Core of a category - Knowlpedia

Core idea

The core of a category $\mathcal C$ , written $\mathcal C^\simeq$ or $\operatorname{Core}(\mathcal C)$ , is the groupoid with the same objects as $\mathcal C$ and with

\operatorname{Hom}_{\mathcal C^\simeq}(X,Y) =\{f\in\operatorname{Hom}_{\mathcal C}(X,Y):f\text{ is an isomorphism}\}.

It is the largest subcategory of $\mathcal C$ containing every object and only invertible morphisms.

Functoriality

Every functor $F:\mathcal C\to\mathcal D$ preserves isomorphisms, so restriction gives a functor

F^\simeq:\mathcal C^\simeq\longrightarrow\mathcal D^\simeq.

Consequently, taking the core is functorial.

Interpretation

Passing to the core forgets all noninvertible maps but retains automorphism groups. Passing further to isomorphism classes forgets those automorphisms too. For example, the core of the category of smooth manifolds has diffeomorphisms as its morphisms, while the original category also contains arbitrary smooth maps.

References

Emily Riehl, Category Theory in Context, Dover, 2016. Author-hosted text. Relevant: §1.1, categories and their maximal subgroupoids.