An Introduction to Model Categories
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1 An Introduction to Model Categories Brooke Shipley (UIC) Young Topologists Meeting, Stockholm July 4, 2017
2 Examples of Model Categories (C, W ) Topological Spaces with weak equivalences f : X Y if π (X ) = π (Y ).
3 Examples of Model Categories (C, W ) Topological Spaces with weak equivalences f : X Y if π (X ) = π (Y ). Chain complexes with quasi-isomorphisms f : C D if H (C) = H (D).
4 Examples of Model Categories (C, W ) Topological Spaces with weak equivalences f : X Y if π (X ) = π (Y ). Chain complexes with quasi-isomorphisms f : C D if H (C) = H (D). Simplicial abelian groups with weak equivalences f : A B if H (NA) = H (NB).
5 Definition of Model Categories Definition: A model category is a category C with 3 classes of maps W, C, and F, satisfying 5 axioms as below. Weak equivalences, denoted, Cofibrations, denoted, and Fibrations, denoted.
6 Definition of Model Categories Definition: A model category is a category C with 3 classes of maps W, C, and F, satisfying 5 axioms as below. Weak equivalences, denoted, Cofibrations, denoted, and Fibrations, denoted. closed under composition
7 Definition of Model Categories Definition: A model category is a category C with 3 classes of maps W, C, and F, satisfying 5 axioms as below. Weak equivalences, denoted, Cofibrations, denoted, and Fibrations, denoted. closed under composition acyclic cofibrations A B acyclic fibrations X Y
8 Axioms for Model Categories C has all finite colimits and limits.
9 Axioms for Model Categories C has all finite colimits and limits. (2 of 3) If two of f, g, gf are weak equivalences, then so is the third.
10 Axioms for Model Categories C has all finite colimits and limits. (2 of 3) If two of f, g, gf are weak equivalences, then so is the third. W, C, F are closed under retracts.
11 Axioms for Model Categories C has all finite colimits and limits. (2 of 3) If two of f, g, gf are weak equivalences, then so is the third. W, C, F are closed under retracts. Lifting: Lifts exist in the following squares: A X A X B Y B Y
12 Axioms for Model Categories C has all finite colimits and limits. (2 of 3) If two of f, g, gf are weak equivalences, then so is the third. W, C, F are closed under retracts. Lifting: Lifts exist in the following squares: A X A X B Y B Y Factorization: Any map f : X Y factors in two ways X Z Y X W Y.
13 Homotopy Category, Quillen Pair, Quillen Equivalence The homotopy category of a model category (C, W ) is defined by inverting the weak equivalences. Ho(C) = C[W 1 ]
14 Homotopy Category, Quillen Pair, Quillen Equivalence The homotopy category of a model category (C, W ) is defined by inverting the weak equivalences. Ho(C) = C[W 1 ] Given C, D model categories and an adjunction: C F D, then (F, G) is a Quillen pair if F preserves cofibrations and G preserves fibrations. Then there is an induced adjunction: Ho(C) LF Ho(D) RG G
15 Homotopy Category, Quillen Pair, Quillen Equivalence The homotopy category of a model category (C, W ) is defined by inverting the weak equivalences. Ho(C) = C[W 1 ] Given C, D model categories and an adjunction: C F D, then (F, G) is a Quillen pair if F preserves cofibrations and G preserves fibrations. Then there is an induced adjunction: Ho(C) LF Ho(D) RG If (LF, RG) induces an equivalence on the homotopy categories, then (F, G) is a Quillen equivalence. C QE D and Ho(C) = Ho(D). G
16 Examples The projective model structure on ch + : W = quasi-isomoprhisms F = epimorphisms in positive degree C = monomorphisms with projective cokernel.
17 Examples The projective model structure on ch + : W = quasi-isomoprhisms F = epimorphisms in positive degree C = monomorphisms with projective cokernel. The injective model structure on ch : W = quasi-isomoprhisms C = monomorphisms in negative degree F = epimorphisms with injective kernel.
18 Examples The projective model structure on ch + : W = quasi-isomoprhisms F = epimorphisms in positive degree C = monomorphisms with projective cokernel. The injective model structure on ch : W = quasi-isomoprhisms C = monomorphisms in negative degree F = epimorphisms with injective kernel. Both extend to model structures on Ch: Ch Proj QE Ch Inj and Ho(Ch Proj ) = Ho(Ch Inj )
19 Counter-examples and Examples There are examples of model categories C, D with Ho(C) = Ho(D), but there is no Quillen pair inducing this equivalence. So, C QE D.
20 Counter-examples and Examples There are examples of model categories C, D with Ho(C) = Ho(D), but there is no Quillen pair inducing this equivalence. So, C QE D. (N, Γ) form a Quillen pair and a Quillen equivalence sab QE ch + and Ho(sAb) = Ho(ch + )
21 Counter-examples and Examples There are examples of model categories C, D with Ho(C) = Ho(D), but there is no Quillen pair inducing this equivalence. So, C QE D. (N, Γ) form a Quillen pair and a Quillen equivalence sab QE ch + and Ho(sAb) = Ho(ch + ) (Schwede-S. 2003) N induces a functor on simplicial rings, and is part of a Quillen equivalence, s Rings QE DGA + and Ho(s Rings) = Ho(DGA + )
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