Spahn HTT, A.3.2 (Rev #1, changes)

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

is a weak equivalence in $S$.

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

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 $F(f)=f$.

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.