Higher algebra and terminology about 2-objects

It is well known that one way to build higher category theory is to use some induction process, where an $n$-category has as $0$-cells some $n-1$ categories, such that for two $0$-cells $\mathcal{A}$ and $\mathcal{B}$, $Hom(\mathcal{A}, \mathcal{B})$ admits a structure of $n-1$ categories.

Now, this process of generalization is obviously not unique, and one could also consider that what should be seen as an $n$-category is an internal $n-1$-category inside the category of all (small) $n-1$-categories. The latter are usually called $n$-fold categories (for the case $n = 2$, the terminology double category is also in use).

However, it appears that people are usually calling "$2$-objects" (say, a $2$-group), a $1$-object (say, a group object) inside $Cat$. Yet, this construction is copying the one of a double-category, so these objects should in my opinion rather be called double-objects.

Hence, a double-group is a a group object inside $Cat$: this shows that a double group (what is usually called a $2$-group) is nothing else than a model of the syntactic category of a group object inside Cat, so it's just a group object. Now, if a 1-group is a group object made of $0$ and $1$-cells, why not calling a $2$-group a group object made of $1$-cells and $2$-cells in a $2$-category? If one is doing that, then taking the underlying syntactic category of the category one is working on will not "remove" any information, so it seems that such terminology would be better.

That is, given a (strict) $2$-category, and two $0$-cells $\mathcal{A}$, $\mathcal{B}$, shouldn't the terminology of a $2$-object be a $1$-object inside $Hom(\mathcal{A}, \mathcal{B})$? In particular, this would also allows to sketch what is a $2$-group using a $2$-category, and not simply a $1$-category. Thus, in this respect, a $2$-group "really is" a concept of level $2$.

Ex: Let $\mathcal{A}$, $\mathcal{B}$ be two categories with $\mathcal{B}$ cartesian (we also impose that it has a terminal object $1$). A $2$-group is a group object in $Hom(\mathcal{A}, \mathcal{B})$.

More explicitly, Let $term: \mathcal{A} \rightarrow \mathcal{B}$ be the constant functor sending everyone into $1$ in $\mathcal{B}$. We call such functor a terminal functor because it is a terminal object in $Hom(\mathcal{A}, \mathcal{B})$. Now, for any functor $m: \mathcal{A} \rightarrow \mathcal{B}$, we define the functor $G \times G: \mathcal{A} \rightarrow \mathcal{B}$ to be the functor sending every object $A$ of $\mathcal{A}$ to $(G \times G)(A) = G(A) \times G(A)$ and every arrow $f$ to $(G \times G)(f) = G(f) \times G(f)$. One can verify that such functor is indeed a product inside the hom category, where the projection maps $\pi_1$, $\pi_2$ are natural transformations whose components are exactly the projection maps in $\mathcal{B}$. In particular, we can define a binary operation on $G$ as a natural transformation $m: G \times G \Rightarrow G$ respecting the associativity square, and a unit as a natural transformation $u: term \Rightarrow m$ respecting the relations of a unit. Obviously, one could also define an "inverse natural transformation" as in the case of a simple group object.

What is a $2$-group here? Well, it is a collection of group objects in $\mathcal{B}$ whose operations are all compatible between each others. In particular, if $\mathcal{A}$ is a category with one object, a $2$-group is just a group object in $\mathcal{B}$. To do so, is like associating a group object to any object of $\mathcal{A}$, and a homomorphism of group object for each arrows. One could also gives lots of other example, like, a presheaf of group is nothing else than a $2$-group between the opposite of a site and Sets, etc.

Now, one could make a remark and say that what I did is just a group object inside some functor category. Obviously, this is true, but the problem is that when one is looking at a functor category as a syntactic $1$-category, structurally speaking, there is no way to "remember" that our objects are actually functors and our arrows are natural transformations. That is, there is no way to remember that our objects actually admits an evaluation that is preserved by its arrows.

So, should we switch terminology?

• A funny thing to remark: Grp can be seen as the $2$-group object (in the above sense) from Sets to itself. This is not surprising, considering that all that was done can be considered as a weak form of 2-sketch: a sketch of some functor category. – sure Jul 6 '15 at 19:11
• You might find interesting the microcosm principle. You might also find interesting the observation that a lax monoidal functor from the contractible category to a monoidal category is an algebra object in the target. – Theo Johnson-Freyd Jul 7 '15 at 1:05
• Those being said, is there a non-rhetorical question here? This comes across strongly as being too opinionated. – Theo Johnson-Freyd Jul 7 '15 at 1:06