Localization of an ∞-category

In mathematics, specifically in higher category theory, a localization of an ∞-category is an ∞-category obtained by inverting some maps.

An ∞-category is a presentable ∞-category if it is a localization of an ∞-presheaf category in the sense of Bousfield, by definition^[1] or as a result of Simpson.^[2]

Definition

Let S be a simplicial set and W a simplicial subset of it. Then the localization in the sense of Dwyer–Kan is a map

${\displaystyle u:S\to W^{-1}S}$

such that

• ${\displaystyle W^{-1}S}$ is an ∞-category,
• the image ${\displaystyle u(W_{1})}$ consists of invertible maps,
• the induced map on ∞-categories

  ${\displaystyle u^{*}:\operatorname {Hom} (W^{-1}S,-){\overset {\sim }{\to }}\operatorname {Hom} _{W}(S,-)}$

is invertible.^[3]

When W is clear form the context, the localized category ${\displaystyle S^{-1}W}$ is often also denoted as ${\displaystyle L(S)}$.

A Dwyer–Kan localization that admits a right adjoint is called a localization in the sense of Bousfield.^[4] For example, the inclusion ∞-Grpd ${\displaystyle \hookrightarrow }$ ∞-Cat has a left adjoint given by the localization that inverts all maps (functors).^[5] The right adjoint to it, on the other hand, is the core functor (thus the localization is Bousfield).

Properties

Let C be an ∞-category with small colimits and ${\displaystyle W\subset C}$ a subcategory of weak equivalences so that C is a category of cofibrant objects. Then the localization ${\displaystyle C\to L(C)}$ induces an equivalence

${\displaystyle L({\underline {\operatorname {Hom} }}(X,C)){\overset {\sim }{\to }}{\underline {\operatorname {Hom} }}(X,L(C))}$

for each simplicial set X.^[6]

Similarly, if C is a hereditary ∞-category with weak fibrations and cofibrations, then

${\displaystyle L({\underline {\operatorname {Hom} }}(I,C)){\overset {\sim }{\to }}{\underline {\operatorname {Hom} }}(I,L(C))}$

for each small category I.^[7]

See also

• ∞-topos