nLab
core

Contents

Definition

Definition

For CC \in Cat a category, its core

core(C)core(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:CatGrpd. 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:GrpdCatU \colon Grpd \to Cat from groupoids to categories.

Proof

Given a category CC and a groupoid AA, a functor

AC 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

ACore(C)C. A \to Core(C) \to C \,.
Remark

The left adjoint to U:GrpdCatU \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

\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 \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.

Revised on November 27, 2012 16:59:37 by Urs Schreiber (131.174.40.3)