Lecture 2: Homotopical Algebra Nicola Gambino School of Mathematics University of Leeds Young Set Theory Copenhagen June 13th, 2016 1
Homotopical algebra Motivation ◮ Axiomatic development of homotopy theory ◮ Addressing size issues in localizations Key notion ◮ Model category 2
Outline Part I: Model categories Part II: Groupoids Part III: Simplicial sets 3
Part I: Model categories 4
� � � � � � � � Lifting problems Fix a category C . Definition. Let p : B → A and i : X → Y . ◮ We say p has the right lifting property w.r.t. i if for every diagram X B p i � A Y there exists a diagonal filler X B p i Y A Notation: i ⋔ p 5
� � � � � � � � � � Examples ◮ In Set , if i injective and p surjective then i ⋔ p X B p i Y A ◮ In Top , a map p : B → A is a Hurewicz fibration if i X ⋔ p X × { 0 } B i X p X × I A for all X . The case X = {∗} is a path-lifting property. 6
� � � � � � � � � � � Lifting problems: special cases 1. If A = 1, then we have an ‘extension property’ (cf. injective objects): X B i Y 2. If X = 0, then we have a ‘lifting property’ (cf. projective objects): B p Y A Note. General case is a combination of these: X B p i Y A 7
� � � Weak factorisation systems Definition. A weak factorisation system on C is a pair ( L , R ) of classes of maps such that: 1. L = { i | ( ∀ p ∈ R ) i ⋔ p } 2. R = { p | ( ∀ i ∈ L ) i ⋔ p } 3. Every f : X → Y admits a factorisation f X Y p i B with i ∈ L and p ∈ R . Example ◮ ( Inj , Surj ) is a weak factorisation system on Set . 8
� � � Model structures Definition. A Model structure on C consists of three classes of maps, � � Weq , Fib , Cof , such that 1. Weq satisfies 3-for-2, i.e. for all h X Z g f Y 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
Model structures (II) Examples 1. Any category C admits a model structure where Weq = { isomorphisms } , Fib = { all maps } , Cof = { all maps } 2. The category Top admits a model structure where Weq = { homotopy equivalences } , Fib = { Hurewicz fibrations } 3. The category Top admits a model structure where Weq = { weak homotopy 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
� � � � � � Model structures: factorisations Remark 1. Every f : X → Y admits two factorisations f X Y p i B 1. i ∈ Weq ∩ Cof , p ∈ Fib 2. i ∈ Cof , p ∈ Weq ∩ Fib Example. The ‘path object’ factorisation ∆ A A × A A r ( s , t ) P with r ∈ Weq ∩ Cof and ( s , t ) ∈ Fib . 11
� � � � � � � � � � Model structures: lifting problems Remark 2. We have diagonal fillers X B p i Y A in two cases: 1. i ∈ Weq ∩ Cof , p ∈ Fib 2. i ∈ Cof , p ∈ Weq ∩ Fib Example. We have diagonal fillers for A E p r A × A P ( s , t ) for all p ∈ Fib . 12
Part II: Groupoids 13
Example: groupoids The category Gpd ◮ objects: groupoids, i.e. categories in which every arrow is invertible ◮ maps: functors Examples 1. Sets and bijections. 2. A group G is a groupoid with one object, ∗ , and Map ( ∗ , ∗ ) = G . 3. Every topological space has a fundamental groupoid , Π 1 ( X ) of points and homotopy classes of maps. 14
� � � � � � � Isofibrations Definition. A functor p : B → A between groupoids is a isofibration if it has the following path lifting property: ∃ β � B b 0 ∃ b 1 ❴ p � a 1 A a 0 ∀ α Note. p : B → A is isofibration iff b { 0 } B i 0 p � A J a has a diagonal filler, where J = 0 1 15
� � � � � � The model structure on groupoids Theorem. The category Gpd admits a model structure ◮ Weq = equivalence of categories ◮ Fib = isofibrations ◮ Cof = functors injective on objects Note. The ( Weq ∩ Cof , Fib )-factorisation of f : A → B is given by f A B i p { ( x , y , β ) | β : fx → y } In particular ∆ A A × A A r ( s , t ) A J 16
Part III: Simplicial sets 17
Simplicial sets The simplicial category ∆ has ◮ objects: [ n ] = { 0 < . . . < n } , non-empty finite linear orders ◮ morphisms: order-preserving functions Definition. A simplicial set is a functor A : ∆ op → Set [ n ] �→ A n The category SSet ◮ objects: simplicial sets ◮ maps: natural transformations 18
Simplicial sets as spaces Idea. We think of a simplicial set as a set of instructions to construct a space: ◮ For n ≥ 0, define the topological standard n -simplex | ∆ n | = { ( x 0 , . . . , x n ) ∈ R n +1 | x i ≥ 0 , x 0 + . . . + x n = 1 } ◮ For A ∈ SSet , define its geometric realization � � � A n × | ∆ n | R ( A ) = / ≃ [ n ] ∈ ∆ This gives a functor R : SSet → Top . Note. For [ n ] ∈ ∆, there is ∆ n ∈ SSet such that R (∆ n ) ∼ = | ∆ n | This is called the (simplicial) standard n -simplex . 19
Examples: nerve of a groupoid Given a groupoid G , its nerve is the simplicial set NG : ∆ op → Set defined by ( NG ) n = set of strings of n -composable arrows in G f 1 f 1 f n � x 1 � x 2 � . . . � x n } = { x 0 Note. ◮ NG captures objects and maps of G , not composition. ◮ This gives a functor N : Gpd → SSet . 20
� � � � The category of simplicial sets SSet is a presheaf category ⇒ ◮ it has all small limits and colimits ◮ it is locally cartesian closed: all slices are cartesian closed. Equivalently: B SSet / B Σ f ∆ f Π f f ⊣ ⊣ SSet / A A 21
� � � � Kan fibrations Definition. A map p : B → A is a Kan fibration if every diagram Λ n � B k h n p k � A ∆ n k is obtained by removing from ∆ n its interior and has a diagonal filler. Here, Λ n the interior of the face opposite the k -th vertex, and h n k the inclusion. Examples. Λ 2 � B Λ 1 � B 1 0 h 1 h 1 p p 0 � 0 � � A � A ∆ 1 ∆ 2 Note. p : B → A is an isofibration in Gpd ⇒ Np : NB → NA Kan fibration. 22
� � � � The model structure on simplicial sets Theorem. The category SSet admits a model structure where ◮ Weq = weak homotopy equivalences ◮ Fib = Kan fibrations ◮ Cof = monomorphisms Note. The fibrant objects are the Kan complexes : Λ n � B Λ 2 � B Λ 1 � B 1 0 k e.g. h n h 1 h 1 k 0 � 0 � ∆ n ∆ 1 ∆ 2 Note. G groupoid ⇒ NG Kan complex (using the composition and inverses) Kan complexes can be seen as weak ∞ -groupoids. 23
Summary Part I: Model structures Part II: Groupoids Part III: Simplicial sets Tomorrow: the type theory T has models in groupoids and simplicial sets. 24
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