6, we set up some material on Stable Homotopy Theory.
















 §.§ Related works
 
Model structures on comma categories have been discussed  by  Stanculescu <cit.>, Toën <cit.> and others. But they consider instead the comma categories (↓), where  is a left Quillen functor. Considering other motivations, Stanculescu focused on the dual notion of what we call Quillen-Segal object, in that one would demand  (_0) →_1 to be a weak equivalence. But in the case of algebras,  we will capture the homotopy theory of free algebras.  He also mentioned that it could be interesting to study the comma category  and consider what we call Quillen-Segal object. 

The general philosophy of this paper is also close to that of Hollander <cit.>, Hirschowitz-Simpson <cit.>, Jardine <cit.>, Joyal-Tierney <cit.>,  Stanculescu <cit.> and many others. By this we mean that the theory of higher stacks is a theory of fibrant objects. And these fibrant objects satisfy some generalized Segal conditions with respect to a certain Quillen-Segal theory.


Part of this paper was done when I was a postdoctoral fellow in the research group of Rick Jardine, and I would like to thank him for his support during those years. I would like to thank David White for some helpful comments  and for referring me to a paper of  Mark Hovey that was inspirational. 




§ QUILLEN-SEGAL THEORIES

The goal of this section is to provide a general framework that is commonly used to produce a model structure on some category  such that the fibrant objects satisfy some generalized Segal conditions. 


Let  be a general model category, that will be in many cases a monoidal model category in the sense of Hovey <cit.>. We will denote by () its category of morphisms, also called the arrow category. Hovey <cit.> provides the injective and projective model structures on (). In both model structure the weak equivalences are the objective-wise weak equivalences. We will denote them by  and  and they will be reviewed in Section <ref>. The category () is a diagram category and these model structures are particular case of Reedy model structures which have been widely discussed in the literature (see  <cit.>,<cit.>,<cit.>,<cit.>).



Let  be an arbitrary category. 


  * An -valued Quillen-Segal theory  on  is a family of functors called Segal data functors: {_i: →() }_i ∈ I,
for some small set I.

  * Say that an object c ∈ satisfies the generalized Segal conditions, if for every i ∈ I _i(c) ∈() is a weak equivalence in . 

  * Inductively, an  n-fold -valued Quillen-Segal theory  on  is a family of functors, called Segal n-data functors: {_i: →^n() }_i ∈ I,
for some small set I.





The first example comes of course from Γ-spaces and Segal n-categories. We shall outline the idea for classical Segal categories. Let  be the category of simplicial sets and set I= (Δ) and let = (Δ^op, ) be the category of simplicial spaces.  For n ∈Δ, let _n: →() be the functor that takes a simplicial space  and  projects the nth Segal map: p_n: (n) →(1) ×_(0)⋯×_(0)(1).
As mentioned before, it's the same formulas for Segal spaces <cit.>. 

Leinster <cit.> introduced up-to-homotopy algebras as as colax monoidal functor : (Δ^+,+,0) → (M, ⊗ ,I) such that for every n, m ∈Δ^+ the colaxity map φ_n,m: (n+m) →(n) ⊗(m) and φ_0: () → I are weak equivalences. In this case the theory is given by the functor _n,m defined by _n,m()=φ_n,m and _0()= φ_0. We've generalized this notion to Segal enriched category <cit.>. And the homotopy theory is difficult because precisely the functors _n,m are hardly right adjoints for a general tensor products ⊗≠×. We will discuss it in <cit.>.





 §.§ Injective and projective  theory



Let  be a model category. 



  * An injective  -valued Quillen-Segal theory  on  is a family of right Quillen functors, called Segal data functors: 
    {_i: →}_i ∈ I

for some small set I.

  * A projective -valued Quillen-Segal theory on  is a family of right Quillen functors, called Segal data functors: 
    {_i: →}_i ∈ I

for some small set I.

  * We will say that there is an -valued  Quillen-Segal theory on  if there is a projective or an injective -valued Quillen-Segal theory on .



We remind the reader that part of being a right Quillen functor means that there is a left adjoint Υ_i: → to each of the Segal data functor _i. It also implies that Υ_i is automatically a left Quillen, in that it preserves the cofibrations and the trivial cofibrations in .





If  is model category we will denote by , , , the three classes of weak equivalences, cofibrations, and fibrations, respectively.





Let  be a cofibrantly generated model category and let =(_i) be a Quillen-Segal theory on .
Assume that one the following conditions holds.



  *  has a model structure (,,) which is cellular and left proper such that every _i ∈(, _inj) is right Quillen.

  *  has a model structure (,,) which is combinatorial and left proper such that every _i ∈(, _inj) is right Quillen.

Then there is a new model structure _B()=(_B,_B,_B) on  which is a left Bousfield localization, such that:


  * ={_i: →}_i ∈ I defines an Quillen-Segal theory such that every _i is also right Quillen with respect to the model structure _B();

  * every fibrant object in this new model structure satisfies the generalized Segal conditions.




We will give  the proof at the end of the section.  For a projective Quillen-Segal theory, we need to recall the definition of a tractable model category as in Barwick <cit.>.

A cofibrantly generated model category  is tractable if  is locally presentable and both cofibrations and trivial cofibrations are generated by a set of morphisms between cofibrant objects. 



Let  be a tractable  model category and let  be a category.
Assume that one the following conditions holds.



  *  has a model structure (,,) which is cellular and left proper.

  *  has a model structure (,,) which is combinatorial and left proper.

Then for any projective theory ={_i: →}_i ∈ I on  with  respect to the model structure (,,), there is a new model structure _B()=(_B,_B,_B) on  which is a left Bousfield localization such that:


  * ={_i: →}_i ∈ I defines a projective theory with respect to the model structure _B() and

  * every fibrant object in this new model structure satisfies the generalized Segal conditions.



The guiding principle in proving Theorem <ref> and Theorem <ref> is to develop some techniques that allow one to perform two tasks: the first task is to be able to detect when an object satisfies the Segal conditions; and  the second task is to have a functorial process that takes an object c and creates an object (c) that satisfies the Segal conditions. For the first task, this amounts to know when a morphism f=_i(c) ∈ is a weak equivalence in . There are three known techniques to detect this  when  is cofibrantly generated and we outline them briefly hereafter.


  * f will be a weak equivalence if we can show that f has the RLP with respect to the set of generating cofibrations  of . In that case f is in fact a trivial fibration. 

  * If f is a map between fibrant objects, f will be a weak equivalence if we can show that it satisfies the  Homotopy Extension Lifting Property (HELP) with respect to the elements of .

  * If f :X → Y, we can use function complexes  and check whether  (C,f): (C,X) →(C,Y) is a weak equivalence of simplicial sets, as C runs through the set of domain and the codomain of maps in .

With these techniques in mind, it suffices to provide a set of maps  and consider the left Bousfield localization with respect to  such  that by adjunction, being -local as in <cit.> forces our map f= _i(c) to be a weak equivalence in  through one of the above techniques. A -localization in the sense of <cit.> will allow by the small object argument of Quillen to perform the second task i.e, produce a functor  with a natural transformation →  (a fibrant replacement functor) such that (c) satisfies the generalized Segal conditions. 
 

These techniques are now standard in Homotopy theory and go back to Bousfield-Friedlander <cit.>, Jardine <cit.>, Joyal <cit.>, Kan <cit.> and others. Simpson <cit.> used the first two techniques for Segal categories where the functor  is a Segalification functor. Hovey <cit.> used the third technique for spectra, where the functor  is weakly equivalent to the Ω-spectrification as in Bousfield-Friedlander <cit.>. 



In this paper we use the first two techniques like Simpson to prove Theorem <ref> and Theorem <ref>. As a consequence we get, among other things, a generalized Segalification functor that encompasses the Ω-spectrification. Hovey's method can also be used in our context and we will explain later the connection between his approach and ours.
We will produce two sets of maps _inj and _proj in  that will be used in the small object argument to force and to detect, by adjunction, the RLP and the HELP against elements of . The set _inj is suitable for an injective Quillen-Segal theory and _proj is for  a projective Quillen-Segal theory. 










 §.§ Lifting properties and detection of the Segal conditions



  §.§.§ Homotopy theory of the arrow category

Let's review briefly the injective  and the projective model structure on the category  that can be found in Hovey <cit.>. Let =[0 → 1] be a the walking morphism category. Then () is the functor category (, ) and these model structures are special cases of Reedy model structures (see <cit.>). Indeed, one can consider the category =[0 → 1] as inverse category, and in that case the Reedy structure is the injective model structure. We can also consider =[0 → 1] as a direct category and  we get the projective model structure (see <cit.>,<cit.> for details).  It's important to observe  need not to be a cofibrantly generated to get the Reedy model structure.  


Given two objects of (), f: X_0 → X_1 and  g: Y_0 → Y_1, a map α: f → g in () consists of two morphisms α_i : X_i → Y_i such that we have a commutative square  in  :
 
    (0,18)*+X_0="W";
    (0,0)*+ X_1="X";
    (30,0)*+Y_1="Y";
    (30,18)*+Y_0="E";
    @->^-α_1"X";"Y";
    @->^-f"W";"X";
    @->^-α_0"W";"E";
    @->^-g"E";"Y";



Let α: f → g be a map in (). With the previous notation we will say that:


  * α is a injective cofibration if α_0 and α_1 are cofibrations in . 

  * α is a  projective  cofibration if


  * α_0 is a cofibration  in  and 

  * the unique induced morphism X_1 ∪^X_0 Y_0 → Y_1 is a  cofibration in .
 

  * α is a level-wise weak equivalence (resp. level-wise fibration) if α_0 and α_1 are weak equivalences (resp. fibrations) in .

  * α is an injective fibration if α_1: X_1 → Y_1 is a fibration and if the induced map : 

    X_0 → X_1 ×_Y_1 Y_0

is a fibration. 



Let  be a model category.  Then with the previous definitions, the following hold.


  * The three classes of injective fibrations, injective cofibrations and level-wise weak equivalences determine a model structure on () called the injective model structure.

  * The three classes of projective fibrations, projective cofibrations and level-wise weak equivalences determine a model structure on () called the injective model structure.

  * If  is cofibrantly generated (resp combinatorial) then so are the injective and projective model structures.



See Hovey <cit.>.



  §.§.§ Localizing sets


We borrow here some notation in Hovey <cit.>.

Let  be a cofibrantly generated model category with   and  the respective sets of generating cofibrations and trivial cofibrations. 


  * For any morphism s : A → B of  we will denote by α_s: s →_B the map in () corresponding to the following commutative square.
 
    (0,18)*+A="W";
    (0,0)*+B="X";
    (30,0)*+B="Y";
    (30,18)*+B="E";
    @->^-"X";"Y";
    @->^-s"W";"X";
    @->^-s"W";"E";
    @->^-"E";"Y";



  * We will denote by α_ the set of all α_i for i ∈, that is, α_= {α_i}_i ∈.

  * Similarly will denote by  α_ the set of all α_j for j ∈ : α_= {α_j}_j ∈.






Define the localizing injective set as the set α_.



Let s: A → B be a morphism of  and let _∅ be the identity morphism of  the initial object of 


  * If  s: A → B is a weak equivalence in  then α_s: s →_B is a level-wise weak equivalence in ().

  * If  s: A → B is a (trivial) cofibration  in  then α_s: s →_B is a level-wise (trivial) cofibration in (). 

  * If A is cofibrant in , and s is a cofibration,  then the unique map _∅→ s is a level-wise cofibration in .





Like any isomorphism of , the identity _B  is a simultaneously a cofibration, a trivial cofibration, and a weak equivalence.  The maps s and _B are  the components defining the map α_s.  Therefore α_s is a level-wise weak equivalence (resp. (co)fibration) if and only if s is a weak equivalence (resp. (co) fibration).

The third assertion is clear because B is also cofibrant since by composition the unique map ∅→ A s B is a cofibration.




  
Relative cylinder object For a cofibration s: A → B the map α_s is related to a relative cylinder object for the cofibration s (see <cit.>). Consider the pushout of s along itself with the pushout data B s A s B and write B ∪^A B for the pushout-object. From the commutative square that defines the map α_s:s (s,_B)_B, we get a unique map ϕ:B ∪^A B →  B using the universal property of the pushout. In particular everything  commutes below.

 
    (0,20)*+A="W";
    (0,0)*+B="X";
    (40,0)*+B="Y";
    (40,20)*+B="E";
    (20,6)*+B ∪^A B="Z";
    @->_-"X";"Y";
    @->^-s"W";"X";
    @->^-s"W";"E";
    @->^-"E";"Y";
    
    @->^-ϵ_0"X";"Z";
    @->^-ϕ"Z";"Y";
    @->^-ϵ_1"E";"Z";


Now use the axiom of the model category  to factor the map ϕ as a cofibration followed by a trivial fibration:

    B ∪^A B δ E_s  [∼]p B.

This diagram determines a relative cylinder object for the cofibration s : A → B. 
The maps ϵ_1 and ϵ_2 are cofibrations because cofibration are closed under cobase change; and by composition δ∘ϵ_0 and δ∘ϵ_1 are also cofibrations. Moreover by 3-for-2 with respect to the equality p ∘ [δ∘ϵ_i] = _B, they are also weak equivalences. It turns out that  δ∘ϵ_1 and δ∘ϵ_2 are trivial cofibrations.


Let s: A → B be a cofibration as previously. 


  * Denote by j_0 the composite δ∘ϵ_0 will write j_1= δ∘ϵ_1. Both are trivial cofibrations.

  * Let ζ_s: s (s,j_0) j_1 be the map in  given by the commutative square:
 
    (0,18)*+A="W";
    (0,0)*+B="X";
    (30,0)*+E_s="Y";
    (30,18)*+B="E";
    @->^-j_0_∼"X";"Y";
    @->^-s"W";"X";
    @->^-s"W";"E";
    @->^-j_1_∼"E";"Y";


  * Let ℓ_s: j_1 (_B, p)_B be the level wise weak equivalence in  given by the commutative square: 
 
    (0,18)*+B="W";
    (0,0)*+E_s="X";
    (30,0)*+B="Y";
    (30,18)*+B="E";
    @->>^-p_∼"X";"Y";
    @->^-j_1"W";"X";
    @->^-_B"W";"E";
    @->^-_B"E";"Y";





Define the localizing projective set as the set ζ_= {ζ_i }_i ∈.



With the previous notation the following hold. 


  * For every cofibration s, ζ_s is a cofibration in 

  * For every s we have α_s= l_s ∘ζ_s

  * If s: A → B is a cofibration with A cofibrant then,  ℓ_s is a level-wise weak equivalence between cofibrant objects in  and thus . 

  * Let Υ: → be a left Quillen functor with respect to either  _inj or _proj. If s is a cofibration between cofibrant objects, then  Υ(ℓ_s) is a weak equivalence in .

  * Assume that  Υ: → is a left Quillen functor with respect to either  _inj or _proj. Then for any cofibration  s between cofibrant objects, Υ(α_s) is a weak equivalence if and only if Υ(ζ_s) is a weak equivalence.




Assertion (1) and (2) are  immediate by construction. 
Observe that the cofibrant object in the projective model structure _proj are the cofibrations s : A → B whose domain is cofibrant. Moreover any projective cofibration is  also an injective cofibration, thus s is also a cofibrant object in the injective model structure. Now clearly ℓ_s is a level-wise weak equivalence. This gives Assertion (3).  


Assertion (4) is a consequence of Ken Brown's Lemma <cit.>. Indeed, s and _B are cofibrant and ℓ_s is weak equivalence between cofibrant objects, so by Ken Brown's Lemma any left Quillen functor must send it to a weak equivalence between cofibrant objects.

Assertion (5) is a consequence of Assertion (4) and the 3-for-2 property of weak equivalences in the model category .



Let s:A → B be a cofibration and let  u and v be two elements in _(B, X). Say that f and g are homotopic relative to s if there is a map h : E_s → X such that h∘ j_0= u and h ∘ j_1= v.



It's important to observe the following.


  * Note that the maps j_0 and j_1 restrict to the same map on A. Therefore we have an equality h ∘ (j_0 ∘ s) =  h ∘ (j_1 ∘ s).

  * Any map h: E_s → X defines tautologically a homotopy between u:=h∘ j_0 and v:=h ∘ j_1.






  
Homotopy lifting problem

Let f:X_0 → X_1 be an arbitrary morphism and let s: A → B be a cofibration in .  Consider a lifting problem of solid arrows defined by s and f through a map θ: s (θ_0,θ_1) f  in  as follows. 

    (0,18)*+A="W";
    (0,0)*+B="X";
    (30,0)*+X_1="Y";
    (30,18)*+X_0="E";
    @->^-θ_1"X";"Y";
    @->^-s"W";"X";
    @->^-θ_0"W";"E";
    @->^-f"E";"Y";
    
    @–>^-"X";"E";

A solution  up-to-homotopy to this problem consists of a map r: B  X_0 such that:


  * r ∘ s = θ_0 i.e, r solves strictly the upper triangle; and 

  * f ∘ r and θ_1 are homotopic relative to s, in that, there exists a map h : E_s → X_1 such that h ∘ j_0= θ_1 and h ∘ j_1=f ∘ r. In other words r solves the lower triangle up-to a relative homotopy.
 
Say that f has the Homotopy Extension Lifting  Property with respect to s if there is a solution up-to-homotopy to any lifting problem defined by s and f.



Let f : X_0 → X_1 be an object in  and let f →∗ be the unique map going to the terminal object. Consider a lifting problem of solid arrows defined by ζ_s and f →∗ as follows. 
 
    (0,18)*+s="W";
    (0,0)*+j_1="X";
    (30,0)*+∗="Y";
    (30,18)*+f="E";
    @->^-!"X";"Y";
    @->_-ζ_s=(s,j_0)"W";"X";
    @–>^-(γ_0, γ_1)"X";"E";
    @->^-(θ_0,θ_1)"W";"E";
    @->^-!"E";"Y";


Then we have an equivalence between the following.


  * A solution γ=(γ_0,γ_1): j_1 → f to this lifting problem.

  * An up-to-homotopy solution to the lifting problem in  defined by s and f by the map θ: s (θ_0,θ_1) f.




One simply writes down everything. 
Given a solution γ=(γ_0,γ_1): j_1 → f, we get a solution up-to-homotopy by letting r=γ_0 and h=γ_1 and conversely given a solution up-to-homotopy r: B → X_0 and h: E_s → X_1 defines a map γ=(r,h): j_1 → f that is a solution to the lifting problem.  





  §.§.§ Detecting the Segal conditions


With the sets of maps α_ and α_ we are able to detect in () whether a map f: X_0 → X_1 is a trivial fibration or  fibration by lifting property just like we do with the set  and  in . Hovey <cit.> showed  that α_ is a subset of the generating set of cofibrations in the injective model structure on (). 



Let  be a  model category and let S be a set of maps. 


  * A morphism f: X_0 → X_1 has the RLP in   with respect to S, if and only if the unique map  f →∗ going to the terminal object has the RLP in () with respect to the set α_S = {α_s}_s ∈ S.

  * More generally, if g: Y_0 → Y_1, then a map β: f → g in () has the RLP with respect to α_S = {α_s}_s ∈ S, if and only if the induced map 
X_0 → X_1 ×_Y_1 Y_0 has the RLP in  with respect to the set S.

  * In particular a morphism f: X_0 → X_1 is a trivial fibration , if and only if, f →∗ has the RLP with respect to the set α_.



The last  assertion is also mentioned in Rosicky <cit.>.






    (-60,20)*+A="A";
    (20,30)*+X_0="B";
    (-20,-10)*+B="C";
    (60,0)*+Y_0="D";
    
    
    @->^"A";"B";
    @->_s"A";"C";
    @->^β_0"B";"D";
    @->_"C";"D";
    
    (-60,-30)*+B="X";
    (20,-20)*+X_1="Y";
    (-20,-60)*+B="Z";
    (60,-50)*+Y_1="W";
    @–>^γ_1        "X";"Y";
    @->_s"A";"X";
    @->^β_1"Y";"W";
    @->^"C";"Z";
    @->^"B";"Y";
    @->^Ł__UZ"D";"W";
    @=__B"X";"Z";
    @->_"Z";"W";
    
    (20,30)+(11,-25)*+X_1 ×_Y_1 Y_0 ="E";
    @->^"B";"E";
    @.>^"E";"D";
    @.>^"E";"Y";



It suffices to prove Assertion (2). 
Consider a lifting problem defined by α_s and β:
 
    (0,18)*+s="W";
    (0,0)*+_B="X";
    (30,0)*+g="Y";
    (30,18)*+f="E";
    @->^-(γ_0, γ_1)"X";"Y";
    @->_-(s,_B)"W";"X";
    @->^-(θ_0,θ_1)"W";"E";
    @->^-(β_0,β_1)"E";"Y";


This commutative square corresponds to a commutative cube in . And if we unfold it, we see that the maps γ_0 : B → Y_0 and θ_1 : B → X_1 determine a commutative square with the maps g and β_1; in that β_1 ∘θ_1 = g ∘γ_0. The universal property of the pullback square gives a unique map δ: B → X_1 ×_Y_1 Y_0 that makes everything compatible. In particular θ_1 factors through δ. If we put this in the original cube, everything below commutes.  

    (-60,20)*+A="A";
    (20,30)*+X_0="B";
    (-20,-10)*+B="C";
    (60,0)*+Y_0="D";
    
    
    @->^θ_0"A";"B";
    @->_s"A";"C";
    @->^β_0"B";"D";
    @->_γ_0       "C";"D";
    
    (-60,-30)*+B="X";
    (20,-20)*+X_1="Y";
    (-20,-60)*+B="Z";
    (60,-50)*+Y_1="W";
    @–>^    θ_1"X";"Y";
    @->_s"A";"X";
    @->^β_1"Y";"W";
    @=^"C";"Z";
    @->_-f"B";"Y";
    @->^g"D";"W";
    @=__B"X";"Z";
    @->_γ_1"Z";"W";
    
    (20,30)+(11,-25)*+X_1 ×_Y_1 Y_0 ="E";
    @->^"B";"E";
    @.>^"E";"D";
    @.>^"E";"Y";
    
    
    @.>^δ"C";"E";


As we see, we get a commutative square that corresponds to a lifting problem defined by the map s:A → B and the map X_0 → X_1 ×_Y_1 Y_0:
 
    (0,18)*+A="W";
    (0,0)*+B="X";
    (30,0)*+X_1 ×_Y_1 Y_0="Y";
    (30,18)*+X_0="E";
    @->^-δ"X";"Y";
    @->_-s"W";"X";
    @->^-θ_0"W";"E";
    @->^-"E";"Y";


Now it suffices to observe that this lifting problem has a solution if and only if our original lifting problem has a solution. Indeed if χ :  B → X_0 is a solution to the previous lifting problem, then the map (χ,θ_1): _B → f in () determines a solution to the original lifting problem. And reciprocally given a solution to the original lifting problem (χ,θ_1): _B → f, then the component χ: B → X_0 is a solution to the lifting problem defined by s and X_0 → X_1 ×_Y_1 Y_0.




  
HELP Lemma There are various versions of the HELP Lemma and we refer the reader to Boardman-Vogt <cit.>, May <cit.>, Simpson <cit.>, Vogt <cit.> and the references therein. We will use the  version used by Simpson <cit.>. This version is  for tractable model categories but the argument remains exactly the same for a cofibrantly generated model category where the cofibrations are generated by a set of maps between cofibrant objects.


Let  be a tractable model category with a set of cofibration . Then any morphism f between fibrant objects and that satisfies the HELP with respect to every element in  is a weak equivalence. 


Equivalently we have:

Let  be a tractable model category with a set of cofibration . If f is a morphism in   such that the unique map f →∗ is a level fibration in  and  has the RLP with respect to every element in ζ_,  then f is a weak equivalence.


See <cit.>.




 §.§ Localizing a Quillen-Segal theory

We now give the proof of Theorem <ref> and Theorem <ref>. 


Let ={_i: →() }_i ∈ I be an -valued Quillen-Segal theory on  such that for every i ∈ I we have a left adjoint Υ_i: →. 


  * Define the injective localizing set for  as:

    _inj():= ∐_i ∈ IΥ_i(α_)


  * Define the projective localizing set for  as:

    _proj():= ∐_i ∈ IΥ_i(ζ_)





With the previous notation we have the following. 


  * If  is an injective theory then every element in _inj is a cofibration in 

  * If  is a projective theory then every element in _proj is a cofibration in 




Indeed, every element in α_ (resp. ζ_) is a injective (reps. projective) cofibration in ; therefore its image by the left Quillen functor Υ_i is a cofibration in .



The hypotheses of the theorem allow us to consider the left Bousfield localization  with respect to the set _inj (see Hirschhorn <cit.> and Barwick <cit.>). Then thanks to the previous proposition every element in _inj is a cofibration and a weak equivalence; thus a new trivial cofibrations. It follows that for any fibrant object c in this new model structure the unique map c →∗ in  must be Υ_i(α_)-injective for every i ∈ I. And by adjunction we find that  the map _i(c) →∗ in  is α_-injective and _i(c) is a trivial fibration by Lemma <ref>. In particular for every i ∈ I, _i(c) is a weak equivalence (between fibrant objects); thus c satisfies the generalized Segal conditions. 



The proof is the same as in the injective case. 
Consider the left Bousfield localization  with respect to the set _proj. Then thanks to the previous proposition every element in _proj is a cofibration and a weak equivalence; thus a new trivial cofibrations. It follows that for any fibrant object c in this new model structure the unique map c →∗ in  must be Υ_i(ζ_)-injective for every i ∈ I. And by adjunction we find that  the map _i(c) →∗ in  is ζ_-injective and _i(c) is a weak equivalence by the HELP Lemma <ref>. In particular for every i ∈ I, _i(c) is a weak equivalence (between fibrant objects); thus c satisfies the generalized Segal conditions. 






§ QUILLEN-SEGAL ALGEBRAS




 §.§ Comma categories and Quillen-Segal objects



  §.§.§ Comma categories

We recall here some definitions and properties on comma categories. These results are well known in Category Theory (see for example <cit.>, <cit.>, <cit.>). We include some of them for completeness.

Let :→ be a functor. The comma category (↓) is the category described as follows.


  * Objects are triples [] = [_0, _1, π_: _0 →(_1)] ∈××()

  * Given two objects [] = [_0, _1, π_] and [] = [_0, _1, π_], a map σ : [] → [] is a pair σ=[σ_0,σ_1] ∈_(_0, _0) ×_(_1, _1) such that we have a commutative square in :

    (0,18)*+_0="W";
    (0,0)*+(_1)="X";
    (30,0)*+(_1)="Y";
    (30,18)*+_0="E";
    @->^-(σ_1)"X";"Y";
    @->^-π_"W";"X";
    @->^-σ_0"W";"E";
    @->^-π_"E";"Y";

That is:

    _(↓)([],[])={ [σ_0, σ_1] ∈_(_0,_0) ×_(_1,_1)    |   π_∘σ_0 = (σ_1)∘π_}


 

We have three obvious functors :


  * Π_0: →, with Π_0([])= _0;

  * Π_1: →, with  Π_1([])= _1;

  * Π_: →(), with Π_([])= π_.




It's sometimes convenient to adopt another notation for  because we don't see the category . So we will also write :=.



  §.§.§ Quillen-Segal algebras and objects

We are now able to formulate the definition of Quillen-Segal algebras and  Quillen-Segal objects.. 

Let  be a model category and let : → be a functor.  

An object []=[_0, _1, π_] ∈ is a Quillen-Segal -object if [] satisfies the Segal condition, that is if  π_: _0 →(_1) is a weak equivalence in . 



Let  be a model category and let Ø be a monad on  or an operad enriched over .

A Quillen-Segal Ø-algebra is a Quillen-Segal -object []=[_0, _1, π_] ∈  for the forgetful functor : () →.


In the previous definition, Ø is not restricted to be a monad or an operad. Indeed Ø can be a PROP, properad, cyclic operad etc. These objects can be viewed themselves as algebras over some  monad Ø' defined on another category ' (see <cit.>, <cit.>, <cit.>).  Moreover if we fix such `operator-object' Ø, its category of algebras with coefficient in , is equivalent to a category of a monad  defined on  another category ',  where ' is in many cases  a product of copies of  (see for example <cit.>) or more generally a diagram category (,).

 


  * The previous definition makes sense in any category with weak equivalences. 

  * Let () ↪() be the subcategory of weak equivalences in . Then the category of Quillen-Segal objects is precisely the full subcategory of  of objects that are sends to () through the functor Π_ : →()









§ PROPERTIES OF COMMA CONSTRUCTIONS

We give here the required definitions and properties on comma categories that are necessary to deploy the homotopy theory on . Most of the material is classical in category theory and many results are straightforward.


 §.§ Embedding
 

A direct consequence of the definition is the following:

We have a functor ι : defined as follows. 


  * For ∈,  ι()=[(), , _()]. 

  * The functor ι takes a map θ : → of  to the map ι(θ)=[(θ), θ].

  * The composite Π_1 ∘ι is the identity. In other words  is a retract of  in the category  of small categories. 

  * The functor ι is injective on objects and we have an isomorphism of hom-sets

    _(,) ≅_(ι(), ι())

In particular the functor ι: → exhibits  as a full subcategory of  





The fact that ι is injective on objects is clear since  appears alone in the triple [(), , _()].  

On morphisms we have ι(θ_1) = ι(θ_2) ⇔ [(θ_0), θ_0] = [(θ_1), θ_1]  ⇔θ_1 = θ_2 and  (θ_0)=(θ_1); which means that ι is also injective on morphisms (= ι is  faithful). 


Let's now show that ι is also surjective on morphisms (= ι is  full). 
If σ= [σ_1, σ_1] is a map in  from ι() to ι(), we have:

    _()∘σ_0 = (σ_1) ∘_()⇔σ_0 = (σ_1) ⇔σ = [(σ_1), σ_1] = ι(σ_1)
 with σ_1: →.



 §.§ Arrow-category as comma category.

Recall that = [0 → 1]  is the walking-morphism category and therefore  is the functor category (,). If we put together this fact and the previous definition of comma category we get the following obvious results.  


Let : → be the identity functor. Then the following hold.


  * The category of morphisms  () is isomorphic to the comma category :=. And the functor Π_: →() is an isomorphism.

  * The category of  morphisms  () is isomorphic to the functor category (, ).  

  * The functor Π_0 : → is the source-functor and corresponds to the evaluation functor _0: (, ) →.

  * The functor Π_1 : → is the target-functor and corresponds to the evaluation functor _1: (, ) →.






If we apply Proposition <ref> for the functor = _ then  is the category of (). In particular one gets the following results that can also be found in Hovey  <cit.>.


Let  be any category. Then the following hold.



  * We have an embedding ι_ : () that takes an object X ∈ to the identity _X and a morphism f: X → X' to the map _X →_X' whose components are both equal to f. 

  * The functor ι_ is left adjoint to the functor _0: () →. We will denote by L_0= ι_. 

  * If there is an initial object ∅∈, then the functor _1: () → has also a left adjoint L_1: →(). 
We have L_1(X)=∅→ X. 





 §.§ Adjunctions, limits and colimits





  §.§.§ Adjunctions

We give here the adjunctions that will be needed to get the various model structures on our comma category. The constructions and the results that will follow are also classical in category theory. We include the proof for completeness. Recall that we have an embedding ι: → and a functor Π_1: →. Our goal in this section is to establish the following theorem.


Let : → be a functor. With the previous notation the following hold. 



  * The functor Π_1: → is left adjoint to the embedding ι : →. In particular  is equivalent to a full reflective subcategory of .

  * The functor Π_1: → has also a left adjoint L_1: → with L_1()= [∅, , ∅→()], where ∅→() is the unique map.

  * If furthermore  has a left adjoint : →, then


  * The functor Π_: →()  has a left adjoint 

    Γ : () →


  * The functor Π_0: → has a left adjoint ^+ : → such that the composite Π_1 ∘^+ is the functor . 

We have ^+= Γ∘ L_0. 





Assertion (1) is the content of Proposition <ref>. 

Assertion (2) can be easily verified. 

Assertion (3) is given by Proposition <ref>  and Proposition <ref>.






Let : → be a functor. Then the functor  ι : → is right adjoint to the  functor  Π_1: →.  

In particular  is equivalent to a full reflective subcategory of .



Let [] be an object of  and  be an object of . We wish to show that we have a functorial isomorphism of hom-sets :

    _(Π_1([]), ) ≅_([], ι())
 
Recall that Π_1([])= _1 and that by definition 
a map σ=[σ_0,σ_1] : [] →ι() consists of two maps σ_0 ∈_(_0, ()) and  σ_1 ∈_(_1, ) such that :
 
    _()∘σ_0 = (σ_1) ∘π_⟺σ_0 = (σ_1) ∘π_⟺σ = [(σ_1) ∘π_, σ_1].
 
 It follows from the above equivalence that the function
    φ : _(_1, ) →_([], ι()),
 defined by φ(σ_1)=[(σ_1) ∘π_, σ_1] is surjective.  The presence of σ_1 alone in the couple [(σ_1) ∘π_, σ_1] implies that φ is also injective. Therefore  φ is an isomorphism as required. The functoriality in both variables [] and  is clear. 



With the previous notation the following hold.


  * For any [] ∈, the unit  η_1([]): [] →ι(Π_1([])) of the previous adjunction is given by the pair [π_, __1]. 

  * For any ∈ the counit ε_1(): Π_1(ι()) → of the adjunction is the identity.





Clear.



Let : → be a functor that posses a left adjoint : →. Let η : _→ be the unit of this adjunction.
Then we have a functor 
    Γ : () →

 defined as follows.


  * If f : X_0 → X_1, then Γ(f) = [X_0, (X_1), η_X_1∘ f] with η_X_1 : X_1 →(X_1)

  * If g : Y_0 → Y_1 and  α=(α_0,α_1) : f → g is a map in (), then Γ(α)= [α_0, (α_1)].

In particular


  * Π_0(Γ(f))=X_0, Π_0(Γ(α))=α_0;

  * Π_1(Γ(f))=  (X_1), Π_1(Γ(α))=  (α_1)

  * Π_(Γ(f))= η_X_1∘ f : X_0 → X_1 →(X_1)





For the second assertion  it suffices to show that we have an equality 
    η_Y_1∘ g ∘α_0 = (α_1) ∘η_X_1∘ f .
 
We get this equality by observing that the naturality of η : _→ and part of α being a map in  imply that all three squares below are commutative.


    (0,18)*+X_0="W";
    (0,0)*+(X_0)="X";
    (30,0)*+(X_1)="Y";
    (30,18)*+X_1="E";
    @->^-(f)"X";"Y";
    @->^-η_X_0"W";"X";
    @->^-f"W";"E";
    @->^-η_X_1"E";"Y";
        
    (0,18)*+X_1="W";
    (0,0)*+(X_1)="X";
    (30,0)*+(Y_1)="Y";
    (30,18)*+Y_1="E";
    @->^-(α_1)"X";"Y";
    @->^-η_X_1"W";"X";
    @->^-α_1"W";"E";
    @->^-η_Y_1"E";"Y";
        
    (0,18)*+X_0="W";
    (0,0)*+Y_0="X";
    (30,0)*+Y_1="Y";
    (30,18)*+X_1="E";
    @->^-g"X";"Y";
    @->^-α_0"W";"X";
    @->^-f"W";"E";
    @->^-α_1"E";"Y";


And we get:

    η_Y_1∘ g ∘α_0    = η_Y_1∘ (g ∘α_0)
       = η_Y_1∘ (α_1 ∘ f)
       = (η_Y_1∘α_1) ∘ f
       = ((α_1) ∘η_X_1) ∘ f



We are now able to establish the following adjunction. 


The functor Γ : () → is left adjoint to the functor Π_ : →().



The proof is tedious but straightforward so we shall give an outline of it. 

Let f: X_0 → X_1 be an object of () and let = be an object of . We wish to establish that we have a functorial isomorphism of hom-sets:

    _(Γ(f), ) ≅_()(f, π_)


We define a function ϕ: _()(f, π_) →_(Γ(f), ) as follows.  



  * Let α: f → a map in () displayed by the  commutative square:
 
    (0,18)*+X_0="W";
    (0,0)*+ X_1="X";
    (30,0)*+(_1)="Y";
    (30,18)*+_0="E";
    @->^-α_1"X";"Y";
    @->^-f"W";"X";
    @->^-α_0"W";"E";
    @->^-"E";"Y";

From the adjunction ⊣ there is a functorial isomorphism of hom-sets:

    ϱ : _(X_1, (_1))  _( X_1, _1)


  * Consider ϱ(α_1) : (X_1) →_1,  the adjoint transpose of α_1 : X_1 →(_1).

  * We define ϕ(α)=[α_0, ϱ(α_1)] ∈_(X_0, _0) ×_( X_1, _1).

 Let's check  that the pair ϕ(α)=[α_0, ϱ(α_1)] defines indeed a morphism in , that is, we must prove that we have an equality: (ϱ(α_1)) ∘Π_Γ(f) = ∘α_1.

By definition, Π_Γ(f)= η_X_1∘ f and from the adjunction ⊣, we have: 
    α_1= (ϱ(α_1)) ∘η_X_1.

Putting all together we have a diagram in which everything commutes: 

 
    (0,18)+(0,10)*+X_0="W";
    (0,0)*+ X_1="X";
    (0,0)+(20,10)*+(X_1)="O";
    (30,0)+(20,0)*+(_1)="Y";
    (30,18)+(20,10)*+_0="E";
    @->^-α_1"X";"Y";
    @->^-f"W";"X";
    @->^-α_0"W";"E";
    @->^-"E";"Y";
    
    @->^-η_X_1"X";"O";
    @->^-Π_Γ(f)"W";"O";
    @->^-(ϱ(α_1))"O";"Y";

The inner commutative square gives a map Π_Γ(f) →  in (). This means that ϕ(α)=[α_0, ϱ(α_1)] is a map in .  


This function is clearly an injection since ϱ is an isomorphism. And we have an inverse function ϕ^-1 that takes a map σ=[σ_0,σ_1]: Γ(f) → to the map 
    ϕ^-1(σ)= [σ_0, ϱ^-1(σ_1)]: f →.



Let's now put together the various adjunctions and see how they interact. 

Let  : → be a functor. Then the following hold.


  * We have a commutative diagram.

    (0,18)*+="W";
    (0,0)*+="X";
    (30,0)*+()="Y";
    (30,18)*+="E";
    @<-^-_0"X";"Y";
    @->_-Π_0"E";"X";
    @->^-"W";"X";
    @->^-ι"W";"E";
    @->^-Π_"E";"Y";


  * If moreover   has a left adjoint : → then we have a commutative diagram of left adjoints functors.

    (0,18)*+="W";
    (0,0)*+="X";
    (30,0)*+()="Y";
    (30,18)*+="E";
    @->^-L_0"X";"Y";
    @->^-^+"X";"E";
    @<-^-"W";"X";
    @<-^-Π_1"W";"E";
    @<-^-Γ"E";"Y";

With ^+= Γ∘ L_0.





  §.§.§ Limits and colimits

Given a diagram D: → we have three induced diagrams:


  * D_0=Π_0(D): → 

  * D_1=Π_1(D): → 

  * π_D=(D):→().

For every j ∈ we have D(j)= [D_0(j), D_0(j), π_D(j)]. Our goal here is to see how we compute limits and colimits in the comma category :=.




Let : ⇆: be an adjunction between complete and cocomplete categories. 
Then with the previous notation the following hold. 


  * There is an induced map in : 
    π_: D_0 →( D_1)
 

And the triple [ D_0,  D_1, π_] equipped with the obvious maps is the colimit of the diagram  D: →.

  * As the functor  preserves limits, there is also an induced map in   

    π_lim: lim D_0 →(lim D_1)

And the triple [lim D_0, lim D_1, π_lim] equipped with the obvious maps is the limit of the diagram  D: →.

The category  is also complete and cocomplete.


An immediate consequence is that: 



  * The functors Π_0: → and Π_1: → preserve limits and colimits. In particular they preserve pushouts.

  * The functor : →() preserves limits.



In other words the functor (Π_0,Π_1): →× creates limits and colimits (see <cit.>)



For every j ∈ denote by ϵ_1(j): D_1(j) → D_1 the canonical map in   going to the colimit and consider its image (ϵ_1(j)): (D_1(j)) →( D_1). Then the maps 
    {(ϵ_1(j)) ∘π_D(j) : D_0(j) →( D_1) }_j ∈

 determine a natural transformation from D_0 to the constant diagram of value ( D_1). Indeed,  for every structure map l:j → j' in  we can apply the functor  to the equality ϵ_1(j) = ϵ_1(j') ∘ D_1(l) and get:

    (ϵ_1(j) )= (ϵ_1(j'))  ∘(D_1(l)).

Moreover, we also have a commutative diagram in :


    (0,18)*+D_0(j)="W";
    (0,0)*+(D_1(j))="X";
    (30,0)*+(D_1(j'))="Y";
    (30,0)+(50,0)*+( D_1) ="Z";
    (30,18)*+D_0(j')="E";
    @->^-(D_1(l))"X";"Y";
    @->^-π_D(j)"W";"X";
    @->^-D_0(l)"W";"E";
    @->^-π_D(j')"E";"Y";
    @.>^-(ϵ_1(j'))"Y";"Z";

Using the universal property of the colimit of D_0, we find a unique map  
    π_: D_0 →( D_1)

that makes everything compatible.  If we write ϵ_0(j) : D_0(j) → colim D_1, the canonical map in  going to the colimit, then the pair [ϵ_0(j),ϵ_1(j)] determines a map in : 

    [D_0(j), D_0(j), π_D(j)] →  [ D_0,  D_1, π_].


It is tedious but straightforward to check that [ D_0,  D_1, π_] equipped with these maps satisfies the universal property of the colimit of the diagram D. This proves Assertion (1).  



For Assertion (2) we proceed in a dual manner using the fact that  preserves limits like any right adjoint functor.  Using this, we know that (lim D_1) equipped with the obvious maps is the limit of the diagram (D_1).  

Let p_0(j): lim D_0 → D_0(j) be the canonical projection so that for every structure map l : j → j' we have p_0(j') = D_0(l) ∘ p_0(j). Applying  to this equality yields:

    (p_0(j')) = (D_0(l)) ∘(p_0(j)).

Then the maps: {π_D(j) ∘ p_0(j): lim D_0 →(D_1(j)) }_j ∈,
determine a natural transformation from the constant diagram of value lim D_0, to the diagram (D_1).
 
By the universal property of the limit (lim D_1) we find a unique map 

    π_lim: lim D_0 →(lim D_1)

that makes everything compatible. In particular for every structure map l : j → j' we have a commutative diagram:

    (0,18)*+D_0(j)="W";
    (0,0)*+(D_1(j))="X";
    (30,0)*+(D_1(j'))="Y";
    (30,18)+(-70,0)*+lim D_0 ="Z";
    (30,0)+(-70,0)*+(lim D_1) ="T";
    (30,18)*+D_0(j')="E";
    @->^-(D_1(l))"X";"Y";
    @->^-π_D(j)"W";"X";
    @->^-D_0(l)"W";"E";
    @->^-π_D(j')"E";"Y";
    @.>^-p_0(j)"Z";"W";
    @.>^-(p_1(j))"T";"X";
    @.>^-π_lim"Z";"T";

It's not hard to check that the object [lim D_0, lim D_1, π_lim] equipped with the maps defined by the maps 

    [p_0(j), p_1(j)] : [lim D_0, lim D_1, π_lim] → D(j)

 satisfies the universal property of the limit for the diagram D.





  §.§.§ Accessibility of comma categories


We list below some technical results on locally presentable categories. Good references on the subject include <cit.>, <cit.>, <cit.>. 



Let  : → be an accessible functor between accessible categories. 
Then the category = (↓) is also accessible.



This is a special case of Theorem 2.43 in the book of Adámek and Rosický <cit.>  for the comma category  _↓≅.



Let : ⇄ : be an adjunction between locally presentable categories. Then the category  is also locally presentable. 



By the previous result we know that  is an accessible category. So it remains to show that  is also cocomplete. But this is given by Proposition <ref>. 






§ INJECTIVE AND PROJECTIVE MODEL STRUCTURES

In this section we consider a cofibrantly generated model category  without requiring the axioms of a monoidal model category yet.  


 §.§ Detecting the Segal conditions for comma categories

Consider the adjunction : ⇄(): Γ given in Proposition <ref>. 

Define the injective localizing set   _= {Γ(α_i) }_i ∈ = Γ (α_) as the image of α_ under the left adjoint Γ. 

 An immediate consequence of Lemma <ref> is:

In the category  the following hold. 



  * An object []=[_0,_1,] is -injective, that is [] →∗ has the RLP with respect to , if and only if, the map : _0 →(_1) is a trivial fibration in . In particular [] satisfies the Segal condition.

  * For any ∈, ι()=[(), , _()] is -injective. 




We proceed by adjunction. [] →∗ has the  RLP with respect to , if and only if, the map : _0 →(_1) has the RLP with respect to α_, if and only if  is a trivial fibration, thanks to Lemma <ref>. This gives the first assertion. The second assertion is clear since every identity morphism is a trivial fibration. 



Let  be a tractable model category. Define the projective localizing set   _proj= {Γ(ζ_i) }_i ∈ = Γ (ζ_) as the image of ζ_ under the left adjoint Γ. 

The following proposition will be used for the projective model structure.

Let  be a tractable model category. Let []=[_0,_1,] be an object in the category  and assume that:


  * [_0,_1,] is _proj-injective, that is [] →∗ has the RLP with respect to _proj;

  *  is a map between fibrant object i.e, the unique map →∗ is a level fibration.

Then  satisfies the Segal conditions i.e,  is a weak equivalence. 



Just proceed by adjunction with Lemma <ref>.






 §.§ Injective and projective model structures on 

We are going to mimic Hovey's result and have an injective model structure on  such that when =_ we recover Hovey's theorem. Stanculescu <cit.> and Toën <cit.> considered model structures on the (equivalent) category (↓).  

 
Recall that from the adjunction ⊣ there is a functorial isomorphism of hom-sets:

    ϱ : _(_0,  (_1))  _( (_0), _1).

Then if σ=[σ_0,σ_1]: → is a map in  we have two commutative squares that are mutually adjoint:

    (0,18)*+_0="W";
    (0,0)*+(_1)="X";
    (30,0)*+(_1)="Y";
    (30,18)*+_0="E";
    @->^-(σ_1)"X";"Y";
    @->^-π_"W";"X";
    @->^-σ_0"W";"E";
    @->^-π_"E";"Y";
    (45,9)*+⟺="Z";
    
    (0,18)*+(_0)="W";
    (0,0)*+_1="X";
    (30,0)*+_1="Y";
    (30,18)*+(_0)="E";
    @->^-σ_1"X";"Y";
    @->^-ϱ(π_)"W";"X";
    @->^-(σ_0)"W";"E";
    @->^-ϱ(π_)"E";"Y";


Let σ: → be a map in . With the previous notation we will say that:


  * σ is a injective (trivial) cofibration if σ_0 and σ_1 are (trivial) cofibrations in . 

  * σ is a level-wise weak equivalence (resp. level-wise fibration) if:


  * σ_0 is a weak equivalence (resp. fibration)  in  and

  * σ_1 is a weak equivalence (resp. fibration) in .


  * σ is an injective (trivial) fibration  if:


  * σ_1: _1 →_1 is a  (trivial) fibration in  and 

  * the induced map _0 →(_1) ×_(_1)_0
is a (trivial) fibration in . 



  * σ is an projective (trivial) cofibration  if:


  * σ_0: _0 →_0 is a  (trivial) cofibration in  and 

  * the induced map _1 ∪^(_0)(_0) →_1 is a (trivial) cofibration in .




We need the following result to establish the Reedy model structure. The proof is straightforward but we include it for completeness. 

Let : → be a right Quillen functor. Then a map σ: → is an injective trivial fibration if and only if, it's an injective fibration and a level-wise weak equivalence. 



To prove the only if part, it suffices to show that σ is a level-wise weak equivalence.  If σ=[σ_0,σ_1] is an injective trivial fibration, then σ_1 is a trivial fibration in  by definition, and therefore (σ_1) is also trivial fibration in , since  preserves trivial fibrations. Now in any model category, trivial fibrations are closed under pullback, and we see that the canonical map p: (_1) ×_(_1)_0 →_0 is also a trivial fibration.  The other part of being an injective trivial fibration means that the map _0 →(_1) ×_(_1)_0 is a trivial fibration. The map σ_0 is the composite the previous map followed by the map p, and both are trivial fibrations, thus σ_0 is a trivial fibrations as well.  In the end σ_1 and σ_0 are trivial fibrations in the respective model category, in particular each of them is a  weak equivalence.   


For the if part, we simply need to show that the map _0 →(_1) ×_(_1)_0 is a weak equivalence since it's already a fibration.  The argument is based on the 3-for-2 property of weak equivalences in . Indeed, assume σ=[σ_0,σ_1] is an injective fibration and a level-wise weak equivalence. Then σ_1 is a trivial fibration in  by definition, and therefore (σ_1) and then  its pullback p are also a trivial fibrations.  As mentioned above, we have a factorization σ_0 as the map _0 →(_1) ×_(_1)_0 followed by the map p: (_1) ×_(_1)_0 →_0. Since σ_0 and p are weak equivalences, then by 3-for-2, the map 
_0 →(_1) ×_(_1)_0 is also a weak equivalence as desired. 


The dual statement is:

 A map σ: → is a projective trivial cofibration if and only if, it's a level-wise weak equivalence and a projective cofibration. 

The proof is dual to the previous one so we just need to adapt it. 

To prove the only if part, it suffices to show that σ is a level-wise weak equivalence.  If σ=[σ_0,σ_1] is an projective trivial cofibration, then σ_0 is a trivial cofibration in  by definition, and therefore (σ_0) is also trivial cofibration in , since  preserves trivial cofibrations. Now in any model category, trivial cofibrations are closed under cobase change. It follows that the canonical map q: _1 →_1 ∪^(_0)(_0) is a trivial cofibration in . The  other part of being an projective trivial cofibration means that the map _1 ∪^(_0)(_0) →_1 is a trivial cofibration. The map σ_1 is the composite of the map q followed  by the previous map, and both are trivial cofibrations. Then σ_0 is a trivial cofibrations as well.  In the end σ_0 and σ_1 are trivial cofibrations in the respective model category, in particular each of them is a  weak equivalence.   


For the if part, we simply need to show that the map  _1 ∪^(_0)(_0) →_1 is a weak equivalence since it's already a cofibration.  The argument is also based on the 3-for-2 property of weak equivalences in . Indeed, assume σ=[σ_0,σ_1] is a projective cofibration and a level-wise weak equivalence. Then σ_0 is a trivial cofibration in  by definition,  therefore (σ_0) and  its cobase change q are also a trivial cofibrations in .  As mentioned above, we have a factorization σ_1 as the map q followed by the map _1 ∪^(_0)(_0) →_1. Since σ_1 and q are weak equivalences, then by 3-for-2, the map 
    _1 ∪^(_0)(_0) →_1
 is also a weak equivalence as desired.




 §.§ Factorizations

We generalize here to comma categories the required factorizations that are necessary to get the Reedy model structure. We simply follow the classical method.



  §.§.§ Injective factorizations


Let : → be a right Quillen functor. Then with the previous definition the following hold. 


  * Any map σ: → can be factored as an injective cofibration followed by a an injective trivial fibration.

  * Any map σ: → can be factored as an injective trivial cofibration followed by an injective fibration. 


If we decide to prove each assertion separately, this will be very long. So we will use a generic language of (weak) factorization systems (see for example <cit.>). In any model category we have two weak factorizations systems (, ∩) and (∩, ), where ,  and  are respectively the classes of cofibrations, fibrations and weak equivalences.  Then we have two different factorization  systems  on  and  on  that determine the model structure on each model category. The functor  is such that ( ) ⊆ for each of the  corresponding factorization systems. 


Let  σ=[_0,_1]: → be a map in . 

Use the axiom of the model category  with respect to the factorization system  to write σ_1 as _1= r(_1) ∘ l(_1):

    _1 _1 = _1l(_1)  E_1 r(_1)_1,
 
with r(_1) ∈ and l(_1) ∈. The image under  of this factorization, gives a factorization  (_1)= (r(_1)) ∘(l(_1)), with (r(_1)) ∈ since  ( ) ⊆.

Form the pullback square in  defined by the pullback data: 
    (E_1) (_1) _0,

and let p_1 : (E_1) ×_(_1)_0 →_0 and p_2:   (E_1) ×_(_1)_0  →(_1) be the canonical maps. Then p_1 ∈ because  is closed under pullbacks.  
The universal property of the pullback square gives a unique map δ : _0 →(E_1) ×_(_1)_0,
such that everything below commutes. 

    (0,30)*+_0="W";
    (0,0)*+(_1)="X";
    (60,0)*+(_1)="Y";
    (60,30)*+_0="E";
    @->^-π_"W";"X";
    @->^-σ_0"W";"E";
    @->^-π_"E";"Y";
    
    (30,20)*+(E_1) ×_(_1)_0="U";
    (30,0)*+(E_1)="V";
    @->^-(l(_1))"X";"V";
    @->>^-(r(_1))"V";"Y";
    @->^-p_2"U";"V";
    @->>^-p_1"U";"E";
    @->^-δ"W";"U";


Now we use the factorization system  to factor the map δ:

    δ : _0 →(E_1) ×_(_1)_0 = _0  l(δ)  m_0 r(δ)(E_1) ×_(_1)_0,
 
with r(δ) ∈ and l(δ) ∈.
Let =[_0,_1,π_] be the object of  defined by

    _0= m_0,   _1=E_1,   π_=p_2 ∘ r(δ).


We  have a map l(σ) : → given by the couple [l(δ),  l(σ_1)] ∈× and a map r(σ) : → given by the couple [p_1 ∘ r(δ),  r(σ_1)]  with [r(δ),  r(σ_1)] ∈×,  such that = r() ∘ l().  This gives the assertions. 






  §.§.§ Projective factorizations


Let : → be a right Quillen functor. Then with the previous definition the following hold. 

  * Any map σ: → can be factored as a projective cofibration followed by a  projective trivial fibration.

  * Any map σ: → can be factored as a projective trivial cofibration followed by a projective fibration. 



Consider a factorization system  on  and a factorization system  on  such that () ⊆ and () ⊆.  


Let  σ=[_0,_1]: → be a map in . 

Use the axiom of the model category  with respect to the factorization system  to write σ_0 as _0= r(_0) ∘ l(_0):

    _0 _0 = _1l(_0)  m_0 r(_0)_0,
 
with r(_0) ∈ and l(_0) ∈. 

The image under  of this factorization, gives a factorization  (_0)= (r(_0)) ∘(l(_0)), with (l(_0)) ∈ since  ( ) ⊆. 

Form the pushout square in  defined by the pushout data: 
    _1 (_0)  (m_0),

and let i_2 : _1 →_1 ∪^(_0)(_0) and i_1:  (m_0)   →_1 ∪^(_0)(_0) be the canonical maps. Then i_2 ∈ because  is closed under pushouts.
The universal property of the pushout square gives a unique map 

    ζ : _1 ∪^(_0)(m_0) →_1,

such that everything below commutes. 

    (0,30)*+(_0)="W";
    (0,0)*+_1="X";
    (60,0)*+_1="Y";
    (60,30)*+(_0)="E";
    @->^-ϱ()"W";"X";
    @->^-_1"X";"Y";
    @->^-ϱ()"E";"Y";
    
    (30,10)*+_1 ∪^(_0)(m_0)="U";
    (30,30)*+(m_0)="V";
    @->^-(l(_0))"W";"V";
    @->^-(r(_0))"V";"E";
    @->^-i_2"X";"U";
    @->^-i_1"V";"U";
    @->^-ζ"U";"Y";

Now we use the factorization system  to factor the map ζ:

    δ : _1 ∪^(_0)(m_0) →_1 = _1 ∪^(_0)(m_0)  l(ζ)  E_1 r(ζ)_1,
 
with l(ζ) ∈ and r(ζ) ∈.
Let =[_0,_1,π_] be the object of  defined by

    _0= m_0,   _1=E_1,   π_= ϱ^-1 (l(ζ) ∘ i_1) ∈_(m_0,(E_1)).


We  have a map l(σ) : → given by the couple [l(_0),  l(ζ) ∘ i_2], with  [l(_0),  l(ζ)] ∈×, and a map r() : → given by the couple [ r(_0),  r(ζ)] ∈× ;  such that = r() ∘ l().  These gives the assertions. 






 §.§ Lifting properties

On the model category  we will use here again the generic notation  for both factorization systems. And similarly we will denote by  both factorization systems on . In both cases we have () ⊆ and () ⊆. Recall that =[_0,_1]: → is an injective (trivial) cofibration in  if [_0,_1] ∈×. And β=[β_0,β_1]: [] → [] is an injective (trivial) fibration if [ δ , β_1] ∈×, where δ:_0 →(_1) ×_(_1)_0 is the induced map.


  §.§.§ Injective lifting properties


Let : → be a right Quillen functor. Then with the previous definition the following hold. 


  * Any lifting problem defined by an injective cofibration and  an injective trivial fibration has a solution.

  * Any lifting problem defined by an injective trivial cofibration and  an injective fibration has a solution.




The proof is the same for both assertions by considering the appropriate factorization system on  and . 

Consider a lifting problem in  defined by σ : [_0,_1] and β: [β_0,β_1] as follows.
 
    (0,18)*+="W";
    (0,0)*+="X";
    (30,0)*+="Y";
    (30,18)*+="E";
    @->^-(γ_0, γ_1)"X";"Y";
    @->_-(_0,_1)"W";"X";
    @->^-(θ_0,θ_1)"W";"E";
    @->^-(β_0,β_1)"E";"Y";

The image of this lifting problem under : → is a lifting problem defined by _1 and β_1. Therefore if [_1, β_1]∈×, then there is a solution s_1: _1 →_1 of this lifting problem in . Then by functoriality of , the map (s_1) is a solution to the induced lifting problem defined by (_1) and (β_1) in . Part of (s_1) being a solution gives an equality (γ_1)= (β_1) ∘(s_1). And  [γ]=[γ_0,γ_1] being a morphism in  gives the equality π_∘γ_0 = (γ_1) ∘. 


Now consider the map (s_1) ∘∈_(_0,(_1)) and the map γ_0 ∈_(_0,_0). Then by the above, it's not hard to see that these maps complete the pullback data 
    (_1) (_1) _0
 into a commutative square (π_∘γ_0= (β_1) ∘(s_1) ∘). Therefore, by the universal property of the pullback square there is a unique map: ζ:_0 →(_1) ×_(_1)_0,
making everything compatible. In particular γ_0 and (s_1) ∘  factor through ζ. 


Our original lifting problem in  defined by [σ] and [β] is represented by a commutative cube in . If we unfold it, we find that everything commutes in the diagram hereafter:

    (-60,20)*+_0="A";
    (20,30)*+_0="B";
    (-20,-10)*+_0="C";
    (60,0)*+_0="D";
    
    
    @->^θ_0"A";"B";
    @->__0"A";"C";
    @->^β_0"B";"D";
    @->_γ_0       "C";"D";
    
    (-60,-30)*+(_1)="X";
    (20,-20)*+(_1)="Y";
    (-20,-60)*+(_1)="Z";
    (60,-50)*+(_1)="W";
    @–>^    θ_1"X";"Y";
    @->_"A";"X";
    @->^β_1"Y";"W";
    @->^"C";"Z";
    @->_-"B";"Y";
    @->^π_"D";"W";
    @->_(_1)"X";"Z";
    @->_γ_1"Z";"W";
    
    (20,30)+(11,-25)*+(_1) ×_(_1)_0 ="E";
    @->>^δ"B";"E";
    @.>^"E";"D";
    @.>^"E";"Y";
    
    
    @.>^ζ"C";"E";
    
    @->_-(s_1)"Z";"Y";


As we see, we get a commutative square that corresponds to a lifting problem defined by the map _0:_0 →_0 and the map δ: _0 →(_1) ×_(_1)_0:
 
    (0,18)*+_0="W";
    (0,0)*+_0="X";
    (30,0)*+(_1) ×_(_1)_0="Y";
    (30,18)*+_0="E";
    @->^-ζ"X";"Y";
    @->_-_0"W";"X";
    @->^-θ_0"W";"E";
    @->^-δ"E";"Y";


Now it suffices to observe that this lifting problem has a solution if and only if our original lifting problem has a solution. Indeed, if s_0 : _0 →_0 is a solution to the previous lifting problem, then we have map [s]=[s_0,s_1]: → that is a solution to the original problem. Conversely given a solution [s]=[s_0,s_1]: → to the original lifting problem , then the component s_0: _0 →_0 is a solution to the lifting problem defined by _0 and δ: _0 →(_1) ×_(_1)_0. 


Finally one clear sees that the lifting problem defined by _0 and δ has a  solution s_0 ∈_(_0,_0) since [_0,δ] ∈×.







  §.§.§ Projective lifting properties


Let : → be a right Quillen functor. Then with the previous definition the following hold. 


  * Any lifting problem defined by a projective cofibration and a projective trivial fibration has a solution.

  * Any lifting problem defined by a projective trivial cofibration and  projective fibration has a solution.




We proceed in a dual manner as in the proof of Proposition <ref> with the same notation for both factorizations systems on  and .  

Consider a lifting problem in  defined by σ : [_0,_1] and β: [β_0,β_1] as follows.
 
    (0,18)*+="W";
    (0,0)*+="X";
    (30,0)*+="Y";
    (30,18)*+="E";
    @->^-(γ_0, γ_1)"X";"Y";
    @->_-(_0,_1)"W";"X";
    @->^-(θ_0,θ_1)"W";"E";
    @->^-(β_0,β_1)"E";"Y";

The image of this lifting problem under : → is a lifting problem defined by _0 and β_0. Therefore if [_0, β_0]∈×, then there is a solution s_0: _0 →_0 of this lifting problem in . Then by functoriality of , the map (s_0) is a solution to the induced lifting problem defined by (_0) and (β_0) in . Part of (s_0) being a solution gives an equality (θ_0)= (s_0) ∘(_0). And  [θ]=[θ_0,θ_1] being a morphism in  gives by adjunction a morphism ϱ() [(_0), θ_1]ϱ() in (). In particular we have an the equality ϱ() ∘(θ_0) = θ_1 ∘ϱ(). 


Now consider the map (s_0) ∘∈_((_0),_1) and the map θ_1 ∈_(_1,_1). Then using the previous equalities, it's not hard to see that these maps complete the pushout data 
    (_0) (_0)(_0) ϱ()_1
 into a commutative square (θ_1 ∘ϱ()= ϱ() ∘(s_0) ∘(_0)). Therefore, by the universal property of the pushout square, there is a unique map: ξ:_1 ∪^(_0)(_0) →_1,
making everything compatible. In particular  ϱ() ∘(s_0) and θ_1 factor through ξ. 


The original lifting problem in  defined by [σ] and [β] is represented by adjunction by a commutative cube in . If we unfold it, we find that everything commutes in the diagram hereafter:

    (-60,20)*+(_0)="A";
    (20,30)*+(_0)="B";
    (-20,-10)*+(_0)="C";
    (60,0)*+(_0)="D";
    
    
    @->^(θ_0)"A";"B";
    @->_(_0)"A";"C";
    @->_(s_0)"C";"B";
    @->^(β_0)"B";"D";
    @->_(γ_0)       "C";"D";
    
    (-60,-30)*+_1="X";
    (20,-20)*+_1="Y";
    (-20,-60)*+_1="Z";
    (60,-50)*+_1="W";
    @–>^    θ_1"X";"Y";
    @->_ϱ()"A";"X";
    @->^β_1"Y";"W";
    @->^ϱ()"C";"Z";
    @->_-"B";"Y";
    @->^ϱ()"D";"W";
    @->__1"X";"Z";
    @->_γ_1"Z";"W";
    
    (-40,-20)*+_1 ∪^(_0)(_0) ="E";
    @->_-i_1"X";"E";
    @^(->^δ"E";"Z";
    @->^ξ"E";"Y";
    
    
    @.>^"C";"E";


As we see, we get a commutative square that corresponds to a lifting problem defined by the maps δ: _1 ∪^(_0)(_0) →_1 and β_1: _1 →_1:
 
    (0,18)*+_1 ∪^(_0)(_0)="W";
    (0,0)*+_1="X";
    (30,0)*+_1="Y";
    (30,18)*+_1="E";
    @->^-γ_1"X";"Y";
    @->_-δ"W";"X";
    @->^-ξ"W";"E";
    @->^-β_1"E";"Y";


As in the injective case, it suffices to observe that this lifting problem has a solution if and only if our original lifting problem has a solution. Indeed, if s_1 : _1 →_1 is a solution to the previous lifting problem, then we have map [s]=[s_0,s_1]: → that is a solution to the original problem. Conversely given a solution [s]=[s_0,s_1]: → to the original lifting problem , then the component s_1: _1 →_1 is a solution to the lifting problem defined by δ: _1 ∪^(_0)(_0) →_1 and β_1. 


Finally one clear sees that the lifting problem defined by δ and β_1 has a  solution s_1 ∈_(_1,_1) since [_0,δ] ∈×.





 §.§ The model structures

We are able to state our theorems. 


  §.§.§ Injective model structure



Let : → be a right Quillen functor. Then with the previous definitions, the following hold. 
There is a model structure on the category = which may be described as follows. 


  * A map σ: → is a weak equivalence if and only if it's a level-wise weak equivalence.

  * A map σ: → is cofibration if it's an injective cofibration. 

  * A map σ: → is fibration if it's an injective fibration. 

  * We will denote this model category  by . 




The class of level-wise weak equivalences clearly satisfies the 2-out-of-3 property. The three classes of cofibrations, fibrations and weak equivalences are closed under composition and retracts. With Proposition <ref>,  Proposition <ref> and Lemma <ref>, one can easily verify that the axioms of a model structure hold (see for example <cit.>).


Let : → be a right Quillen functor. 
 

  * We have a Quillen adjunction  : ⇄ : ι, where  is left Quillen. 

  * We also have a Quillen adjunction  L_1: ⇄ :, where L_1 is left Quillen. 

  * We have a Quillen adjunction Γ: ⇄:, where Γ is left Quillen.

  * The functors  and  preserve the weak equivalences.



The functor : → preserves (trivial) cofibrations and (trivial) fibrations. So clearly it's simultaneously a left Quillen functor and a right Quillen functor. This gives the first two assertions.  


For Assertion (3), it suffices to observe that if σ=[_0, _1] is an injective trivial fibration then ()=[_0, (_1)] is an injective (trivial) fibration in  by definition ( being a right Quillen functor). Therefore  is right Quillen which means automatically that Γ is left Quillen.  

The last assertion follows from the definition of a level-wise weak equivalences. 




  
Cofibrantly generated
Let's now assume that  and  are cofibrantly generated in the sense of <cit.>. We will denote by  and  the respective sets of the generating cofibrations and trivial cofibrations for . And similarly let  and  be the respective sets of generating cofibrations and trivial cofibrations of . Given  s: A → B ∈(), we've introduced in Notation <ref> a map α_s: s (s,)_B in (). 

If  and  are cofibrantly generated (resp. combinatorial), then  is cofibrantly generated (resp. combinatorial). 


  * The set _=L_1() ∐Γ(α_) is a set of generating cofibrations in . 

  * The set _=L_1() ∐Γ(α_) is a set of generating trivial cofibrations  in . 




Corollary <ref> says that the category  is locally presentable. So we just need to prove that it is cofibrantly generated if we want to prove that it's a combinatorial model category.  


A map =[_0,_1] has the RLP with respect to all maps in L_1() ∐Γ(α_) if and only if it is simultaneously L_1()-injective and Γ(α_)-injective. One the one hand, using the adjunction L_1 ⊣,  =[_0,_1] is L_1()-injective if and only if _1 is -injective, if and only if, _1 is a trivial fibration.  


On the other hand, using the adjunction Γ⊣,  : [_0,_1] is Γ(α_)-injective if and only if ()=[_0,(_1)] is α_-injective. By Lemma <ref>, ()=[_0,(_1)] is α_-injective if and only if  δ:_0 →(_1) ×_(_1)_0 is -injective, if and only if δ is a trivial fibration. This gives Assertion (1). The second assertion is proved the same way. 



  §.§.§ Localizing the injective model structure

The functor : → preserves the weak equivalences. Let _ be the class of level-wise weak equivalences and let _ be the class of weak equivalences in . Denote by _L=^-1(_). Clearly _⊆_L.

Elements in _L will be called new weak equivalences. 


For any i ∈, Γ(α_i) is an injective cofibration and a new weak equivalence. 



If i: U → V, then α_i: i (i,_V)_V is an injective cofibration in ()  whose components are i and _V. Then Γ(α_i): Γ(i) [i, (_V)]Γ(_V) is an injective cofibration in  whose components are i and (_V),  by definition of Γ (see Proposition <ref>). (Γ(α_i)) = (_V) is an isomorphism, thus a weak equivalence and: 

    (Γ(α_i)) ∈_⇔Γ(α_i) ∈^-1(_)= _L.
 

We are now in the hypotheses of Smith's theorem to localize the injective model structure to get a model structure on  with the class of new weak equivalences _L and the same set of generating cofibrations _. We will use Smith's theorem and its consequences that can be found in  Beke <cit.>. 


Let : → be a right Quillen functor between combinatorial model categories. 


  * The data (_, _L) define a combinatorial model structure on  that will be denoted by ^+ and which may be described as follows.


  * A map =[_0,_1] is a weak equivalence if it's in _L, that is if _1 is a weak equivalence in .

  * The cofibrations are the injective cofibrations.

  * The fibrations are the maps satisfying the RLP with respect to every map that is simultaneously a cofibration and a weak equivalence. 


  * We have a Quillen equivalence : ^+  ⇄ : ι, where  is left Quillen. 

  * Any fibrant object =[_0,_1,] in ^+ satisfies the Segal condition, that is : _0 →(_1) is a weak equivalence.



Assertion (1) is a direct consequence of a result of  Beke <cit.>, which is itself a consequence of Smith recognition theorem.  

The Quillen equivalence is given by Proposition 4.4 in Beke <cit.> since ι:→ exhibits  as a full reflective subcategory and the functor  preserves the weak equivalences. This gives Assertion (2).  


For Assertion (3) we proceed by adjunction. If  is fibrant in ^+ , then the  unique map →∗ has the RLP with respect to all trivial cofibrations. And by Proposition <ref>, all elements of Γ(α_) are trivial cofibrations in ^+,  therefore →∗ has the RLP with respect to every element in Γ(α_) i.e,  is Γ(α_)-injective.  Now by Proposition <ref>, we know that  satisfies the Segal condition ( is a trivial fibration) and the assertion follows. 

The following gives an explicit description of the fibrations.

With the previous definition we have the following. 


  * The set _^+=_∐Γ(α_) is a set of generating trivial cofibrations.

  * A map θ: [_0,_1] is fibration in  ^+, if it's an injective fibration such that the induced map δ:_0 →(_1) ×_(_1)_0 is a trivial fibration in .




The second assertion is a consequence of the first considering the fact that a map θ  is Γ(α_)-injective if and only if the map δ is -injective, thus a trivial fibration in  ( see Lemma <ref>). 

 
To get the first assertion, we shall prove that (_) ∩_L ⊆(_^+).

Let : [_0,_1] be an element in  (_) ∩_L. Apply the small object argument to factor  as a relative -cell complex followed by a -injective map, = r() ∘ l(), where r()=[r()_0, r()_1] and l()=[l()_0, l()_1]. If we look at this factorization in  it gives a factorization of _1= r()_1 ∘ l()_1 with l()_1 ∈ cell() and  r()_1 is -injective map, thus a fibration. By assumption ∈_L which means that _1 is a weak equivalence in  and so is l()_1; therefore by 3-for-2, r()_1 is also a weak equivalence in . So we find that r()_1 is a trivial fibration i.e, r() is L_1()-injective.  


On the other hand, part of r() being -injective implies that r() is in particular Γ(α_)-injective since Γ(α_) ⊂. So in the end r() is _-injective, which means that r() is an injective trivial fibration. The map  is an injective cofibration so it posses the LLP with respect to r(). Putting this together with the factorization = r() ∘ l(), the retract argument as in <cit.> implies that   is a retract of l() ∈ cell(), thus ∈(). 






  §.§.§ Projective model structure



Let : → be a right Quillen functor. Then with the previous definitions, the following hold. 
There is a model structure on the category = which may be described as follows. 


  * A map σ: → is a weak equivalence if and only if it's a level-wise weak equivalence.

  * A map σ: → is cofibration if it's a projective cofibration. 

  * A map σ: → is fibration if it's a projective (= level) fibration. 

We will denote this model category  by . 



The class of level-wise weak equivalences clearly satisfies the 2-out-of-3 property. The three classes of cofibrations, fibrations and weak equivalences are closed under composition and retracts. With Proposition <ref>,  Proposition <ref> and Lemma <ref>, one can easily verify that the axioms of a model structure hold.


Let : → be a right Quillen functor. 
 

  * We have a Quillen adjunction  : ⇄ : ι, where  is left Quillen. 

  * We also have a Quillen adjunction  L_1: ⇄ :, where L_1 is left Quillen. 

  * We have a Quillen adjunction Γ: ⇄:, where Γ is left Quillen.

  * The functors  and  preserve the weak equivalences.



The functor : → preserves (trivial) cofibrations and (trivial) fibrations. So clearly it's simultaneously a left Quillen functor and a right Quillen functor. This gives the first two assertions.  


For Assertion (3), it suffices to observe that if σ=[_0, _1] is a projective trivial fibration then ()=[_0, (_1)] is a projective (trivial) fibration in  by definition ( being a right Quillen functor). Therefore  is right Quillen which means automatically that Γ is left Quillen.  

The last assertion follows from the definition of a level-wise weak equivalences. 




  
Cofibrantly generated
Let's now assume that  and  are cofibrantly generated as before. We have a similar theorem as in the injective model structure. 

If  and  are cofibrantly generated (resp. combinatorial), then  is cofibrantly generated (resp. combinatorial). 


  * The set _=L_1() ∐Γ(L_0())= L_1() ∐^+() is a set of generating cofibrations in . 

  * The set _=L_1() ∐Γ(L_0())=L_1() ∐^+() is a set of generating trivial cofibrations  in . 




Like in the injective case, Corollary <ref> says that the category  is locally presentable. So  it remains to prove that it is cofibrantly generated to prove that it's a combinatorial model category.  


A map =[_0,_1] has the RLP with respect to all maps in L_1() ∐Γ(L_0()) if and only if it is simultaneously L_1()-injective and Γ(L_0())-injective. One the one hand, using the adjunction L_1 ⊣,  =[_0,_1] is L_1()-injective if and only if _1 is -injective, if and only if, _1 is a trivial fibration.  


On the other hand, using the adjunction ^+ ⊣ (_0 ∘),  =[_0,_1] is ^+()-injective if and only if _0()=_0 is -injective, if and only if _0 is a trivial fibration. This gives Assertion (1). The second assertion is proved the same way. 



  §.§.§ Localizing the projective model structure

Given  s: A → B ∈(), we've introduced in Notation <ref> a map ζ_s: s (s,)_B in () and we've defined the universal projective localizing set ζ_. Let _proj= Γ(ζ_)={Γ(ζ_i) }_i ∈. 
 

Like in the injective model structure, we consider the same class _L=^-1(_) of  new weak equivalences. 


For any i ∈, Γ(ζ_i) is a projective cofibration and a new weak equivalence. 



If i: U → V, then ζ_i: i (i,j_0) j_1 is an projective cofibration in () by construction (see Proposition <ref>). The components of ζ_i are i and j_0. With the notation introduced in Notation <ref>, Γ(ζ_i): Γ(i) [i, (j_0)]Γ(j_1) is therefore a projective cofibration in  whose components are i and (j_0),  since Γ is left Quillen (see Proposition <ref>). (Γ(ζ_i)) = (j_0) is a trivial cofibration because j_0 is trivial cofibration and  is left Quillen. In particular, (j_1)=(Γ(ζ_i)) is a weak equivalence; that is:

    (Γ(ζ_i)) ∈_⇔Γ(ζ_i) ∈^-1(_)= _L.
 

Just like in the injective case, we are in the hypotheses of Smith's theorem to localize the projective model structure to get a model structure on  with the class of new weak equivalences _L and the same set of generating cofibrations _. 

Let  be a tractable model category and let : → be a right Quillen functor between combinatorial model categories.


  * The data (_, _L) define a combinatorial model structure on  that will be denoted by ^+ and which may be described as follows.


  * A map =[_0,_1] is a weak equivalence if it's in _L, that is if _1 is a weak equivalence in .

  * The cofibrations are the projective cofibrations.

  * The fibrations are the maps satisfying the RLP with respect to every map that is simultaneously a cofibration and a weak equivalence. 


  * We have a Quillen equivalence : ^+  ⇄ : ι, where  is left Quillen. 

  * Any fibrant object =[_0,_1,] in ^+ satisfies the Segal condition, that is : _0 →(_1) is a weak equivalence.



Assertion (1) is a direct consequence of Proposition 4.2 in Beke <cit.>, which is itself a consequence of Smith recognition theorem.  

The Quillen equivalence is given by Proposition 4.4 in Beke <cit.> since ι:→ exhibits  as a full reflective subcategory and the functor  preserves the weak equivalences. This gives Assertion (2).  


For Assertion (3) we proceed by adjunction. If  is fibrant in ^+ , then the  unique map →∗ has the RLP with respect to all trivial cofibrations. And by Proposition <ref>, all elements of Γ(ζ_) are trivial cofibrations in ^+,  therefore →∗ has the RLP with respect to every element in Γ(ζ_) i.e,  is Γ(ζ_)-injective.  Now by Proposition <ref>, we know that  satisfies the Segal condition ( is a weak equivalence between fibrant objects) and the assertion follows. 



The identity functor : ^+ →^+ is a left Quillen functor which is a Quillen equivalence.



It's the identity functor and we have the same class of new weak equivalences. Moreover any projective cofibration is an injective cofibration. 


Unlike the injective model structure, it's difficult to give an explicit description of the fibrations in general without any assumption of (left)  properness on . This will require some work that will make the paper very long. We will do this in a future work.


We will close this section with some observations.


  * If  is tractable, we can show that ^+ is the left Bousfield localization with respect Γ(α_). Indeed, elements in Γ(α_)) are maps between cofibrant objects so there is no need to take a cofibrant approximation of these maps to define their image under a left derived functor.

  * There are other model structure on  that one can get from the injective and the projective model structure on  with the adjunction Γ⊣. They will be considered in a future work.  






§ GENERATING THE SEGAL CONDITIONS






 §.§ Factorization lemma and the small object argument
 

Recall that for any object []=[_0,_1,] we have _1= ([]). In the adjunction ι: ⇄ :, the unit η_1([]): [] →ι(_1) is given by the pair [π_, __1] ∈_(_0,(_1)) ×_(_1,_1) (see Proposition <ref>).  


Consider the set = {Γ(α_i) }_i ∈ introduced previously. 
Our goal here is to apply the small object argument (see <cit.>, <cit.>) with respect to  and show that the unit  
    η_1([]): [] →ι(_1)
 can be factored as a relative -cell complex followed by a -injective map. Now observe that since the codomain of each α_i is the identity, the codomain of Γ(α_i) is also the identity. Moreover as Γ is a left adjoint, it preserves any kind of colimits; in particular it distributes over coproducts. It follows that the codomain of any coproduct ⊔_NΓ(α_i) ≅Γ(⊔_Nα_i) of elements of  is a essentially a coproduct of identities. Following  Proposition <ref>, if we take the pushout of such a product along an attaching map Domain(⊔_NΓ(α_i)) → [], we won't change the object _1 since  preserves colimits and the pushout of an isomorphism is an isomorphism.  
 

It follows from this description that the factorization given by the small object argument, amounts to factor the map : _0 →(_1) as a relative -cell complex followed by an -injective map, that is a trivial fibration, leaving _1 essentially unchanged.


We shall now describe explicitly the pushout of a single element of . 

Given a morphism θ_1: B →(_1) in  we will denote by ϱ(θ_1):  B →_1 its adjoint-transpose in the adjunction : ⇄ :.

Let  θ=(θ_0,θ_1) : s → be a map in () whose adjoint-transpose φ(θ)=[θ_0, ϱ(θ_1)]:Γ(s) → [] is  represented by the commutative square:
 
    (0,18)*+A="W";
    (0,0)*+( B)="X";
    (30,0)*+(_1)="Y";
    (30,18)*+_0="E";
    @->^-(ϱ(θ_1))"X";"Y";
    @->_-η_B ∘ s"W";"X";
    @->^-θ_0"W";"E";
    @->^-"E";"Y";

Then the following hold.


  * The pushout of  Γ(α_s): Γ(s) →Γ(_B) along φ(θ) is the map 
    l(Γα_s)=[l_s(), __1]: [] → [] ∪^Γ(s)Γ(_B)
 displayed by the commutative square:
 
    (0,18)*+_0="W";
    (0,0)*+(_1)="X";
    (30,0)*+(_1)="Y";
    (30,18)*+_0 ∪^A B="E";
    @=^-"X";"Y";
    @->^-"W";"X";
    @->^-l_s()"W";"E";
    @->^-r_s()"E";"Y";

where l_s() is the pushout of s along θ_0 and r_s() is the canonical map induced by universal property of the pushout.  

  * The pushout-object []= [] ∪^Γ(s)Γ(_B) is given by []=[_0, _1, π_] with 

    _0=_0 ∪^A B,   _1= _1,   π_=r_s()


  * We have a factorization of : 
_0 _0 ∪^A B   (_1)
that induces in turn a factorization of the unit 

    η_1([]): [] ι(_1)= []  [] ι(_1)






This directly follows from Proposition <ref> where we explain how we compute colimits, and in particular pushouts in  . The pushout square is represented by a commutative cube in  that we display below for the reader's convenience. 


    (-30,20)*+A="A";
    (20,20)+(0,10)+(0,-4)*+_0="B";
    (-30,0)*+( B)="C";
    (20,0)+(0,10)+(0,-4)*+(_1)="D";
    @->^θ_0"A";"B";
    @->_η_B ∘  s"A"+(0,-2);"C";
    @.>^<<<<<<<<<<<<<"B"+(0,-2);"D";
    @.>^>>>>>>>>>(ϱ(θ_1))"C";"D";
    
    (-30,20)+(-15,-10)+(40,0)+(0,5)*+B="E";
    (20,20)+(-15,-10)+(40,0)+(0,10)+(0,5)+(0,-4)*+_0 ∪^A B ="F";
    (-30,0)+(-15,-10)+(40,0)+(0,5)*+( B)="G";
    (20,0)+(-15,-10)+(40,0)+(0,10)+(0,5)+(0,-4)*+(_1)="H";
    @->^"E";"F";
    @->_>>>>η_B"E"+(0,-2);"G";
    @->^r_s()"F";"H";
    @->^"G";"H";
    
    @->^s"A";"E";
    @->^l_s()"B";"F";
    @=^"C";"G";
    @=^"D";"H";
  

In this cube the lower and upper faces are pushout squares in . Clearly we have a factorization of :

    _0 _0 ∪^A B   (_1).


The map B →_0 ∪^A B is the canonical map going to the colimit and it gives a factorization of θ_1:

    θ_1 = B _0 ∪^A B   (_1).

The remaining assertions are clear. 




 §.§ Identifying a pushout

We also need a lemma that reverses the process.


Let []=[_0,_1,] be an object of  and consider a factorization of = r() ∘ l() displayed as follows:

    : _0 →(_1) = _0  E_0   (_1)

Let []=[_0, _1, π_] be the object of  defined by 

    _0=E_0,   _1= _1,   π_=r().

Then the following hold.


  * The factorization induces a tautological map: ψ_(l,r): [] → [] in :

    (0,18)*+_0="W";
    (0,0)*+(_1)="X";
    (30,0)*+(_1)=(_1)="Y";
    (30,18)*+E_0=_0="E";
    @=^-"X";"Y";
    @->^-"W";"X";
    @->^-l()"W";"E";
    @->^-r()"E";"Y";
 

  * If l() is a cobase of some map s : A → B then the map ψ_(l,r) is a cobase change of the map Γ(α_s).




The first assertion is clear. For Assertion (2) we just need to define the attaching map Γ (s) → [] in  that will be used to get a pushout data:

    Γ(_B) Γ (s) → [].

 Now by hypothesis the map l() fits in a pushout square: 
 
    (0,18)*+A="W";
    (0,0)*+B="X";
    (30,0)*+E_0≅_0 ∪^A B="Y";
    (30,18)*+_0="E";
    @->^-"X";"Y";
    @->^-s"W";"X";
    @->^-"W";"E";
    @->^-l()"E";"Y";


We extend this commutative square with the map r(): E_0 →(_1).  This gives a commutative square that defines a map θ : s → in the category (). The adjoint transpose of θ gives the attaching map φ(θ)=[θ, ϱ(θ_1)]: Γ(α_s) → [] as in the previous lemma. And by the previous lemma one clearly sees that ψ_(l,r) is the cobase change of Γ(α_s) along φ(θ). 






 §.§ Cell complexes
 The last piece of ingredient we need is the following lemma which is a consequence of the previous ones. It's the general form of the second assertion in Lemma <ref>.


Let S be a small set of maps of  and denote by α_S = {α_s}_s ∈ S. Let []=[_0,_1,] be an object of  and consider a factorization of = r() ∘ l() displayed as follows:

    : _0 →(_1) = _0  E_0   (_1)

Let []=[_0, _1, π_] be the object of  defined by 

    _0=E_0,   _1= _1,   π_=r().


If the morphism l() is a relative S-cell complex, then the map ψ_(l,r) : [] → [] in Lemma <ref> is a relative Γ(α_S)-cell complex.


This is just a repetition of the second assertion of Lemma <ref>. Indeed, the map l() is a transfinite composition of maps l()_k; where each of them is a cobase change of some map s_k ∈ S.  In particular for each k we have a factorization of  : =r()_k∘ l()_k.

By the same construction as in Lemma <ref>,  this factorization defines maps ψ_(l_k,r_k): []→_k whose transfinite composite is the map ψ_(l,r).  Thanks to Lemma <ref>  we know that each ψ_(l_k,r_k) is a pushout of Γ(α_s_k).

We see that ψ_(l,r) is a transfinite composition of maps that are pushouts of elements of Γ(α_S), thus it's a relative Γ(α_S)-cell complex as claimed.



In the previous lemma we can generalize the statement when the map _0  E_0 is a retract of a map l'():_0'  E'_0 that is a relative S-cell complex. In that case the map ψ_(l,r) : [] → [] would be a retract of relative Γ(α_s)-cell complex. 

To see this, first observe that from the definition of a retract, there is map a:E_0 → E'_0 and a retraction b:E_0' → E_0 such that b∘ a= _E_0.  Similarly we have a map u:_0 →_0' and a retraction v:'_0 →_0 such that v ∘ u = __0. If we set '= r() ∘ b ∘ l'_ we get an object ['] ∈ to which we can apply the construction of the previous lemma and get a map ψ_'(l',r').  
It's not hard to see that the map  is a retract of π_' and that ψ_(l,r) is a retract  of  ψ_'(l',r').




 §.§ Factorization Theorem

We give here one of the central results of this section. We show that the unit in the adjunction ⊣ι can be factored as as relative  -cell complex followed by a level-wise trivial fibration. 

Let  be a cofibrantly generated model category with  the generating set of cofibrations . Consider the sets α_=  {α_i}_i ∈  and let = Γ(). Then for any object ∈ the following hold.



  * We can factor the unit η_1,: →ι(()) as a relative -cell complex followed by a map that is a level-wise trivial fibration:

    ι(()).


  * The map  ι(()) is the unit of the adjunction for the object . In particular the unit η_1, is a level-wise weak equivalence in .


 




Apply the small object argument in , to get a factorization of  as a relative -cell complex followed by an -injective map; that is a trivial fibration: 

    : _0 →(_1) = _0 l() E_0    [∼]r()(_1)


By the same construction as in Lemma <ref> we define the object ∈ with:

    _0=E_0,   _1= _1,   π_=r().

In particular ()=_1 and the unit of the adjunction for the object  is the map η_1,=[r(), __1]: →ι(()) which is a level-wise trivial fibration since r and __1 are.  


Let's denote simply by ψ, the map ψ(l,r)=[l(),__1]: → constructed  in Lemma <ref>. Now it's clear that the composite η_1,∘ψ is just η_1, since:

    η_1,∘ψ=[r(), __1] ∘ [l(),__1]= [r()∘ l(), __1]= [, __1]=η_1,.

This completes the proof of the Lemma.





§ STABLE HOMOTOPY THEORY

In this section we assume that = and that : → is an endofunctor which is right Quillen. The example that we shall keep in mind is = Ω: _∗→_∗. The left adjoint : → is to be thought as the suspension functor Σ = S^1 ∧ -. We will write ^n for the composite of  with itself n times and similarly we have ^n; with the convention ^0= _= ^0. 


There many references in the literature on stable homotopy theory. Classical ones include Adams <cit.>, Bousfield-Friedlander <cit.>, Lima <cit.>. Modern foundations are to be found in  Elmendorf-Kriz-Mandell-May (EKMM) <cit.>, Hovey <cit.>, Schwede <cit.> and the many references therein. In the world of ∞-categories we shall refer the reader to Lurie <cit.>.



A spectrum X is a sequence of pointed simplicial sets (X_n)_n ∈ together with basepoint preserving maps γ: S^1 ∧ X_n → X_n+1.  A map f: X → Y of spectrum is a sequence (f_n) of maps f_n:X_n → Y_n in _∗ such that the following commutes for every n ≥ 0.
 
 
    (0,18)*+S^1 ∧ X_n="W";
    (0,0)*+ X_n+1="X";
    (30,0)*+Y_n+1="Y";
    (30,18)*+S^1 ∧ Y_n="E";
    @->^-f_n+1"X";"Y";
    @->^-γ_X"W";"X";
    @->^-∧ f_n"W";"E";
    @->^-γ_Y"E";"Y";


If we take a general left adjoint T that generalizes the suspension functor, we find the definition of spectrum given by Hovey <cit.>.
With the adjunction (S^1 ∧ -) ⊣Ω, the previous definition is equivalent to the one below that we shall work with (see Adams <cit.>).

A spectrum X is a sequence of pointed simplicial sets (X_n)_n ∈ together with basepoint preserving maps ε: X_n →Ω(X_n+1).  A map f: X → Y of spectrum is a sequence (f_n) of maps f_n:X_n → Y_n in _∗ such that the following commutes for every n ≥ 0.
 
 
    (0,18)*+ X_n="W";
    (0,0)*+Ω (X_n+1)="X";
    (30,0)*+Ω(Y_n+1)="Y";
    (30,18)*+ Y_n="E";
    @->^-Ω(f_n+1)"X";"Y";
    @->^-ε_X"W";"X";
    @->^-f_n"W";"E";
    @->^-ε_Y"E";"Y";


The category of (pre)spectra will be denoted by .

For every n we have an object [X_n] ∈↓Ω given by [X_n]=[X_n, X_n+1, ε]. This gives a sequence of objects in  ↓Ω. In virtue of this, we will study spectra through sequences of objects in . 


We have a natural forgetful functor P: →∏_n∈ that maps X to ([X_n])_n ∈.  Projecting the nth factor yields a functor P_n: → that maps X ↦ε_n.  As n runs through , we have a Quillen-Segal theory on the category . Our goal here is to show that we can transfer the model structure from ∏_n∈ to the category  with the classical argument of right induced model structure (see for example <cit.>, <cit.>). On  we have an injective model structure _inj(Theorem <ref>) and the projective model structure _proj (Theorem <ref>).
Each model structure will induce a model structure on  called projective and injective model structure just like for (unbouded)  chain complexes (see <cit.>). But first we need to set up some definitions and properties.  


 §.§ -sequences and spectra


As pointed out by Adams, the indexing set for spectra can be either  or . We will index our sequences over  and mention explicitly when we index over . Many of the constructions that will follow hold for any ordinal ⊆, possibly finite. We will denote by  the discrete category associated to the set . Similarly we have  and  the discrete category associated to  and  respectively.




  * So far if  is an object of , we've written =[_0, _1, π_: _0 →(_1)]. But when we consider sequences (_n) there will be multiple indexes, so we will write instead =[^0, ^1, : ^0 →(^1)] to put the indexing integer as _n^0 . We've change the letter π into  because the letter π refers to homotopy groups. 

  * In the upcoming definitions we will be talking about -sequences but this simply means -indexed sequences and not a λ-sequence as in <cit.>. 






Let :→ be a right Quillen functor. 



  * A -sequence with coefficient in  is  an object =(_n)_n ∈∈∏_n∈, in that for every n ∈, 
    _n=[_n^0, _n^1, _n: _n^0 →(_n^1)].

Equivalently  determines a functor : _disc→, that maps n ↦_n.

  * Say that a -sequence  is linked if for every n there is an isomorphism:

    τ_n:_n^1 ≅_n+1^0.

We will denote by (,τ) the linked sequence. 

  * Say that a morphism : (,τ) (_n) (,τ') is a linked morphism, if for every n, we have τ'_n ∘_n^1 = _n+1^0 ∘τ_n, or equivalently  _n+1^0=τ'_n ∘_n^1 ∘τ_n^-1. In other words the following commutes.

 
    (0,18)*+_n^1="W";
    (0,0)*+_n+1^0="X";
    (30,0)*+_n+1^0="Y";
    (30,18)*+_n^1="E";
    @->^-_n+1^0"X";"Y";
    @->^-τ_n_≅"W";"X";
    @->^-_n^1"W";"E";
    @->^-τ'_n_≅"E";"Y";


  * Say that a linked sequence (,τ) is strictly linked if for every n, τ_n is the identity morphism.







Pictorially, we can represent an unlinked -sequence by a `stair diagram':

    (-30,18)*+_n^0="W";
    (0,0)*+_n+1^0="X";
    (30,0)*+(_n+1^1)="Y";
    (0,18)*+(_n^1)="E";
    @->^-_n+1"X";"Y";
    @.^-?"E";"X";
    @->^-_n"W";"E";
 
When the sequence is linked we can move up the lower stair and get a map (τ_n) ∘_n : _n^0 →(_n+1^0). Each of this map has an adjoint ( _n^0) →_n+1^0. If  is the suspension functor given by = S^1 ∧ -, then the previous map takes the form S^1 ∧_n^0 →_n+1^0, and we see that we have a spectrum as in Definition <ref> above, when  the sequence is indexed over n ∈. 
The sequence (_n^0,(τ_n) ∘_n) is the same spectrum as in Definition <ref>. 


With the previous notation we have:


  * The category of -sequences is equivalent to the functor category (, )= ()^.

  * Linked -sequences and linked morphisms form a category Sp_^+() whose objects are called -prespectrum.

  * Strictly linked -sequences form a subcategory Sp_() ⊂ Sp_^+(). 




For a linked -sequence (,τ), define the associated -prespectrum to be the sequence (_n^0,(τ_n) ∘_n).



The inclusion Sp_()  Sp_^+() is an equivalence of categories. A quasi inverse is the functor Θ: Sp_^+() → Sp_()  defined as follows. 


  * Θ maps (,τ) to the strictly linked sequence =(_n) given by 
    _n= [_n^0, _n+1^0,(τ_n) ∘_n ].


  * Θ maps =(_n) to =(_n) given by: 

    _n= (_n^0,_n^1) :=(_n^0, τ'_n ∘_n^1 ∘τ_n^-1).





Clearly for every n we have _n^1=_n+1^0 by construction. This means that Θ((,τ)) ∈ Sp_(). By the same reasoning Θ() is a linked morphism if  is. These constructions are clearly functorial and Θ is well defined.

The proposition will be proved as soon as we establish that  ↦ is an isomorphism of hom-sets: 

    Θ: _(,) ≅_(,).

And this is clear since we have an inverse  that takes (_n^0,_n^1) ↦ (_n^0, τ'_n^-1∘_n^1 ∘τ_n)=(_n^0, τ'_n^-1∘_n+1^0 ∘τ_n).  


With the previous proposition we will focus on the strictly linked sequences to simplify the constructions.

In the standard situation of spectra,  is a category of pointed objects, and therefore the initial object ∅ and the terminal object ∗ are uniquely isomorphic to form a zero object in . In the upcoming results we will only use this property when really needed. So we shall write distinctly ∅ for the initial object and ∗ for the terminal object. 




With the previous definition we will adopt the following notation. 


  * The above  forgetful functor P: Sp_() → will take the form P: Sp_() → ()^.

  * Similarly we will denote by P_n: Sp_() →, the composite of P followed by the nth projection: P_n()=_n.





The category Sp_() is complete and cocomplete and the following hold. 



  * The functor P:  Sp_() → creates limits and colimits which are computed level-wise.


  * The functor P has a left adjoint and this adjunction is monadic.
 

We differ the proof for the moment because we need the following intermediate result that will simplify the proof of the proposition.



The projection functor P_n has a left adjoint  Υ_n: →  defined as follows.


  * For =[_0,_1, ] ∈ we define Υ_n()=(_k)_k ∈∈∏_k∈,  by the formulas:


  * _k= L_1(∅)= [∅, ∅, ∅!(∅)] if k<n-1

  * _k= L_1(_1)= [∅, _0, ∅!(_0)] if k=n-1

  * _k= = [_0,_1, _0 (_1)] if k=n


  * _k= ^+(^k-(n+1)(_1))= [^k-(n+1)(_1), (^k-(n+1)(_1)),  η_^k-(n+1)(_1)] if k ≥ n+1; where η is the unit of the adjunction ⊣. Equivalently we have the inductive formula:
 
    _k= ^+(_k-1^1)=[_k-1^1,(_k-1^1), η__k-1^1],    k ≥ n+1 .



  * If =[_0; _1] ∈_(,) we define Υ_n()=(_k)_k ∈ by the formulas:


  * _k= [_∅, _∅] if k<n-1

  * _k= [_∅,_0]  if k=n-1

  * _k== [_0,_1] if k=n

  * _k= ^+(^k-(n+1)(_1))= [^k-(n+1)(_1), ( ^k-(n+1)(_1)) ] if k ≥ n+1. In a short form we have the inductive formula:

    _k= ^+(_k-1^1)=[_k-1^1, (_k-1^1)],    k ≥ n+1.




Before we give the proof, let's take a moment to see what Υ_n really does. If we are given an object =[_0,_1, _0 (_1)], by adjunction the map  corresponds to a unique map (_0) ϱ()_1. Then  Υ_n() is the prespectrum given by the sequence:

    ∅, ⋯, ∅, (_0)_n th,_1_(n+1)th, (_1)_(n+2)th, ^2(_1),⋯, ^k-(n+1)(_1)_k ≥ (n+3)⋯,

with the connecting morphisms:
 
    ⋯, ∅_∅∅, (_0) ϱ()_1, (_1) (_1), ^2(_1) ^2(_1),⋯_^k-(n+1)(_1), ⋯ .
 
 
In our formalism, we consider the adjoint definition of the spectrum given by the following sequence and its connecting morphisms.

    ∅, ⋯, ∅, _0_n th,_1_(n+1)th, (_1)_(n+2)th, ^2(_1),⋯, ^k-(n+1)(_1)_k ≥ (n+3)⋯.


    η_∅, ⋯, ∅!(_0), _0 (_1), _1 η__1(_1), (_1) η_(_1)^2(_1),⋯η_^k-(n+1)(_1) , ⋯ .
 



It's clear that Υ_n() is a linked sequence and that Υ_n() is a linked morphism. The constructions are clearly functorial so the previous data define a functor.  

The lemma will follow as soon as we show that for every ∈ and for every ∈, there is a functorial isomorphism of hom-sets:

    φ: _(,_n) ≅_(Υ_n(), ).

 This is straightforward but we give the proof for the reader's convenience. 
If ∈_(,_n) we define  φ()= (_k) ∈_(Υ_n(), ) as the linked morphism given by the formulas:


  * _k: L_1(∅) →_k is adjoint map to the unique morphism ∅!_k^1, in the adjunction L_1 ⊣ of Theorem <ref>, if k<n-1.  The components of _k are _k^0=∅!_k^0  and _k^1= ∅!_k^1 the unique morphisms from the initial object. 

  * _k=[_∅,_0]: L_1(_0) →_n-1 is the adjoint map to _0: _0 →_n^0=_n-1^1  if k=n-1. Thus this morphism is uniquely determined by _0. 

  * _k=: →_n if k=n

  * Inductively for k ≥ n+1, _k: _k= ^+(_k-1^1) →_k is the adjoint map to _k-1^1:  _k-1^1 →_k-1^1=_k^0.


The  function φ is clearly  1-1 because the nth component is . And we have an inverse function φ^-1:_(Υ_n(), ) →_(,_n) that projects the nth component. And the lemma follows. 




 For n ∈, define the nth Dirac mass functor 
 
    δ_n: →(,)

as the left adjoint to the nth evaluation _n: (,) → defined by _n()=_n. 

For an object ∈, δ() is given by:


  * δ_n()_k= if k = n.

  * δ_n()_k= ∅ if k ≠ n.
 


  * We have an functorial isomorphism of hom-sets: 

    _(, _n) ≅_()^(δ(), ).


  * For every =(_n) ∈ ()^ we have ≅∑_n δ_n(_n) =  ∐_n δ_n(_n).   






  
Proof of Proposition <ref>
We can give the proof of Proposition <ref> as follows. 

That P:  Sp_() → ()^ creates limits and colimits is obvious and we leave it as an exercise for the reader. So Assertion (1) is clear.  


Since  is complete, coproducts exist and by a classical argument we can find a left adjoint to P as follows. For =(_n) ∈∏_n∈, let Υ: ∏_n∈→ be the functor defined by:

    Υ()= ∑_nΥ_n(_n)= ∐_n Υ_n(_n).
 
For every ∈ we have the following isomorphism of hom-sets:


    _(Υ(), )     = _(∐_n Υ_n(_n), ) ≅∏_n _(_n, P_n()) 
       ≅∏_n _(_n, _n) =  ∏_n _(_n, _n())  
       ≅∏_n _()^(δ(_n), )
       ≅_()^(∐_nδ(_n), )
       ≅_()^(, )

Finally this adjunction is monadic by Beck monadicity theorem (see <cit.>). Indeed, P reflects isomorphisms because if =(_k) is a  linked morphism such that each _k has an inverse _k^-1 in , then the sequence (_k^-1) is a linked morphism which is the inverse of . Moreover as mentioned before the functor P creates colimits in particular it creates coequalizers of P-split pairs. And with the left adjoint Υ we are in the hypotheses of Beck's theorem. 
This ends the proof of the Proposition.




 §.§ Homotopy of -sequences

The category of -sequences is the diagram category . By Theorem <ref> 	and Theorem <ref>, there is an injective and projective model structure on . We will then consider the homotopy theory of the diagram category (, _inj) and (, _proj).  Model structures on diagram categories have been studied for decades and they are well known in the literature (see for example <cit.>, <cit.>, <cit.>). If  is simply cofibrantly generated we will consider the projective model structure on . And if  is combinatorial then we will also consider the injective model structure on the diagram category .

As before, given a subset ⊆ we have a restriction →(,) and the constructions that will follow hold for the diagram category  (,)




Let : → be a morphism in  and consider a model structure on .


  * Say that  is a level weak equivalence if for every n, _n: _n →_n is a level-weak equivalence.

  * Say that  is a level (trivial) cofibration  if for every n, _n: _n →_n is a level (trivial) cofibration.

  * Say that  is a level (trivial) fibration  if for every n, _n: _n →_n is a level (trivial) cofibration.
 



Let  be a combinatorial model category and consider the category  with the injective model structure or the projective model structure.


  * There is an injective model structure on the category  where the cofibrations and the weak equivalences are level-wise.

  * There is a projective model structure on the category  where the cofibrations and the weak equivalences are level-wise.

Both model categories are combinatorial and left proper if  is in addition left proper.



The existence of these model structures can be found for example in <cit.>. The left properness follows from the fact that _inj and _proj are left proper since  pushouts are computed object wise. We've seen that  is locally presentable, therefore  is also locally presentable.





  §.§.§ Strict model structures for prespectra

We wish to transfer the model structure from  to . The ingredient that is needed is the following lemma.

Let : [_0,_1] be a map in . Then the following hold. 


  * For every n, Υ_n() : Υ_n() →Υ_n() is a level (trivial) cofibration if  is a level (trivial) cofibration

  * For every n and for any pushout square as follows 
 
    (0,18)*+Υ_n()="W";
    (0,0)*+Υ_n()="X";
    (30,0)*+Υ_n() ∪^Υ_n()="Y";
    (30,18)*+="E";
    @->^-"X";"Y";
    @->^-Υ_n()"W";"X";
    @->^-"W";"E";
    @->^-u"E";"Y";

the map u:→Υ_n() ∪^Υ_n() is a level trivial cofibration if  is.  In particular  it's a level weak equivalence. 




By definition given  : [_0,_1], 
Υ_n()=(_k)_k ∈ is given by the formulas:


  * _k= [_∅, _∅] if k<n-1

  * _k= [_∅,_0]  if k=n-1

  * _k== [_0,_1] if k=n

  * _k= ^+(^k-(n+1)(_1))= [^k-(n+1)(_1), ( ^k-(n+1)(_1)) ] if k ≥ n+1.

Since  is a left Quillen functor, it preserves the cofibrations and the trivial cofibrations. Therefore for every k≥ n+1, ^k-(n+1)(_1) and ( ^k-(n+1)(_1)) are (trivial) cofibrations if _1 is a (trivial) cofibration. This means that _k is a level (trivial) cofibration if _1 is for k ≥ n+1. 

For k ≤ n, the components of _k are one of the maps _∅, _0 or _1. So clearly _k is a level (trivial) cofibration if _0 and _1 are (trivial) cofibrations. This gives Assertion (1).  


Assertion (2) is clear because colimits and in particular pushouts in the category   are computed in . Finally since _inj is a model category, the cobase change of any trivial cofibration is again a trivial cofibration.



  
Projective model structure 
In what follows, we endow  with either the injective or the projective model structure. 


Let  be a cofibrantly generated model category and consider the projective model structure on . Then there is a right induced model structure on the category  which is cofibrantly generated which may be described as follows. 


  * A map  is a weak equivalence if P() is a weak equivalence in _proj

  * A map  is a fibration if P() is a fibration in _proj

  * A map  is a cofibration if  it posses the LLP with respect to any map that is simultaneously a fibration and a weak equivalence.

The adjunction Υ: ⇄ : P is a Quillen adjunction and for every n, _n= ∘_n ∘ P_n: → is right Quillen.



This is a classical argument of Quillen <cit.>. See also Goerss-Jardine <cit.> or Beke <cit.> <cit.>. The main ingredient is precisely Lemma <ref> above.



If in addition  is left proper (resp. combinatorial) then   is left proper (resp. combinatorial).



Left properness follows from the fact that pushouts in  are computed object wise in , and clearly _inj and _proj is left proper if  is. To prove that  is combinatorial boils down to show that it's locally presentable. This can be proved directly but we can take a shortcut using the fact that the adjunction Υ⊣ P is monadic and the monad is clearly finitary i.e, it preserves filtered colimits. 



If  is combinatorial, we have a similar theorem for the injective model structure on .



  
Bousfield-Friedlander strict model structure
We can be more specific when we consider the “projective-projective” model category (,_proj)_proj. We do not require  to be cofibrantly generated. And the argument holds for a bounded below ordinal ⊂. 



Let : → be a right Quillen functor and let _proj be  the projective model structure   as in Theorem <ref>.
Then there is a model structure on  called the projective strict model structure which may be described as follows. 



  * A map : → is a weak equivalence if for every n, _n: _n →_n is a weak equivalence in _proj, that is a level weak equivalence. 

  * A map : → is a fibration if for every n, _n: _n →_n is a fibration in _proj, that is a level fibration. 

  * A map : → is a cofibration if for every n, _n: _n [_n^0,_n^1]_n n is a cofibration in _proj, that is for every n , _n^0 is a cofibration and the canonical map δ:_n+1^1 ∪^(_n^0)(_n^0) →_n+1^1 is a cofibration in .

  * If  is cofibrantly generated (resp. left proper, resp. combinatorial) then  is cofibrantly generated (resp. left proper, resp. combinatorial). We will denote it by _proj.

 


One get the factorization axioms and the lifting properties inductively because  is bounded below and is a direct Reedy category.  The main reason is that given a map _n: _n [_n^0,_n^1]_n in , when we factor it in the projective  model structure we always start with a factorization of _n^0. We also do the same thing when we construct a solution for lifting properties. Proceeding this way we can factor any linked morphism  being careful to take the factorization of _n^1 and apply them to _n+1^0=_n^1 to build in  a factorization of _n+1=[_n+1^0,_n+1^1]. One gets inductively a the required factorization (resp lifting) at each step by linked morphisms.The same  argument is used by Rosický <cit.>. 


If  is cofibrantly generated (resp. combinatorial, resp left proper) then this model structure coincides with the right induced model structure transferred from (,_proj)_proj with Theorem <ref>. 



If =_∗ with the standard Kan-Quillen model structure, then the “projective-projective” model structure on  of Theorem <ref>  coincides with the strict model structure of Bousfield-Friedlander that is also obtained by Hovey <cit.>, Schwede <cit.>.


Clear. 





  §.§.§ Stable homotopy category

We have a projective Quillen-Segal theory  on _proj with _n: _proj→_proj, defined by _n()=[ε: _n^0 →(_n+1^1]. Each _n is right Quillen and the left adjoint is Υ_n ∘Γ. We assume that   is a tractable and left proper model category such as _∗. In virtue of Theorem <ref> we can localize the theory and get the following. 

Let : → be a right Quillen functor for a tractable model category which is left proper. Then the following hold. 


  * There is a left proper model structure on the category  such that every fibrant object  is a -spectrum.

  * If =_∗ the the homotopy category is equivalent to the stable homotopy category of Bousfield and Friedlander. 



To prove the last assertion of the theorem we need to outline a result.  Let  be the set of generating cofibration of . For every i ∈ we've introduced in Notation <ref> and Notation <ref> two maps α_i and ζ_i in . The map ζ_i is a projective cofibration whereas α_i is an injective cofibration.

For every n, consider the functor Υ_n ∘Γ∈(,). Then for every i ∈ the following hold. 


  * The map ΥΓ(α_i) =(_k) is such that for every k≥ n+1, _k is an isomorphism in 

  * The map ΥΓ(ζ_i) =(_k) is such that for every k≥ n+1, _k is a level trivial cofibration . 

  * Taking = _∗ we find that for every n and for every i ∈ the maps ΥΓ(α_i), ΥΓ(ζ_i) are stable equivalence. In particular since ΥΓ(ζ_i) is already a cofibration, then it's a trivial stable cofibration. 





The first assertions follow directly by construction of Γ (Proposition <ref>) and Υ_n (Lemma <ref>).  


The third assertion follows by definition of the stable homotopy groups (see <cit.>, <cit.>). These groups are defined as direct colimit so what matters is the stabilization for greater k. The last assertion is clear

We are now able to prove Theorem <ref>. 


The existence of the model structure and the fact that every fibrant object is an -spectrum is simply given by Theorem <ref>.  


To prove that that the homotopy category is the usual stable homotopy category we will show that we have the same fibrant objects in the model structure in Hovey <cit.> which is also a left Bousfield localization of the same strict model structure. 


If ∈ is fibrant in our new model category, then  is level fibrant and satisfies the Segal conditions so it's a -spectrum, whence it's a fibrant object in the model structure of Hovey. Conversely assume that ∈ is fibrant in Hovey's model structure. Then in a tautological way  is -local for any stable equivalence . By Lemma <ref>, we know that for every i and for every n, ΥΓ(ζ_i) is a stable weak equivalence and indeed a trivial cofibration. This means  is ΥΓ(ζ_i)-local for every i and for every n, therefore  is fibrant in the model structure of our Theorem <ref>.



If  is the category of pointed simplicial sets, our model structure on  coincides with the stable model structure of Bousfield-Friedlander <cit.>, Hovey <cit.>, Schwede <cit.>. 



Indeed we're localizing the same original model structure and we have the same fibrant objects  as Hovey. It turns out that the new weak equivalences are also the same since we have the same original model structure, thus the same function complexes.



It remains to compare the previous theorems when we consider the various model structure on . We also need to extend these results when  is cellular and left proper. This will be done later. There is much that is left to be done on the subject.




  §.§.§ -chain complexes

In this section we consider that  has a zero object and let : → be right adjoint. 


Let  be the category of linked -sequence as before. Say that ∈ is an -chain complex if for every n the following composite is the zero map:

    _n^0 ε(_n+1^0) (ε)^2(_n+2^0).

Denote by Ch_() the full subcategory of  spanned by -chain complexes. We have a forgetful functor P: Ch_() →

Obviously if  is an abelian category and = we find here the classical definition of chain complexes. We will develop the homotopy theory of them following  the classical case. And these constructions hold for an ordinal ⊆. 
plain

