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

such that
is an ∞-category,
- the image
consists of invertible maps,
- the induced map on ∞-categories

- is invertible.[3]
When W is clear form the context, the localized category
is often also denoted as
.
A Dwyer–Kan localization that admits a right adjoint is called a localization in the sense of Bousfield.[4] For example, the inclusion ∞-Grpd
∞-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
a subcategory of weak equivalences so that C is a category of cofibrant objects. Then the localization
induces an equivalence

for each simplicial set X.[6]
Similarly, if C is a hereditary ∞-category with weak fibrations and cofibrations, then

for each small category I.[7]
See also
References
- ^ Cisinski, Definition 7.11.5. harvnb error: no target: CITEREFCisinski (help)
- ^ Lurie 2009, Theorem 5.5.1.1.
- ^ Cisinski, Definition 7.1.2. harvnb error: no target: CITEREFCisinski (help)
- ^ Markus 2021, Definition 5.1.20. harvnb error: no target: CITEREFMarkus2021 (help)
- ^ Markus 2021, Example just before Proposition 5.1.24. harvnb error: no target: CITEREFMarkus2021 (help)
- ^ Cisinski, Proposition 7.9.2. harvnb error: no target: CITEREFCisinski (help)
- ^ Cisinski, Theorem 7.9.8. harvnb error: no target: CITEREFCisinski (help)
Further reading