This entry is about the concept in category theory. For the core of a ring see there.

For $\mathcal{C}$ any category or, more generally, (n,r)-category, its *core* is the maximal groupoid inside it, generally the maximal sub-(n,0)-category, hence the maximal $\infty$-groupoid inside it.

Since a functor out of a groupoid $\mathcal{G}$, and generally an (n,r)-functor out of an $\infty$-groupoid, necessarily takes values in isomorphisms or generally in equivalences, hence in the core of its codomain,

$Cat_{n,r}
\big(
\mathcal{G}
,\,
\mathcal{C}
\big)
\;\;
\simeq
\;\;
Grpd_{n}
\big(
\mathcal{G}
,\,
core(\mathcal{C})
\big)
\,,$

the core construction is the right adjoint to the full inclusion of groupoids among categories (generally: (n,0)-categories among (n,r)-categories):

$Grpd_n
\underoverset
{\underset{core}{\longleftarrow}}
{\overset{F}{\hookrightarrow}}
{\bot}
Cat_{n,r}
\,.$

(Incidentally, the forgetful functor $F$ from $Grp$ to $Cat$ also has a left adjoint, called the *free groupoid*-construction.)

For $C \in$ Cat a category, its **core** $core(C) \in$ 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

$Core \colon Cat \to Grpd
\,.$

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).

Given a preordered set, regarded as a category, taking its core is the same as partitioning the set into equivalence classes of the preorder.

The core of FinSet is known as the *permutation groupoid* or *symmetric groupoid* or similar.

A *combinatorial species* is a functor from the symmetric groupoid to Set.

The core-functor of def. is right adjoint to the full subcategory-inclusion $U \colon Grpd \to Cat$ of groupoids into categories.

$Grpd
\underoverset
{\underset{core}{\longleftarrow}}
{\hookrightarrow}
{\bot}
Cat
\,.$

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

$A \to C$

(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

$A \to Core(C) \to C
\,.$

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

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 $1$).

The **core** of an $n$-category is the $n$-groupoid consisting only of equivalences at each level; the **core** of an $\infty$-category is similarly an $\infty$-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*.

- Pierre Gabriel, Michel Zisman, ยง1.5.4 of:
*Calculus of fractions and homotopy theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete**35**, Springer (1967) [doi:10.1007/978-3-642-85844-4, pdf]

Discussion in the generality of $(\infty,n)$-categories:

- Jacob Lurie, Around Prop. 1.1.14 in:
*(โ,2)-Categories and the Goodwillie Calculus*(arXiv:0905.0462)

Last revised on December 20, 2023 at 13:10:00. See the history of this page for a list of all contributions to it.