Quillen Adjunctions and Equivalences

Quillen adjunctions are the morphisms in the "category" of model categories, providing the correct notion of functor between homotopy theories. They ensure that adjoint functors between model categories induce adjoint functors on homotopy categories.

DefinitionQuillen Adjunction

An adjunction $F: \mathcal{M} \rightleftarrows \mathcal{N}: G$ between model categories is a Quillen adjunction if one of the following equivalent conditions holds:

$F$ preserves cofibrations and trivial cofibrations
$G$ preserves fibrations and trivial fibrations
$F$ preserves cofibrations and $G$ preserves fibrations

When these conditions hold, $F$ is called the left Quillen functor and $G$ the right Quillen functor.

RemarkWhy This Definition

The definition ensures that the derived functors $\mathbf{L}F$ and $\mathbf{R}G$ are well-defined and form an adjunction: $\mathbf{L}F: \text{Ho}(\mathcal{M}) \rightleftarrows \text{Ho}(\mathcal{N}): \mathbf{R}G$

This is the "correct" way to pass an adjunction to the level of homotopy categories.

DefinitionQuillen Equivalence

A Quillen adjunction $F \dashv G$ is a Quillen equivalence if the derived adjunction is an adjoint equivalence of categories. Equivalently, for all cofibrant $X \in \mathcal{M}$ and fibrant $Y \in \mathcal{N}$ :

$X \to GF(X)$ is a weak equivalence
$FG(Y) \to Y$ is a weak equivalence

Quillen equivalences are the "isomorphisms" in the world of model categories—they establish that two model structures present the same homotopy theory.

ExampleSimplicial Sets and Topological Spaces

The geometric realization and singular complex functors form a Quillen equivalence: $|-|: \mathbf{sSet} \rightleftarrows \mathbf{Top}: \text{Sing}$

This establishes that simplicial sets and topological spaces have equivalent homotopy theories, justifying the use of combinatorial methods in classical homotopy theory.

RemarkDetecting Quillen Equivalences

A Quillen adjunction $F \dashv G$ is a Quillen equivalence if and only if for all cofibrant $X \in \mathcal{M}$ and fibrant $Y \in \mathcal{N}$ , a map $f: FX \to Y$ is a weak equivalence in $\mathcal{N}$ if and only if its adjoint $\tilde{f}: X \to GY$ is a weak equivalence in $\mathcal{M}$ .

This criterion is often easier to verify than checking the unit and counit conditions directly.

ExampleProjective and Injective Model Structures

For the category $\mathbf{Ch}(\mathcal{A})$ of chain complexes over an abelian category $\mathcal{A}$ with enough projectives and injectives, the identity functor establishes a Quillen equivalence between the projective and injective model structures: $\text{id}: \mathbf{Ch}(\mathcal{A})_{\text{proj}} \rightleftarrows \mathbf{Ch}(\mathcal{A})_{\text{inj}}: \text{id}$

Both model structures compute the derived category $\mathbf{D}(\mathcal{A})$ .

DefinitionTotal Derived Functor

For a Quillen adjunction $F \dashv G$ , the total derived functors are: $\mathbf{L}F: \text{Ho}(\mathcal{M}) \rightleftarrows \text{Ho}(\mathcal{N}): \mathbf{R}G$

These form an adjunction on the homotopy categories. When $F \dashv G$ is a Quillen equivalence, the total derived functors form an adjoint equivalence.

ExampleModel Categories of Simplicial Objects

For any model category $\mathcal{M}$ , there is a model structure on $s\mathcal{M}$ (simplicial objects in $\mathcal{M}$ ) where weak equivalences and fibrations are defined levelwise. The constant diagram functor and the functor taking levelwise colimits form a Quillen adjunction: $\text{const}: \mathcal{M} \rightleftarrows s\mathcal{M}: \text{colim}$

This provides a systematic way to study homotopy-coherent diagrams.

Quillen adjunctions and equivalences are the natural morphisms and equivalences in the theory of model categories, allowing us to compare different homotopy theories and transfer results between them systematically.