+-- {: .num_defn} ###### 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. 1. 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 . =-- +-- {: .un_prop #propA.2.2.1} ###### 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 $X$. 1. The coproduct map $f\coprod g$ factors through some cylinder object for $X$. 1. The product map $f\times g$ factors through every path object for $Y$. 1. The product map $f\times g$ factors through some path object for $Y$. =-- +-- {: .num_defn} ###### 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](#propA.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: +-- {: .num_remark} ###### 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 [[nLab:large category|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 functor $Q^* : Func(hC,A) \to Func(C,A)$ is a [[nLab:full and faithful functor]]. 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. =--