# What is the notion of a group object and its action in a 2-category?

It is well known that a group object in a category $$C$$ (with terminal object $$1$$ and such that any two objects of $$C$$ have a product) is defined as an object $$G$$ in $$C$$ with the following morphisms:

$$m:G \times G \rightarrow G$$, $$e:1 \rightarrow G$$, $$\mathit{inv}: G \rightarrow G$$

satisfying some conditions modelled on the group axioms such that $$m$$ behaves as the multiplication map, e behaves as the identity and $$\mathit{inv}$$ behaves as the inverse map.

The group action (right) of a group object $$G$$ on an object $$X$$ in $$C$$ can be defined as a morphism $$\rho:X \times G \rightarrow X$$ in $$C$$ such that the following two diagrams are commutative:

where $$\mathit{id}_X$$, $$\mathit{id}_G$$ are identity morphisms at $$X$$ and $$G$$ respectively and $$\mathit{pr}_1$$ is the first projection on $$X$$ from the product $$X \times 1$$.

My Question are the following:

(1) What is the analogue of the above notion in a $$2$$-category? I couldn't find any literature in this direction.

So I tried to guess it's definition roughly in the following way:

Let $$C$$ be a 2-category (with terminal object $$1$$ and such that any two objects of $$C$$ have a product in the context of 2- category as mentioned in https://ncatlab.org/nlab/show/2-limit#2limits_over_diagrams_of_special_shape ). I define a group object in the 2-category $$C$$ as an object $$G$$ with the following 1-morphisms:

$$m \in C(G \times G ,G)$$, $$e \in C(1,G)$$, $$inv \in C(G,G)$$ satisfying some conditions similar as above but in this case every equality in the conditions will be replaced by an invertible 2-morphism satisfying certain appropriate coherent conditions.

Correspondingly the action of $$G$$ on an object $$X$$ in $$C$$ will be defined exactly in the same way but the above two diagrams will be commutative only upto invertible 2-morphisms.

Is my guess correct?

Even if it is correct but writing all the details (taking care of all the invertible 2 morphisms) seems very complicated to me and seems an inappropriate definition to work with.(Both in the context of strict 2-category and bicategory)

So what should be an appropriate definition for a group object in a 2-category and its corresponding action on an object? (Both in the context of strict 2-category and bicategory)

Secondly,

It is well known that the Strict 2-Group is a group object in Cat(when Cat is considered as a 1 -category or a usual category ).

But then

(2) What is the group object in Cat (when Cat is considered as strict 2-category)?

I would be also very grateful if someone can refer some literatures in this direction.

Thank you.

• 4 upvotes, 4 bookmarks is a good number :D Hope some one answers your question.. May 9 '20 at 5:01
• @PraphullaKoushik I also hope that :D May 9 '20 at 5:15

I'll have a go at answering your question (although higher category-theory is absolutely not my area of expertise).

The conditions for $$(G,m,e,inv)$$ to be a group object is stipulated by the following relations

1. $$m\circ (e\times inv)\circ\Delta=m\circ (inv\times e)\circ\Delta=id_G$$
2. $$m\circ (m\times id_G)=m \circ (id_G\times m)$$
3. $$m\circ(e\times id_G)=m\circ(id_G\times e)=id_G$$

where $$\Delta$$ is the diagonal map and $$(e,id_G)$$ is the obvious map $$G\to G\times G$$.

Writing these out in full is a bit pedantic but it's important. The reason for this is that we can see that adding 2-morphisms to obtain a strict 2-category doesn't change anything about the structure of $$G$$. In particular we don't gain any structure from the 2-category case. Thus the strict 2-groups are actually group objects in $$\mathbf{Cat}$$ viewed as a strict 2-category.

There are other ideas for "group-like" things in higher categories. You can ask for things like weak 2-groups where we consider weak 2-categories and thus only require the composition of morphisms to be associative and unital up to some 2-isomorphism. In particular a weak 2-group is a monoidal category with all morphisms invertible such that for any object $$x$$ we have $$x\otimes x^{-1}$$ and $$x^{-1}\otimes x$$ are only isomoprhic to the tensor unit 1. (This corresponds to (1).) We can also define the notion of a coherent 2-group where we make specific choices for $$x^{-1}$$ and specific isomorphisms $$x\otimes x^{-1}\to 1$$ and $$1\to x\otimes x^{-1}$$ such that these form an adjoint pair. All of this is developed in this paper of Baez's & Lauda's (see definition 20).

In particular, a group object (or a strict 2-group) in a 2-category with finite products is a coherent 2-group $$G$$ except the natural isomorphisms defining the "coherent structure" of $$G$$ are all simply the identity (see Definition 29 of the above paper).

Another reference which spells out different group-like structures in 2-categories as well as how these things act on categories $$X$$ is given in this paper by Morten's & Picken's.

edit: edited for clarity (see comments below)

• @AdittyaChaudhuri Note that in Definition 29 at the end they state that a strict 2-group in a 2-category is "...a group in the underlying category of this 2-category" which is what I was trying to communicate above. I'm glad you are satisfied! :)
– asd
May 10 '20 at 16:40
• Thanks for explaining. May 10 '20 at 16:42
• For the sake of completeness of your answer can you please add the notion of definition 29 in arxiv.org/pdf/math/0307200.pdf along with the reference? Hope this will help the future readers. Thank you. May 10 '20 at 16:54
• @AdittyaChaudhuri Let me know if these edits work and whether you'd like me to make any more changes. And thanks for the feedback!
– asd
May 10 '20 at 17:17
• It looks fine. :) May 10 '20 at 17:30