nLab Grpd

Definition

The (2,1)-category $Grpd$ is the 2-category whose

• objects are small groupoids;

• morphisms are functors;

• 2-morphisms are natural transformations, which are necessarily natural isomorphisms.

This is the full sub-2-category of Cat on those categories that are groupoids.

Properties

Presentation

One may regard $Grpd$ also just as a 1-category by ignoring the natural isomorphisms between functors. This 1-category may be equipped with the natural model structure on groupoids to provide a 1-categorical presentation of the full $(2,1)$-category.

Related categories

• Pos

• Set

• $Grpd$, ∞Grpd

• Cat, (∞,1)Cat

• (∞,n)Cat

References

On the local cartesian closure of the 2-category of groupoids (and the failure of this property for the 1-category version):

• Erik Palmgren, Groupoids and local cartesian closure (2003) [pdf]

Discussion of $Grpd$ as (with some tweaks) categorical semantics for homotopy type theory (the “groupoid model”, precursor to the simplicial set model):

• Martin Hofmann, Thomas Streicher, The groupoid model refutes uniqueness of identity proofs, Proceedings Ninth Annual IEEE Symposium on Logic in Computer Science (1994) [doi:10.1109/LICS.1994.316071]

• Martin Hofmann, Thomas Streicher, The groupoid interpretation of type theory, in: Giovanni Sambin et al. (eds.), Twenty-five years of constructive type theory, Proceedings of a congress, Venice, Italy, October 19-21, 1995. Oxford: Clarendon Press. Oxf. Logic Guides. 36 (1998) 83-111 [ISBN:9780198501275, ps, pdf]

• Martin Hofmann, The groupoid interpretation of type theory, a personal retrospective, talk at HoTT at DMV2015 (2015) [slides]

• Ethan Lewis, Max Bohnet, The groupoid model of type theory, seminar notes (2017) [pdf, pdf]

category: category