Core of a category
In mathematics, especially category theory, the core of a category C is the category whose objects are the objects of C and whose morphisms are the invertible morphisms in C.[1][2][3] In other words, it is the largest groupoid subcategory.
As a functor , the core is a right adjoint to the inclusion of the category of (small) groupoids into the category of (small) categories.[1] On the other hand, the left adjoint to the above inclusion is the fundamental groupoid functor.
For ∞-categories, is defined as a right adjoint to the inclusion ∞-Grpd ∞-Cat.[4] The core of an ∞-category is then the largest ∞-groupoid contained in . The core of C is also often written as . The left adjoint to the above inclusion is given by a localization of an ∞-category.
In Kerodon, the subcategory of a 2-category C obtained by removing non-invertible morphisms is called the pith of C.[5] It can also be defined for an (∞, 2)-category C;[6] namely, the pith of C is the largest simplicial subset that does not contain non-thin 2-simplexes.
References
- ↑ 1.0 1.1 Pierre Gabriel, Michel Zisman, § 1.5.4., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) [1]
- ↑ "Construction 1.3.5.4.". https://kerodon.net/tag/007G.
- ↑ core groupoid in nLab
- ↑ § 3.5.2. and Corollary 3.5.3. of 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.
- ↑ "Construction 2.2.8.9 (The Pith of a 2-Category).". https://kerodon.net/tag/00AL.
- ↑ "5.4.5 The Pith of an (∞,2)-Category". https://kerodon.net/tag/01XA.
Further reading
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