Homotopy and Homotopy Category

The homotopy category of a model category is obtained by formally inverting all weak equivalences. This construction localizes the category at weak equivalences, making them into actual isomorphisms in the homotopy category.

DefinitionLeft and Right Homotopy

Let $\mathcal{M}$ be a model category. For maps $f, g: A \to B$ :

A left homotopy from $f$ to $g$ (when $A$ is cofibrant) is a map $H: A \otimes I \to B$ where $I$ is a cylinder object for $A$ (a factorization of the fold map $A \sqcup A \to A$ as a cofibration followed by a weak equivalence), such that $H|_{A \otimes \{0\}} = f$ and $H|_{A \otimes \{1\}} = g$ .

A right homotopy from $f$ to $g$ (when $B$ is fibrant) is a map $H: A \to B^I$ where $B^I$ is a path object for $B$ (a factorization of the diagonal $B \to B \times B$ as a weak equivalence followed by a fibration), such that evaluating at the endpoints gives $f$ and $g$ .

When $A$ is cofibrant and $B$ is fibrant, left and right homotopy coincide.

DefinitionThe Homotopy Category

The homotopy category $\text{Ho}(\mathcal{M})$ has:

The same objects as $\mathcal{M}$
Morphisms $\text{Hom}_{\text{Ho}(\mathcal{M})}(A, B)$ given by homotopy classes of maps $QA \to RB$ , where $QA$ is a cofibrant replacement of $A$ and $RB$ is a fibrant replacement of $B$

There is a localization functor $\gamma: \mathcal{M} \to \text{Ho}(\mathcal{M})$ that sends weak equivalences to isomorphisms and is universal with this property.

ExampleHomotopy Category of Topological Spaces

For $\mathbf{Top}$ with the Quillen model structure: $\text{Ho}(\mathbf{Top}) \simeq \text{hTop}$ where $\text{hTop}$ is the classical homotopy category of CW complexes (or more generally, spaces with the homotopy type of CW complexes).

This recovers the classical construction from algebraic topology.

RemarkKen Brown's Lemma

A fundamental result states: a functor $F: \mathcal{M} \to \mathcal{N}$ between model categories preserves weak equivalences between fibrant (resp. cofibrant) objects if it preserves fibrations (resp. cofibrations) and trivial fibrations (resp. trivial cofibrations).

This lemma is crucial for constructing derived functors and verifying that functors descend to the homotopy category.

DefinitionDerived Functors

For a functor $F: \mathcal{M} \to \mathcal{N}$ between model categories, the left derived functor $\mathbf{L}F$ is defined by: $\mathbf{L}F(X) = F(QX)$ where $QX$ is a cofibrant replacement. Similarly, the right derived functor $\mathbf{R}F$ uses fibrant replacements: $\mathbf{R}F(X) = F(RX)$

These are well-defined on $\text{Ho}(\mathcal{M})$ and provide the correct homotopy-theoretic version of $F$ .

ExampleDerived Tensor Product

In the category $\mathbf{Ch}(\mathbb{Z})$ of chain complexes with the projective model structure, the derived tensor product is: $X \otimes^{\mathbf{L}} Y = P(X) \otimes P(Y)$ where $P$ denotes a projective (cofibrant) replacement. This construction gives the correct notion of tensor product "up to quasi-isomorphism."

RemarkUniversal Property

The homotopy category satisfies a universal property: for any functor $F: \mathcal{M} \to \mathcal{D}$ that sends weak equivalences to isomorphisms, $F$ factors uniquely (up to natural isomorphism) through $\text{Ho}(\mathcal{M})$ .

This makes $\text{Ho}(\mathcal{M})$ the "correct" target for any construction that should respect homotopy equivalence.

The homotopy category construction is the fundamental reason model categories are useful: they provide an axiomatic framework for constructing localizations at weak equivalences in a controlled way, generalizing classical homotopy theory to abstract categorical settings.