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.¹²³ 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.