2Mon𝒲𝒾𝓀𝒾

Groupoid Core

The groupoid core core(π’žοΈ€) of a category π’žοΈ€ is the β€œgroupoid underlying π’žοΈ€β€, in the sense of Voevodsky.

Definition

Definition Groupoid core of π’žοΈ€ The groupoid core of π’žοΈ€ is the sub category core(π’žοΈ€) of π’žοΈ€ which has the same objects, but only the isomorphisms of π’žοΈ€ as morphisms.

It follows directly from the definition that core(π’žοΈ€) is a groupoid.

Relationship with π’žοΈ€

For any groupoid 𝒒︀, it holds that π‚πšπ­(𝒒︀,π’žοΈ€)≅𝐆𝐫𝐩𝐝(𝒒︀,core(π’žοΈ€))