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 ${\displaystyle C\mapsto \operatorname {core} (C)}$ , 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, ${\displaystyle \operatorname {core} }$ is defined as a right adjoint to the inclusion ∞-Grpd ${\displaystyle \hookrightarrow }$ ∞-Cat.^[4] The core of an ∞-category ${\displaystyle C}$ is then the largest ∞-groupoid contained in ${\displaystyle C}$ . The core of C is also often written as ${\displaystyle C^{\simeq }}$ . 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. Pierre Gabriel, Michel Zisman, § 1.5.4., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) [1]
2. "Construction 1.3.5.4". Kerodon.
3. core groupoid at the nLab
4. § 3.5.2. and Corollary 3.5.3. of Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.
5. "Construction 2.2.8.9 (The Pith of a 2-Category)". Kerodon.
6. "5.4.5 The Pith of an (∞,2)-Category". Kerodon.

Further reading

• https://mathoverflow.net/questions/347477/what-is-the-core-of-a-localization