Model category

Remarks

Some authors ask only for finite limits and colimits, as Quillen did, and some ask the factorizations in axiom 4 to be functorial.

Example: let $\mathcal{C}$ be any category with all small limits and colimits, let $W$ be the isomorphisms, and let $\mathrm{Fib}$ and $\mathrm{Cof}$ be all morphisms. Isomorphisms satisfy two out of three and are closed under retracts. Every square lifts against an isomorphism, which settles axiom 3, and $f = f \circ \id = \id \circ f$ gives axiom 4.

Non-example: in the category of sets, let all three classes be all maps. Then the map $i \colon \set{0, 1} \to \set{0}$ is a cofibration and a trivial fibration. Consider the commutative square with $i$ on both vertical sides, $\id$ on $\set{0, 1}$ on top and $\id$ on $\set{0}$ below. A lift would be a map $h \colon \set{0} \to \set{0, 1}$ with $h \circ i = \id$, which does not exist, so axiom 3 fails.

The main example is the Quillen model structure on topological spaces: weak equivalences are weak homotopy equivalences, fibrations are Serre fibrations, and cofibrations are retracts of relative cell complexes. An object $X$ is fibrant if $X \to \ast$ is a fibration and cofibrant if $\emptyset \to X$ is a cofibration.