Spahn HTT, A.3.2 (Rev #2)

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

Remark

Let $S$ be a monoidal model category. Let $S Cat$ denote the category of small $S$ -enriched categories.

Given a monoidal structure on $S$ also its homotopy category (this was defined to have the same objects and the $hom_hS$ objects consist of the equivalence classes of the morphisms in the $hom_S$ objects wrt. homotopy

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

S\to hS

from $S$ to its homotopy category.

Definition A.3.2.1

Let $S$ be a monoidal model category.

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

In other words: F is a weak equivalence iff

(1) For every pair $X,Y\in C$ , the induced map

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

is a weak equivalence in $S$ .

(2) $F$ is essentially surjective on the level 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 $F$ .

Definition A.3.2.7

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

$F$ is called a quasi-fibration if, for every object $x\in C$ and every isomorphism $f:F(x)\to y$ in $D$ , there exists an isomorphism $\overline f:x\to \overline y$ in $C$ such that $F(f)=f$ .

Theorem 3.2.24

Let $S$ be an excellent model category. Then:

An $S$ -enriched category $C$ is a fibrant object of $sSet Cat$ iff it is locally fibrant: i.e. for all $X,Y\in C$ the hom object $Map_C (X,Y)\in S$ is fibrant.
Let $F:C\to D$ be a $S$ -enriched functor where $D$ is a fibrant object of $sSet Cat$ . Then $F$ is a fibration iff $F$ is a local fibration.

Definition

Let $S$ be a monoidal category. Let $C$ be an $S$ -enriched category.

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

(2) $C$ is called locally fibrant object if for every pair of objects $X,Y\in C$ , the mapping space $Map_C(X,Y)$ is a fibrant object of $S$ .

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

(3.i) $Map_C (X,Y)\to Map_{C^\prime} (FX,FY)$ is a fibration in $S$ for every $X,Y\in C$ .

(3.ii) The induced map $h C\to h C^\prime$ is a quasi-fibration of categories.

Definition A.3.2.16

(excellent model category)

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

(A1) $S$ is combinatorial.

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

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

(A4) $\otimes:S\times S\to S$ is a Quillen bifunctor.

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

Revision on June 29, 2012 at 19:56:11 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.