nLab
core

Definition

Definition

For $C\in$ Cat a category, its core

$\mathrm{core}(C)\in$ Grpd

is the groupoid which is the maximal sub-groupoid of $C$: the subcategory consisting of all objects of $C$ but with morphisms only the isomorphisms of $C$.

This construction extends to a 1-functor

$$\mathrm{Core}:\mathrm{Cat}\to \mathrm{Grpd}.$$Core \colon Cat \to Grpd \,.

Remark

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).

Properties

Universal property

Proposition

The core-functor of def. 1 is right adjoint to the forgetful functor $U:\mathrm{Grpd}\to \mathrm{Cat}$ from groupoids to categories.

Proof

Given a category $C$ and a groupoid $A$, a functor

$$A\to C$$A \to C

(hence a functor out of the underlying category $U(A)$ of $A$) has to send isomorphisms to isomorphisms, hence has to send every morphism of $A$ to an isomorphism in $C$. This means that it factors through the core-inclusion

$$A\to \mathrm{Core}(C)\to C.$$A \to Core(C) \to C \,.

Remark

The left adjoint to $U:\mathrm{Grpd}\to \mathrm{Cat}$ is the localization functor that universally inverts every morphism in $C$. On nerves this is Kan fibrant replacement.

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.

For more on this see also at category object in an (infinity,1)-category.

Related concepts

• core in a 2-category