Spahn HTT, A.3.2 (Rev #3, changes)

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Recall that the homotopy category hSh S of a model category SS was defined to have the same objects as SS and the hom hShom_hS objects consist of the equivalence classes of the morphisms in the hom Shom_S objects wrt. the homotopy equivalence relation. Here two morphisms of SS were called to be homotopic if their product map factors through some path object of their codomain.

Remark

Let SS be a monoidal model category. Let SCatS Cat denote the category of small SS-enriched categories.

Given a monoidal structure on SS also its homotopy category (this was defined to have the same objects and the hom hShom_hS objects consist of the equivalence classes of the morphisms in the hom Shom_S objects wrt. homotopy

are those of SShSh S carries a monoidal structure which is determined up to a unique isomorphism by the requirement that there exists a monoidal functor

S→hSS\to hS

from SS to its homotopy category.

This remains true in case of a monoidal model category under the additional assumption that the projection

[]:S→hS[]:S\to h S
Definition A.3.2.1

Let SS be a monoidal model category.

A functor F:C→C ′F:C\to C^\prime in sSetCatsSet Cat is a weak equivaleence if the induced functor hC→hC ′hC\to h C^\prime is an equivalence of hSh S-enriched categories.

In other words: F is a weak equivalence iff

(1) For every pair X,Y∈CX,Y\in C, the induced map
Map C(X,Y)→Map C ′(F(X),F(Y))Map_C (X,Y)\to Map_{C^\prime} (F(X), F(Y))

is a weak equivalence in SS.

(2) FF is essentially surjective on the level of homotopy categories.

The (which following definition says a functor between categories is called part of the couniversal property of the homotopy category) is a quasi monoidal fibrations functor. if every isomorphism has a lift with respect toFF.

Definition A.3.2.7

Let FF:C\to D$ be a functor between classical categories.

FF is called a quasi-fibration if, for every object x∈Cx\in C and every isomorphism f:F(x)→yf:F(x)\to y in DD, there exists an isomorphism f¯:x→y¯\overline f:x\to \overline y in CC such that F(f)=fF(f)=f.

This implies that a SS enriched category CC gives rise to a hShS-enriched category hChC with the same objects as CC and the hom hChom_hC objects are defined by

hom hC(X,Y):=[hom C(X,Y)]hom_hC (X,Y):=[hom_C (X,Y)]
Theorem 3.2.24

Let SS be an excellent model category. Then:

  1. An SS-enriched category CC is a fibrant object of sSetCatsSet Cat iff it is locally fibrant: i.e. for all X,Y∈CX,Y\in C the hom object Map C(X,Y)∈SMap_C (X,Y)\in S is fibrant.

  2. Let F:C→DF:C\to D be a SS-enriched functor where DD is a fibrant object of sSetCatsSet Cat. Then FF is a fibration iff FF is a local fibration.
Definition Remark

Let SS be a monoidal model category. Let C SCat C S Cat be denote an the category of smallSS -enriched category. categories.

(1) Given A a morphism monoidal structure on f S f S in also itsCChomotopy category is called anequivalencehSh S if carries a monoidal structure which is determined up to a unique isomorphism by the homotopy requirement class that there exists a monoidal functor[f][f] of ff is an isomorphism in hCh C.

(2) CC is called locally fibrant object if for every pair of objects X,Y∈CX,Y\in C, the mapping space Map C(X,Y)Map_C(X,Y) is a fibrant object of SS.

S→hSS\to hS

(3) from AnSS -enriched functor to its homotopy category.F:C→C ′F:C\to C^\prime is called a local fibration if the following conditions are satisfied:

(3.i) Map C(X,Y)→Map C ′(FX,FY)Map_C (X,Y)\to Map_{C^\prime} (FX,FY) is a fibration in SS for every X,Y∈CX,Y\in C.

(3.ii) The induced map hC→hC ′h C\to h C^\prime is a quasi-fibration of categories.
Definition A.3.2.16 A.3.2.1

(excellent Let model category)SS be a monoidal model category.

A model functor category S F:C→C ′ S F:C\to C^\prime is in calledexcellent model categorysSetCatsSet Cat if it is equipped with a symmetric weak monoidal equivaleence structure if and satisfies the following induced conditions functorhC→hC ′hC\to h C^\prime is an equivalence of hSh S-enriched categories.

(A1) In other words: F is a weak equivalence iffSS is combinatorial.

(A2) (1) Every For monomorphism every in pair S X,Y∈C S X,Y\in C , is a cofibration and the collection induced of map cofibrations inSS is stable under products.

(A3) The collection of weak equivalencies in SS is stable under filtered colimits.

Map C(X,Y)→Map C ′(F(X),F(Y))Map_C (X,Y)\to Map_{C^\prime} (F(X), F(Y))

(A4) is a weak equivalence in⊗:S×S→S \otimes:S\times S\to S . is a Quillen bifunctor.

(A5) (2) The monoidal model category S F S F satisfies is essentially surjective on the invertibility level hypothesis. of homotopy categories.

The following definition says a functor between categories is called a quasi fibrations if every isomorphism has a lift with respect to FF.
Definition A.3.2.7

Let FF:C\to D$ be a functor between classical categories.

FF is called a quasi-fibration if, for every object x∈Cx\in C and every isomorphism f:F(x)→yf:F(x)\to y in DD, there exists an isomorphism f¯:x→y¯\overline f:x\to \overline y in CC such that F(f)=fF(f)=f.
Theorem 3.2.24

Let SS be an excellent model category. Then:

  1. An SS-enriched category CC is a fibrant object of sSetCatsSet Cat iff it is locally fibrant: i.e. for all X,Y∈CX,Y\in C the hom object Map C(X,Y)∈SMap_C (X,Y)\in S is fibrant.

  2. Let F:C→DF:C\to D be a SS-enriched functor where DD is a fibrant object of sSetCatsSet Cat. Then FF is a fibration iff FF is a local fibration.
Definition

Let SS be a monoidal category. Let CC be an SS-enriched category.

(1) A morphism ff in CC is called an equivalence if the homotopy class [f][f] of ff is an isomorphism in hCh C.

(2) CC is called locally fibrant object if for every pair of objects X,Y∈CX,Y\in C, the mapping space Map C(X,Y)Map_C(X,Y) is a fibrant object of SS.

(3) An SS-enriched functor F:C→C ′F:C\to C^\prime is called a local fibration if the following conditions are satisfied:

(3.i) Map C(X,Y)→Map C ′(FX,FY)Map_C (X,Y)\to Map_{C^\prime} (FX,FY) is a fibration in SS for every X,Y∈CX,Y\in C.

(3.ii) The induced map hC→hC ′h C\to h C^\prime is a quasi-fibration of categories.
Definition A.3.2.16

(excellent model category)

A model category SS is called excellent model category if it is equipped with a symmetric monoidal structure and satisfies the following conditions

(A1) SS is combinatorial.

(A2) Every monomorphism in SS is a cofibration and the collection of cofibrations in SS is stable under products.

(A3) The collection of weak equivalencies in SS is stable under filtered colimits.

(A4) ⊗:S×S→S\otimes:S\times S\to S is a Quillen bifunctor.

(A5) The monoidal model category SS satisfies the invertibility hypothesis.

Revision on June 29, 2012 at 21:28:20 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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