homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
The (2,1)-category $Grpd$ is the 2-category whose
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.
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.
On the local cartesian closure of the 2-category of groupoids (and the failure of this property for the 1-category version):
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]
Last revised on November 6, 2023 at 11:53:51. See the history of this page for a list of all contributions to it.