nLab groupoid object

Idea

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

• a groupoid internal to $\mathcal{C}$;

• an internal category in $\mathcal{C}$ equipped with an inverse morphism-assigning map.

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

Definitions

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

Definition

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

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

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

We can also define them more explicitly:

Definition

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

Examples

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

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

Related concepts

• monoid, monoid object, monoid object in an (∞,1)-category

• group, group object, group object in an (∞,1)-category

• groupoid, groupoid object, enriched groupoid, simplicial groupoid, groupoid object in an (∞,1)-category

• infinity-groupoid, infinity-groupoid object, groupoid object in an (∞,1)-category

• ring, ring object

References

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

• Charles Ehresmann, Catégories structurées, Annales scientifiques de l’École Normale Supérieure, Série 3, Tome 80 (1963) no. 4, pp. 349-426 (numdam:ASENS_1963_3_80_4_349_0)

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):

• Susan B. Niefield, Dorette A. Pronk, Internal groupoids and exponentiability, Cahiers de topologie et géométrie différentielle catégoriques, LX 4 (2019) (pdf)

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

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