Spahn HTT, A.2.2 (Rev #1)

Definition

Let $X$ be an object in a model category.

A cylinder object is defined to be a factorization of the codiagonal map $X\coprod X\to X$ for $X$ into a cofibration followed by a weak equivalence.
A path object is defined to be a factorization of the diagonal map $X\to X\times X$ for $X$ into a weak equivalence followed by a fibration .

Proposition A.2.2.1

Let $C$ be a model category. Let $X$ be a cofibrant object of $C$ . Let $Y$ be a fibrant object of $C$ . Let $f,g:X\to Y$ be two parallel morphisms. Then the following conditions are equivalent.

The coproduct map $f\coprod g$ factors through every cylinder object for $X$ .
The coproduct map $f\coprod g$ factors through some cylinder object for $X$ .
The product map $f\times g$ factors through every path object for $Y$ .
The product map $f\times g$ factors through some path object for $Y$ .

Definition

(homotopy, homotopy category of a model category)

Let $C$ be a model category.

(1) Two maps $f,g:X\to Y$ from a cofibrant object to a fibrant object satisfying the conditions of Proposition A.2.2.1 are called homotopic morphisms. Homotopy is an equivalence relation $\simeq$ on $hom_C (X,Y)$ .

(2) The homotopy category $h C$ of $C$ is defined to have as objects the fibrant-cofibrant objects of $C$ . The hom objects $hom_{hC}(X,Y)$ are defined to be the set of $\simeq$ equivalence classes of $hom_C (X,Y)$ .

Revision on June 29, 2012 at 19:43:23 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.