nLab groupoid object




A groupoid object internal to an ambient category 𝒞\mathcal{C} is equivalently

This notion is the horizontal categorification of that of a group object.


Let CC be a category with finite limits, and let Grpd\Grpd be the category of small groupoids. We can define groupoid objects representably:


A groupoid object in CC is a functor F:C opGrpdF\colon C^{op}\to \Grpd such that

  1. there is an object g 0C 0g_0\in C_0 such that there is a natural isomorphism F(c) 0C(c,g 0)F(c)_0\simeq C(c,g_0)

  2. there is an object g 1C 0g_1\in C_0 such that there is a natural isomorphism F(c) 1C(c,g 1)F(c)_1\simeq C(c,g_1)

We can also define them more explicitly:


A groupoid object in CC is an internal category AA such that there is an “inverse-assigning morphism” i:A 1A 1i\colon A_1 \to A_1 satisfying certain axioms…


  • A groupoid in Top\Top is a topological groupoid.

  • A groupoid in Diff\Diff is a Lie groupoid. (Note that Diff\Diff does not have all pullbacks, but by suitable conditions on the source and target map we can ensure that the requisite pullbacks do exist.)


For more references see also at internal category and internalization.

The general definition of internal categories seems to have first been formulated in:

  • Alexander Grothendieck, p. 106 (9 of 21) of: FGA Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf, English translation: web version)

following the general principle of internalization formulated in

  • Alexander Grothendieck, p. 370 (3 of 23) in: FGA Technique de descente et théorèmes d’existence en géométrie algébriques. II: Le théorème d’existence en théorie formelle des modules, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 (numdam:SB_1958-1960__5__369_0, pdf, English translation: web version)

The concept of topological groupoids and Lie groupoids goes back to

  • Charles Ehresmann, Catégories topologiques et categories différentiables, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 (pdf, zbMath:0205.28202)

and their understanding as categories internal to TopologicalSpaces and to SmoothManifolds is often attributed to

but it seems that the definition is not actually contained in there, certainly not in its simple and widely understood form due to Grothendieck 61.

Discussion of cartesian closed 2-categories of internal groupoids (mostly in Top, hence for topological groupoids):

A characterisation of internal groupoids as involutive-2-links? appears in:

  • Nelson Martins-Ferreira?, Internal groupoids as involutive-2-links, 2023. (arXiv:2305.13537)

Last revised on April 18, 2024 at 14:39:27. See the history of this page for a list of all contributions to it.