
aeye1 samroopata siddhant

beejganitiya jyaamiti aur beejganitiya topology mein, ganit kii shaakhaaen, samroopata siddhant yaa prerak samroopata siddhant beejganitiya topology kii takneekon koo lagu karane kaa ack tarika haye, vishesh roop seey samroopata, beejganitiya kismon parr aur, adhik aam taur parr, yojanaon parr. yeh siddhant fabian morel aur vladmir vovodsky key kaaran haye. antarnihit vichaar yeh haye kii samroopata siddhant key liye ack vishuddh beejganitiya drishtikon vikasit karna sambhav honah chaahiye, ikai antaral koo pratisthaapit karke, joe ack beejganitiya vividhata naheen haye, joe kii affine rekha key saath haye. iss siddhant ney mishrit uddeshyon kii vyutpann shreni key nirmaan aur milner aur blotch-kaato anumaanon key pramaan jaise shandar anuprayog dekhe hain.
nirmaan.
 homotopy siddhant homotopy shreni sootra _ 1 naamak ack shreni parr adharit haye. saral shabdon mein kahein tou, homotopy shreni, yaa isake bajaay canonical functor sootra _ 2, ack anant shreni kii or chikani sootra _ 4-yojanaon key shreni sootra _ 3 seey saarvbhaumik karyakarta haye joe nisnevic vansh koo santusht kartaa haye, jaise kii affine rekha samvidaatmak hoe jaati haye. yahaan sootra _ 4 kuchh poorv-chuni hui aadhar yojana haye (ee.jee., jatil sankhyaon kaa varnakram sootra _ 6).
saarvbhaumik sampatti key sandarbh mein yeh paribhasha anant shreniyon key binaa sambhav naheen haye. yeye 90 key dashak mein upalabdh naheen they aur mool paribhasha quillen key model shreniyon key siddhant key madhyam seey paarit hoti haye. sthiti koo dekhane kaa ack aur tarika yeh haye kii morel-vovodsky kii mool paribhasha anant shreni sootra _ 1 key liye (samroopata shreni) ack thhos model kaa utpaadan karati haye.
iss adhik thhos nirmaan koo neeche chitrit kiya gayaa haye.
charan 0.
ack aadhar yojana sootra chunein. shaastriya roop seey, sootra 4 koo notherian honey key liye kahaa jaataa haye, lekin maark hoyois jaise kaee aadhunik lekhak ardh-sankshipt ardh-vibhajit aadhar yojanaon key saath kaam karate hain. kissy bhee mamle mein, kaee mahatvapurna parinam kewal ack poorna aadhar kshetra parr gyaat hotey hain, jaise kii jatil sankhyaaen, kewal iss mamle parr vichaar karna poori tarah seey thheek haye.
charan 1.
"charan 1eh nisnevic sheeves. shaastriya roop seey, nirmaan sootra 4 parr chikani yojanaon key shreni sootra 3 parr nisnevic sheeves key shreni sootra 10 key saath shuru hotaa haye. shodhaatmak roop seey, ise (aur ack sateek takniki arth mein) sabhi collimits koo sanlagn karke praapt sootra 3 key saarvbhaumik vistaar key roop mein maanaa jaanaa chaahiye aur nisnevic vansh koo santusht karane key liye majboor karna chaahiye.
"charan 1 beeh saral sheev. adhik aasaani seey maanak samroopata saiddhaantik prakriyaon jaise kii samroopata collimit aur samroopata seemaon koo karane key liye, sootra _ 14 koo saral aavaran kii nimnalikhit shreni key saath pratisthaapit kiya gayaa.
maanaa kii yeh simplexes shreni haye, yaani vaha shreni jisaki vastuein samooh hain.
aur jinake rupankan crum-sanrakshan kaary hain. humm sootra _ 15 koo sootra _ 16 kii shreni koo darshaane dete hain. arthaat, sootra _ 15 sootra _ 10 parr saral vastuon kii shreni haye. aisee vastu koo sootra _ 3 parr "saral sheef" bhee kahaa jaataa haye.
"charan 1seeh fiber functor. kissy bhee chikane sootra _ 4-yojana sootra _ 21, kissy bhee bindu sootra _ 22, aur kissy bhee sheef sootra _ 23 key liye, sootra _ 21 key chhote nisnevic sthal parr sootra _ 23 key pratibandh sootra _ 25 key danthal key liye sootra _ 24 likhein. spasht roop seey, sootra _ 28 jahaan colimit ack étale rupankan sootra _ 31 key madhyam seey vihit samavesh sootra _ 30 key gunankhandan sootra _ 29 parr haye. sangrah sootra _ 32 sootra _ 10 key liye fiber functors kaa ack roodhivaadi parivaar haye.
"charan 1 deeh bund model sanrachana. humm fiber functor key sandarbh mein sootra _ 15 parr ack bund model sanrachana koo paribhaashit karenge. sootra _ 35 koo saral sheev kaa ack rupankan maanaa jaae. humm kahate hainh
iss model sanrachana kii samroopata shreni koo sootra _ 37 seey darshaaya gayaa haye.
charan 2.
iss model sanrachana mein nisnevic avrohan haye, lekin yeh affine rekha koo sikudati naheen haye. ack saral sheef sootra _ 38 koo sootra _ 39-sthaaniya kahaa jaataa haye yadi kissy bhee saral sheef sootra _ 40 key liye maanchitra
sootra _ 42 dwara prerit ack dwibhajan haye. yahaan humm sootra _ 39 koo yonada embedding key madhyam seey ack sheef key roop mein, aur nirantar saral vastu functor sootra _ 44 parr vichaar curr rahey hain.
ack aakritivaad sootra _ 35 ack sootra _ 39-kamjor samatulyata haye yadi kissy bhee sootra _ 39-sthaaniya sootra _ 48 key liye, prerit maanchitra
yeh ack vibhajan haye. sootra 39-sthaaniya model sanrachana sootra 39-kamjor samatulyataon key sambandh mein uparokt model kaa sthaaniyakaran haye.
aupchaarik paribhasha.
anta mein humm samroopata shreni koo paribhaashit curr sakate hain.
dhyaan dein kii nirmaan dwara, kissy bhee ander key liye, ack samroopata haye
homotopy shreni mein.
siddhant key guna.
saral (poorv) sheev key utpaad koo faad aur todein.
kyonki hamane sootra _ 58-homotopy shreni key nirmaan key liye ack saral model shreni key saath shuruat kii, isaliye saral model shreniyon key amoort siddhant seey viraasat mein millie kaee sanrachanaaen hain. vishesh roop seey, sootra _ 60 mein sootra _ 59 nukili saral sheev key liye humm ways utpaad koo colimitformula _ 61 key roop mein banaa sakate hain aur smash utpaad koo samroopata siddhant mein kuchh shaastriya nirmaanon koo punarpraapt karate hue formula _ 62 key roop mein paribhaashit kiya gayaa haye. isake alawa ack saral (poorv) aavaran kaa ack shanku aur ack aakritivaad kaa ack shanku haye, lekin inhen paribhaashit karane key liye saral kshetron kii paribhasha kii aavashyakta hoti haye.
saral kshetra.
iss tathya seey kii humm ack saral model shreni seey shuru karate hain, iska matlab haye kii sootra _ 60 mein saraltaon koo paribhaashit karane vaalaa ack saha-saral kaary-sootra _ 63 haye. yaad rakhein kii beejganitiya aine-simplexes sootra _ 4-schemeformula _ 66 dwara diyaa gayaa haye, inn yojanaon koo sthir poorv-aavaran key roop mein jodate hue aur schiefying sootra _ 60 mein vastuon koo detaa haye, jise humm sootra _ 68 dwara niroopit karate hain. yeye sootra _ 69, i kii chhavi mein vastuen hain.ee. sootra _ 70. phir amoort saral samroopata siddhant kaa upayog karate hue, humm saral gole sootra _ 71 praapt karate hain, phir humm ack saral (poorv) sheef asphormula _ 72 kaa shanku banaa sakate hain aur ack aakriti sootra _ 73 kaa shanku banaa sakate hain joe aarekh sootra _ 74 key collimit key roop mein haye. isake alawa, sootra _ 75 kaa cofiber kewal nilamban sootra _ 76 haye. nukili samroopata shreni mein atirikt roop seey sootra _ 78 dwara diyaa gayaa nilamban functorformula _ 77 haye aur iska daahine sahayak sootra _ 79 jise loop space functor kahaa jaataa haye.
tippani.
vyavastha, vishesh roop seey nisnevic topology, koo beejganitiya key-siddhant koo ack spectrum dwara pratinidhitva yogya banaane key liye chunaa jaataa haye, aur kuchh pahaluon mein blotch-kaato anuman kaa pramaan sambhav banaane key liye.
morel-vovodsky nirmaan key baad anya model shreni sanrachanaon kaa upayog karke yaa nisnevic sheeves (udaaharan key liye, zariski sheeves yaa sirf sabhi preshieves) kii tulanaa mein anya sheeves kaa upayog karke homotopy siddhant key liye kaee alag-alag drishtikon rahey hain. inmein seey pratyek nirmaan ack hii samroopata shreni kaa utpaadan kartaa haye.
siddhant mein dou prakaar key kshetra hainh way joe topology mein-kshetra kii bhoomika nibhaane vaale gunak samooh seey aatey hain, aur way joe saral kshetra seey aatey hain (jinhen nirantar saral aavaran maanaa jaataa haye). yeh dou soochkaankon key saath prerak kshetron key siddhant kii or leey jaataa haye. prerak kshetron key samroopata samoohon kii ganana karane seey kshetron key shaastriya sthir samroopata samooh bhee praapt honge, isaliye iss sambandh mein samroopata siddhant cumm seey cumm shaastriya samroopata siddhant jitna hii jatil haye.
prerak samaantaaen.
eilenberg-mcclane sthaan.
ack abelian samooh sootra _ 80 key liye ack chikani yojana sootra _ 21 kaa sootra _ 81-prerak samroopata sheef hyper alcohology samoohon dwara sootra _ 84 key liye formula _ 83 dwara dee jaati haye. iss samroopata kaa pratinidhitva karane vaalaa ack saral abelian sheef haye joe sootra _ 86 key anuroop haye jise ingit prerak samroopata shreni sootra _ 87 mein ack vastu key roop mein maanaa jaataa haye. phir, ack chikani yojana sootra _ 21 key liye hamaare paas samatulya sootra _ 89 haye joe inn sheev koo prerak eilenberg-mcclane spacepig 3 kaa pratinidhitva kartaa haye.
sthir samroopata shreni.
aeye1-homotopy siddhant mein ack aur nirmaan shreni aiss. aich. ("aiss") haye, joe jee. aim. key saath smash utpaad koo vyutkramaniya banane key liye majboor karke uparokt asthir shreni seey praapt kiya jaataa haye. iss prakriya koo yaa tou tathaakathit jee. aim.-spectrum kaa upayog karke model-shreneegat nirmaan kaa upayog karke yaa vaikalpik roop seey anant-shreniyon kaa upayog karke kiya jaa sakata haye.
s" = vishishtata (r) key liye, vaastavik sankhyaon key kshetra kaa varnakram, ack kaaryak haye
beejganitiya topology seey sthir samroopata shreni mein. functor kii visheshta "x" seey judey vaastavik manifold parr ack chikani yojana "x"/r bhejkar hoti haye. iss functor key paas vaha guna haye joe yeh naksha bhejata haye
ack samatulyata key liye, kyonki sootra _ 92 dou-bindu samooh key samroopata haye. dikhaaya haye kii parinami functor
ack samaantaa haye.

