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

Question

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.

Answer 1

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)