Holmstrom Homotopy category

Construction

The homotopy category $Ho \ C$ of a model category is obtained by “inverting the WEs”. One of the points of the model category axioms is that the Hom sets in the homotopy category actually are sets. The homotopy category is universal for functors from $C$ sending WEs to isomorphisms.

The Hom set $[X,Y]$ between two objects in the homotopy category is isomorphic to $Hom(QX, RY) / \sim$.

Given a model category $C$, the subcategory of cofibrant (fibrant, cofibrant and fibrant) objects has an equivalent homotopy category, the equivalence being induced by the inclusion functor.

For the details, the following notes are useful:

Consider maps from $B$`$ to $`$X$`$ in a model category. If $`$B$`$ is cofibrant and $`$X$ is fibrant, then the left homotopy and the right homotopy relations coincide, and are equivalence relations on the Hom set.

Let $C_{cf}$ be the category of fibrant and cofibrant objects in $C$. The homotopy relation on $C_{cf}$ is an equivalence relation and is compatible with composition. Hence the localisation of $C_{cf}$ exists.

A map in $C_{cf}$ is a WE iff it is a homotopy equivalence.

Here is a reformulation of some of the above:

Theorem: (Hovey page 13…)

Properties

The homotopy category of a model category has all small products and coproducts.

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Sometimes, the term “the homotopy category” refers to the homotopy category of topological spaces (or simplicial sets). See nLab and also the talk of Maltsiniotis in Paris Jan 2009

nLab page on Homotopy category