Lecture 2: Homotopical Algebra

Transcription

1 Lecture 2: Homotoical Algebra Nicola Gambino School of Mathematics University of Leeds Young Set Theory Coenhagen June 13th,

2 Homotoical algebra Motivation Axiomatic develoment of homotoy theory Addressing size issues in localizations Key notion Model category 2

3 Outline Part I: Model categories Part II: Grouoids Part III: Simlicial sets 3

4 Part I: Model categories 4

5 Lifting roblems Fix a category C. Definition. Let : A and i : X Y. We say has the right lifting roerty w.r.t. i if for every diagram X there exists a diagonal filler i i Y X Y A A Notation: i 5

6 Examles In Set, if i injective and surjective then i i X Y A In To, a ma : A is a Hurewicz fibration if i X X {0} i X X I A for all X. The case X = { } is a ath-lifting roerty. 6

7 7 Lifting roblems: secial cases 1. If A = 1, then we have an extension roerty (cf. injective objects): X i Y 2. If X = 0, then we have a lifting roerty (cf. rojective objects): Y A Note. General case is a combination of these: X i Y A

8 Weak factorisation systems Definition. A weak factorisation system on C is a air (L, R) of classes of mas such that: 1. L = {i ( R) i } 2. R = { ( i L) i } 3. Every f : X Y admits a factorisation X f Y i with i L and R. Examle ( Inj, Surj ) is a weak factorisation system on Set. 8

9 Model structures Definition. A Model structure on C consists of three classes of mas, ( Weq, Fib, Cof ), such that 1. Weq satisfies 3-for-2, i.e. for all X h Z f Y g if two out of f, g, h are in Weq, then so is the third. 2. (Weq Cof, Fib) is a weak factorisation system. 3. (Cof, Weq Fib) is a weak factorisation system 9

10 Model structures (II) Examles 1. Any category C admits a model structure where Weq = { isomorhisms }, Fib = { all mas }, Cof = { all mas } 2. The category To admits a model structure where Weq = { homotoy equivalences }, Fib = { Hurewicz fibrations } 3. The category To admits a model structure where Weq = { weak homotoy equivalences }, Fib = { Serre fibrations } Terminology. An object X C is said to be fibrant if X 1 is in Fib cofibrant if 0 X is in Cof. 10

11 Model structures: factorisations Remark 1. Every f : X Y admits two factorisations X f Y i 1. i Weq Cof, Fib 2. i Cof, Weq Fib Examle. The ath object factorisation A A A A r P (s,t) with r Weq Cof and (s, t) Fib. 11

12 Model structures: lifting roblems Remark 2. We have diagonal fillers X i Y A in two cases: 1. i Weq Cof, Fib 2. i Cof, Weq Fib Examle. We have diagonal fillers for A r P (s,t) E A A for all Fib. 12

13 Part II: Grouoids 13

14 Examle: grouoids The category Gd objects: grouoids, i.e. categories in which every arrow is invertible mas: functors Examles 1. Sets and bijections. 2. A grou G is a grouoid with one object,, and Ma(, ) = G. 3. Every toological sace has a fundamental grouoid, Π 1(X ) of oints and homotoy classes of mas. 14

15 Isofibrations Definition. A functor : A between grouoids is a isofibration if it has the following ath lifting roerty: Note. : A is isofibration iff b 0 β A a 0 α b 1 a 1 {0} i 0 J b a A has a diagonal filler, where J =

16 The model structure on grouoids Theorem. The category Gd admits a model structure Weq = equivalence of categories Fib = isofibrations Cof = functors injective on objects Note. The (Weq Cof, Fib)-factorisation of f : A is given by A f i {(x, y, β) β : fx y} In articular A A A A r A J (s,t) 16

17 Part III: Simlicial sets 17

18 Simlicial sets The simlicial category has objects: [n] = {0 <... < n}, non-emty finite linear orders morhisms: order-reserving functions Definition. A simlicial set is a functor A : o Set [n] A n The category SSet objects: simlicial sets mas: natural transformations 18

19 Simlicial sets as saces Idea. We think of a simlicial set as a set of instructions to construct a sace: For n 0, define the toological standard n-simlex n = {(x 0,..., x n) R n+1 x i 0, x x n = 1} For A SSet, define its geometric realization ( ) R(A) = A n n [n] / This gives a functor R : SSet To. Note. For [n], there is n SSet such that R( n ) = n This is called the (simlicial) standard n-simlex. 19

20 Examles: nerve of a grouoid Given a grouoid G, its nerve is the simlicial set NG : o Set defined by (NG) n = set of strings of n-comosable arrows in G = { x 0 f 1 x 1 f 1 x 2... f n x n } Note. NG catures objects and mas of G, not comosition. This gives a functor N : Gd SSet. 20

21 The category of simlicial sets SSet is a resheaf category it has all small limits and colimits it is locally cartesian closed: all slices are cartesian closed. Equivalently: SSet/ f Σ f f Π f A SSet/A 21

22 Kan fibrations Definition. A ma : A is a Kan fibration if every diagram Λ n k h n k n A has a diagonal filler. Here, Λ n k is obtained by removing from n its interior and the interior of the face oosite the k-th vertex, and h n k the inclusion. Examles. Λ 2 1 Λ 1 0 h A h A Note. : A is an isofibration in Gd N : N NA Kan fibration. 22

23 The model structure on simlicial sets Theorem. The category SSet admits a model structure where Weq = weak homotoy equivalences Fib = Kan fibrations Cof = monomorhisms Note. The fibrant objects are the Kan comlexes: Λ n k Λ 2 1 Λ 1 0 h n k n e.g. h h Note. G grouoid NG Kan comlex (using the comosition and inverses) Kan comlexes can be seen as weak -grouoids. 23

24 Summary Part I: Model structures Part II: Grouoids Part III: Simlicial sets Tomorrow: the tye theory T has models in grouoids and simlicial sets. 24