# Notions of Lie 2-groupoids

The term Lie $$2$$-groupoid is used in the literature in more than one context. Some examples are given below:

1. Ginot and Stiénon's paper $$G$$-gerbes, principal $$2$$-group bundles and characteristic classes defines a Lie $$2$$-groupoid to be a double Lie groupoid in the sense of Brown - Determination of a double Lie groupoid by its core diagram satisfying certain conditions.
2. Rajan Amit Mehta and Xiang Tang's paper From double Lie groupoids to local Lie $$2$$-groupoids says that "The simplicial approach to $$n$$-groupoids goes back to Duskin [Higher-dimensional torsors and the cohomology of topoi: the abelian theory] in the discrete case, and the smooth analogue appeared in [Henriques - Integrating L-infinity algebras]." They define Lie $$2$$-groupoid to be a simplicial manifold $$X=(X_k)$$ with some condition on the horn maps.
3. Matias del Hoyo and Davide Stefani's paper The general linear $$2$$-groupoid defines a Lie $$2$$-groupoid to be a Lie $$2$$-category and a $$2$$-category satisfying certain conditions.
4. The n-lab page on Lie $$2$$-groupoids says "a Lie 2-groupoid is a 2-truncated $$\infty$$-Lie groupoid". No further discussion is given there. Clicking the link $$\infty$$-Lie groupoid takes you to the page Lie $$n$$-groupoid, which does not contain many details.

Questions :

1. Are these notions genuinely different notions introduced for different purposes or are they all the same candidate wearing different outfits?
2. Are there other notions of Lie $$2$$-groupoids in the literature?
• Any comment/reference that helps to understand the above notions are welcome as answers... – Praphulla Koushik May 20 '20 at 18:02
• Please do not use $\infty$-category theory language. I really do not know anything about it at present. – Praphulla Koushik May 20 '20 at 18:02
• Sorry for so many typos. It was around 1 AM when I typed this. I misremembered Ginot and Stienon’s paper as some paper by Ginot alone. Thanks for correcting typos @LSpice – Praphulla Koushik May 21 '20 at 2:28

They are essentially all the same thing: some globular, some simplicial (taking the nerve goes from the former to the latter). The only subtlety is perhaps in the requirement on the maps $$d_{2,0}, d_{2,2}$$ to be surjective submersions in the del Hoyo–Stefani paper. This is not unusual for the simplicial approach to n-groupoids in non-finitely complete sites: the Kan condition (which is what these maps are about) shouldn't be just encoded by a certain map being an epimorphism, but being a cover of some sort. This is also the content of Definition 1.2 in Henriques' paper (note that you have linked to the arXiv version 2 of the paper, and that's what I'm referring to. Version 1 has slightly different material and you should also check it out). I don't recall offhand if this is automatic for the globular definition as in Ginot–Stiénon's paper.

So I would say with some confidence that 2.–4. are the same, apart from dealing with a 2-groupoid vs the nerve of a 2-groupoid, and 1. might be very slightly more general, though only in a small technical hypothesis that a certain small class of surjective maps are not required to be submersions a priori. I didn't dig into the techicalities of Brown–Mackenzie (cited by Ginot–Stiénon) to see if one gets submersions automatically. Ideally the nLab page on Lie 2-groupoids would get updated by some friendly soul who wants to put in the work to explain the more elementary viewpoint.

On a historical note, apart from the technical hypothesis, these definitions essentially go back to Charles Ehresmann, who defined (strict) 2-categories, double categories, internal categories (including Lie categories) and so on. Focusing specifically on the case 2-groupoids/double groupoids/Lie groupoids arose slowly, partly driven by Brown, Mackenzie, Pradines, Haefliger, .... and so on.

As far as different notion of 2-groupoids go, one can consider the composition functor on hom-groupoids to be a map of their associated stacks, hence an anafunctor as opposed to an internal functor. This viewpoint is implicit in

Christian Blohmann, Stacky Lie groups, Int. Mat. Res. Not. (2008) Vol. 2008: article ID rnn082, 51 pages, doi:10.1093/imrn/rnn082, arXiv:math/0702399

where the one-object case is considered (the general stacky notion is discussed by Breen in Bitorseurs et cohomologie non abélienne. In The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, pp. 401–476 (1990)). More explicitly this notion is considered in

Chenchang Zhu, Lie $$n$$-groupoids and stacky Lie groupoids arXiv:math/0609420

(with a more general theory in her followup paper 0801.2057) using the language of stacks. In general, what this means is that if you take the viewpoint on internal 2-groupoids as in Definition 2.1 of

A bigroupoid's topology (or, Topologising the homotopy bigroupoid of a space), Journal of Homotopy and Related Structures 11 Issue 4 (2016) pp 923–942, doi:10.1007/s40062-016-0160-0, arXiv:1302.7019.

(there given for topological bigroupoids, but one could repeat the definition mutatis mutandis for Lie 2-groupoids, taking the natural transformations $$a,r,l,e,i$$ to identities), then the hom-groupoid is a Lie groupoid over the square of the manifold of objects, with some properties, and then composition is an anafunctor between Lie groupoids over this manifold. And so on. This provides and extra layer of weakness to the structure. This viewpoint (in the special case of Lie 2-groups) was also used by Chris Schommer-Pries. One should view this as some kind of internal enrichment of a Lie groupoid in the (cartesian monoidal) category of differentiable stacks, much like ordinary 2-groupoids are groupoids enriched in the category of groupoids.

• Ok. Thanks for the answer.. There are so many things to digest here :) I will go one by one and ask if I have any specific question... – Praphulla Koushik May 21 '20 at 5:37