nLab core groupoid

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

Idea

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.)

Definition

Definition

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

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

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).

Examples

Example

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

Example

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

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

Properties

Universal property

Proposition

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

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

Proof

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

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

Remark

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.

Variations and generalizations

$\dagger$-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 $\infty$-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.

Related concepts

• free groupoid, free category

• fiberwise core

• core in a 2-category

• decategorification

References

• Pierre Gabriel, Michel Zisman, §1.5.4 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: (∞,2)-Categories and the Goodwillie Calculus (arXiv:0905.0462)