# Spahn HTT, A.2.2

###### Definition

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

1. 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.

2. 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.

1. The coproduct map $f\coprod g$ factors through every cylinder object for $X$.

2. The coproduct map $f\coprod g$ factors through some cylinder object for $X$.

3. The product map $f\times g$ factors through every path object for $Y$.

4. 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 $hom_C (X,Y)$.

The following remark gives an alternative equivalent definition of the homotopy category of a model category:

###### Remark and Definition

The homotopy category $hC$ (more precisely the projection map $Q:C\to hC$) is couniversal in the following sense:

• for any (possibly large) category $A$ and functor $F : C \to A$ such that $F$ sends all $w \in W$ to isomorphisms in $A$, there exists a functor $F_Q : hC \to A$ and a natural isomorphism
$\array{ C &&\stackrel{F}{\to}& A \\ \downarrow^Q& \Downarrow^{\simeq}& \nearrow_{F_Q} \\ hC }$

The second condition implies that the functor $F_Q$ in the first condition is unique up to unique isomorphism.

As always is the the case with (co)universal properties the object in question can be defined by this property.

Last revised on June 29, 2012 at 22:15:07. See the history of this page for a list of all contributions to it.