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 or as a result of Simpson.

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

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

such that

• $W^{-1}S$ is an ∞-category,
• the image $u(W_1)$ consists of invertible maps,
• the induced map on ∞-categories

  $u^{*}:\mathrm{Hom}(W^{-1}S,-)\stackrel{\sim }{\to }{\mathrm{Hom}}_W(S,-)$

is invertible.

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

A Dwyer–Kan localization that admits a right adjoint is called a localization in the sense of Bousfield. For example, the inclusion ∞-Grpd $\hookrightarrow$ ∞-Cat has a left adjoint given by the localization that inverts all maps (functors). 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 $W\subset C$ a subcategory of weak equivalences so that C is a category of cofibrant objects. Then the localization $C\to L(C)$ induces an equivalence

$L(\underset{\_}{\mathrm{Hom}}(X,C))\stackrel{\sim }{\to }\underset{\_}{\mathrm{Hom}}(X,L(C))$

for each simplicial set X.

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

$L(\underset{\_}{\mathrm{Hom}}(I,C))\stackrel{\sim }{\to }\underset{\_}{\mathrm{Hom}}(I,L(C))$

for each small category I.

See also

• ∞-topos

References

• Cisinski, Denis-Charles (2023) (in en). Higher Categories and Homotopical Algebra. Cambridge University Press. ISBN 978-1108473200. https://cisinski.app.uni-regensburg.de/CatLR.pdf. 
• Lurie, Jacob (2009). Higher Topos Theory. Princeton University Press. ISBN 978-0-691-14048-3. 
• Land, Markus (2021). Introduction to Infinity-Categories. Compact Textbooks in Mathematics. doi:10.1007/978-3-030-61524-6_2. ISBN 978-3-030-61523-9. 
• Daniel Carranza, Chris Kapulkin, Zachery Lindsey, Calculus of Fractions for Quasicategories [arXiv:2306.02218]

Further reading

• https://mathoverflow.net/questions/310731/localization-of-infty-categories
• https://ncatlab.org/nlab/show/localization+of+an+%28infinity%2C1%29-category