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
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 2023, Definition 7.11.5.
- ^ Lurie 2009, Theorem 5.5.1.1.
- ^ Cisinski 2023, Definition 7.1.2.
- ^ Land 2021, Definition 5.1.20.
- ^ Land 2021, Example just before Proposition 5.1.24.
- ^ Cisinski 2023, Proposition 7.9.2.
- ^ Cisinski 2023, Theorem 7.9.8.
- Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.
- Lurie, Jacob (2009). Higher Topos Theory. Princeton University Press. arXiv:math/0608040. 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. Zbl 1471.18001.
- Daniel Carranza, Chris Kapulkin, Zachery Lindsey, Calculus of Fractions for Quasicategories [arXiv:2306.02218]