category object in an (∞,1)-category, groupoid object
internalization and categorical algebra
algebra object (associative, Lie, …)
A groupoid object internal to an ambient category is equivalently
an internal category in equipped with an inverse morphism-assigning map.
This notion is the horizontal categorification of that of a group object.
Let be a category with finite limits, and let be the category of small groupoids. We can define groupoid objects representably:
A groupoid object in is a functor such that
there is an object such that there is a natural isomorphism
there is an object such that there is a natural isomorphism
We can also define them more explicitly:
A groupoid object in is an internal category such that there is an “inverse-assigning morphism” satisfying certain axioms…
A groupoid in is a topological groupoid.
A groupoid in is a Lie groupoid. (Note that does not have all pullbacks, but by suitable conditions on the source and target map we can ensure that the requisite pullbacks do exist.)
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
For more references see also at internal category and internalization.
The general definition of internal categories seems to have first been formulated in:
following the general principle of internalization formulated in
The concept of topological groupoids and Lie groupoids goes back to
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:
Last revised on April 18, 2024 at 14:39:27. See the history of this page for a list of all contributions to it.