core

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).

The core-functor of def. 1 is right adjoint to the forgetful functor $U \colon Grpd \to Cat$ from groupoids to categories.

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*.

Revised on September 12, 2015 12:58:40
by Adeel Khan
(217.187.117.244)