nLab core

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.

Sincefunctor 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):

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

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

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

• A combinatorial species is defined as a presheaf, that is, a contravariant functor to Set, on the core of FinSet.

Every groupoid has a contravariant functor to itself. It preserves the objects and sends the arrows to their inverses.

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

• core in a 2-category

References

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)