nLab core groupoid


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



For ๐’ž\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 โˆž \infty -groupoid inside it.

Since a functor out of a groupoid ๐’ข\mathcal{G}, and generally an (n,r)-functor out of an โˆž \infty -groupoid, necessarily takes values in isomorphisms or generally in equivalences, hence in the core of its codomain,

Cat n,r(๐’ข,๐’ž)โ‰ƒGrpd n(๐’ข,core(๐’ž)), 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 nโŠฅโŸตcoreโ†ชFCat n,r. Grpd_n \underoverset {\underset{core}{\longleftarrow}} {\overset{F}{\hookrightarrow}} {\bot} Cat_{n,r} \,.

(Incidentally, the forgetful functor FF from GrpGrp to CatCat also has a left adjoint, called the free groupoid-construction.)



For CโˆˆC \in Cat a category, its core core(C)โˆˆcore(C) \in Grpd
is the groupoid which is the maximal sub-groupoid of CC: the subcategory consisting of all objects of CC but with morphisms only the isomorphisms of CC.

This construction extends to a 1-functor

Core:Catโ†’Grpd. 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.


Universal property


The core-functor of def. is right adjoint to the full subcategory-inclusion U:Grpdโ†’CatU \colon Grpd \to Cat of groupoids into categories.

GrpdโŠฅโŸตcoreโ†ชCat. Grpd \underoverset {\underset{core}{\longleftarrow}} {\hookrightarrow} {\bot} Cat \,.

Given a category CC and a groupoid AA, a functor

Aโ†’C A \to C

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

Aโ†’Core(C)โ†’C. A \to Core(C) \to C \,.

The left adjoint to U:Grpdโ†’CatU \colon Grpd \to Cat is the localization functor that universally inverts every morphism in CC. On nerves this is Kan fibrant replacement.

Variations and generalizations

โ€ \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 11 or as a dagger category whose morphisms are all linear maps; either way, the core is the same (invertible linear maps of norm exactly 11).

Higher categories

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


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

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