Localization of Categories

Localization of a category at a class of morphisms formally inverts those morphisms, making them isomorphisms. This is the categorical analogue of localizing a ring at a multiplicative set. The derived category $D(\mathcal{A})$ is constructed by localizing the homotopy category $K(\mathcal{A})$ at quasi-isomorphisms.

Definition

Definition5.7Localization of a Category

Let $\mathcal{C}$ be a category and $\mathcal{S}$ a class of morphisms in $\mathcal{C}$ . The localization $\mathcal{C}[\mathcal{S}^{-1}]$ is a category equipped with a functor $Q : \mathcal{C} \to \mathcal{C}[\mathcal{S}^{-1}]$ such that:

$Q(s)$ is an isomorphism for every $s \in \mathcal{S}$ .
$Q$ is universal: for any functor $F : \mathcal{C} \to \mathcal{D}$ sending $\mathcal{S}$ to isomorphisms, there exists a unique $\bar{F} : \mathcal{C}[\mathcal{S}^{-1}] \to \mathcal{D}$ with $\bar{F} \circ Q = F$ .

Definition5.8Calculus of Fractions

A class $\mathcal{S}$ admits a left calculus of fractions if:

$\mathcal{S}$ contains all identities and is closed under composition.
Every diagram $X' \xleftarrow{s} X \xrightarrow{f} Y$ with $s \in \mathcal{S}$ can be completed to a commutative square with $s' \in \mathcal{S}$ .
If $f \circ s = g \circ s$ for $s \in \mathcal{S}$ , there exists $t \in \mathcal{S}$ with $t \circ f = t \circ g$ .

When $\mathcal{S}$ admits a calculus of fractions, morphisms in $\mathcal{C}[\mathcal{S}^{-1}]$ can be represented as "fractions" $f \circ s^{-1}$ .

Examples

ExampleDerived category as localization

$D(\mathcal{A}) = K(\mathcal{A})[\mathrm{qis}^{-1}]$ : the derived category is the localization of the homotopy category at quasi-isomorphisms. The quasi-isomorphisms in $K(\mathcal{A})$ satisfy the calculus of fractions conditions.

ExampleLocalization of a ring

For a commutative ring $R$ and multiplicative set $S$ , the localization $R \to S^{-1}R$ is the localization of the one-object Ab-enriched category $\mathbf{B}R$ at the morphisms $S$ .

ExampleOre localization

A non-commutative ring $R$ with a multiplicative set $S$ satisfying the Ore condition (for each $r \in R$ , $s \in S$ , there exist $r', s'$ with $s'r = r's$ ) admits a localization $S^{-1}R$ . This is the ring-theoretic analogue of the calculus of fractions.

ExampleLocalization of Top at homotopy equivalences

Localizing $\mathbf{Top}$ at homotopy equivalences gives the homotopy category $\mathbf{hTop}$ . However, this localization is difficult to work with directly, which motivates the use of model categories.

ExampleVerdier quotient

For a triangulated category $\mathcal{T}$ and a thick subcategory $\mathcal{S}$ , the Verdier quotient $\mathcal{T}/\mathcal{S}$ is the localization at morphisms whose cone lies in $\mathcal{S}$ .

ExampleGabriel localization

For an abelian category $\mathcal{A}$ and a Serre subcategory $\mathcal{S}$ , the quotient $\mathcal{A}/\mathcal{S}$ is the localization at morphisms whose kernel and cokernel lie in $\mathcal{S}$ . This is used in non-commutative algebraic geometry.

ExampleBousfield localization

In stable homotopy theory, Bousfield localization at a homology theory $E_*$ inverts all $E_*$ -equivalences. The localized category $L_E \mathbf{SHC}$ retains only the $E$ -local information.

ExampleSet-theoretic issues

The localization $\mathcal{C}[\mathcal{S}^{-1}]$ may have proper-class hom-sets even if $\mathcal{C}$ is locally small. The calculus of fractions ensures local smallness for the derived category.

ExampleLocalization and model categories

In a model category $\mathcal{M}$ , the homotopy category $\mathrm{Ho}(\mathcal{M}) = \mathcal{M}[W^{-1}]$ is the localization at weak equivalences $W$ . The model structure provides a concrete description of this localization via fibrant/cofibrant replacements.

ExampleLocalization sequence

For $\mathcal{S} \subseteq \mathcal{T}$ thick, the localization sequence $\mathcal{S} \to \mathcal{T} \to \mathcal{T}/\mathcal{S}$ is analogous to a short exact sequence of triangulated categories. It induces long exact sequences on K-theory.

ExampleDerived category of sheaves

$D^b(\mathbf{Sh}(X))$ is the localization of $K^b(\mathbf{Sh}(X))$ at quasi-isomorphisms. Alternatively, it is the Verdier quotient $K^b(\mathbf{Sh}(X)) / K^b_{\mathrm{ac}}(\mathbf{Sh}(X))$ where the denominator is the subcategory of acyclic complexes.

ExampleLocalization at a single morphism

Localizing a category at a single morphism $f : A \to B$ means formally adding an inverse $f^{-1} : B \to A$ . In the resulting category, $A \cong B$ .

RemarkLocalization and the derived category

The entire construction of the derived category can be summarized as: start with an abelian category $\mathcal{A}$ , form complexes $\mathbf{Ch}(\mathcal{A})$ , mod out by homotopy to get $K(\mathcal{A})$ , and localize at quasi-isomorphisms to get $D(\mathcal{A})$ . The Verdier localization theorem ensures this localization inherits a triangulated structure.