nLab
core

Usual notion

The core $\mathrm{core}(C)$ of a category $C$ is its maximal sub-groupoid: the subcategory consisting of all objects but with morphisms only the isomorphisms.

We usually think of a groupoid as a special kind of category, but we can also think of a category as a groupoid equipped with additional morphisms. (This is possible because Grpd is a reflective subcategory of Cat.) One level decategorified, we usually think in the opposite way: a poset is a set equipped with a partial order, but we can also think of a set as a special kind of poset (specifically, a symmetric one).

Variations and generalizations

$†$-Categories

The core of a dagger category consists of its unitary isomorphisms only. This is why, for example, it makes sense to think of Hilb either as a category whose morphisms are linear maps bounded by $1$ or as a dagger category whose morphisms are all linear maps; either way, the core is the same (invertible linear maps of norm exactly $1$).

Higher categories

The core of an $n$-category is the $n$-groupoid consisting only of equivalences at each level; the core of an $\mathrm{\infty }$-category is similarly an $\mathrm{\infty }$-groupoid: the core of a quasicategory is the maximal Kan complex inside it.