# On 2-actions of strict 2-groupoids?

I'm looking for an opinion if the following makes sense.

A linear representation of a groupoid $\mathcal{G}$ is a functor $$\nabla: \mathcal{G}\longrightarrow \mathsf{Vect}_{\mathbb K},$$ where $\mathsf{Vect}_{\mathbb K}$ is the category of vector spaces over $\mathbb K$.

I presume we could define analogously a linear 2-representation of a strict $2$-grupoid $2\textrm{-}\mathcal{G}$ as a strict 2-functor:

$$2\textrm{-}\nabla: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Vect}_{\mathbb K},$$ where $2\textrm{-}\mathsf{Vect}_{\mathbb K}$ is the $2$-category of 2-vector spaces (the objects are groupoids internal to $\mathsf{Vect}_{\mathbb K}$). I don't know if this definition appears in the literature but I believe it sounds reasonable.

Now, how to make sense of a non-linear version of a 2-representation of a 2-groupoid?

I'm inclined to define a 2-action of a strict $2$-groupoid $2\textrm{-}\mathcal{G}$ is a strict two functor

$$\rho: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Grpd},$$ where $2\textrm{-}\mathsf{Grpd}$ is the 2-category of 2-grupoids (the objects are groupoids internal to $\mathsf{Grpd}$).

To me it sounds $2\textrm{-}\mathsf{Grpd}$ must be the non-linear version of $2\textrm{-}\mathsf{Vect}_{\mathbb K}$.

But I'm not quite certain if this is really coeherent with what an action must mean. I believe category theory must provide some prototype of what an action must be in any setting and I should follow that, but that is beyond my categorical knowledge.

To be precise, my question is, is it "right" to define a 2-action of a strict 2-groupoid as strict 2-functor $\rho: 2\textrm{-}\mathcal{G}\longrightarrow 2\textrm{-}\mathsf{Grpd}$?

I'm looking for arguments which could support it is "natural" to make such definition.

Thanks.

• When do such things occur? I'm very curious. – Yosemite Sam Mar 3 '18 at 16:30
• When you try to integrate some objects which appear in the context of Lie algebroids, VB-algebroids, etc. For example, in this pre-print arxiv.org/pdf/1608.00664.pdf the authors integrate VB-algebroids using strict 2-functors. For me, it appears when I try to integrate non-linear versions of VB-algebroids. – PtF Mar 3 '18 at 17:24
• There's a general account of actions by infinity-groups (ncatlab.org/nlab/show/infinity-action) and indeed infinity-groupoids (ncatlab.org/nlab/show/groupoid+infinity-action). – David Corfield Mar 3 '18 at 22:21
• I think allowing homomorphisms of 2-categories will lead to lead to more examples. – David Roberts Mar 5 '18 at 8:51

A few years ago I collected some results from

in a short beamer that you can find here. It is completely expository (it might have a considerable amount of errors in it!)

For information: As 2-groups are equivalent to crossed modules, you could do a google search for representation theory of those. There is a thesis from Bangor by Forrester-Barker which looked at the linear representation theory of such and other people have explored that as well. One such is an AMS memoir: Infinite-Dimensional Representations of 2-Groups by John Baez, Aristide Baratin, Laurent Freidel, Derek K. Wise, which should be useful.

You could look at 2-actions of a 2-groupoid on any 2-category, and in particular on 2-groupoids. There are links with 2-stacks of 2-groupoids, and that is a very interesting area.

• I appreciate your answer =) – PtF Mar 4 '18 at 22:13

Here are some suggestions on this matter.

In this area one should be aware that there are equivalences of categories

(crossed modules of groupoids) $\sim$ ($2$-groupoids) $\sim$ (double groupoids with connections)

and one has to consider in which of these one should try to formulate results.

In the paper "Modelling and computing homotopy types: I" the distinction is made between "Narrow" and "Broad" algebraic models, and their uses, and for this comparison $2$-groupoids are between the two.

In this paper we (Mackenzie and RB) define actions of a double groupoid on a groupoid, and this seems easily extendable to actions of an $n$-fold groupoid on an $(n-1)$-fold groupoid.

The book Nonabelian Algebraic Topology (NAT) defines cubical $\omega$-groupoids, and it also gives a cubical Dold-Kan theorem. So from chain complexes of vector spaces over $\mathbb K$ one gets a category $\omega$-$\mathsf{Vect}_{\mathbb K}$; so one can start to consider "representations".

There is evidence that such discussions in a globular setting, i.e. in terms of $n$-categories, would be more awkward. Certainly I have found them so. That is why the book NAT strongly uses cubical methods.

These ideas are not well worked out, but I hope give some possibilities.