Higher-dimensional category theory on objects

Question

I would like to know if there exists a satisfying generalization of higher-dimensional category theory on objects, that doesn't forget the inner structure of objects. Usually, what people do is to look for higher dimensional arrows, thus remembering the relations between 0-cells but forgetting that $0$-cells can be "made up".

What I want is to "keep track" of the inner structure of my objects (for example, they can be subcategories or categories).

Example: Cat as a 2-category is such that its 0-cells are small categories, 1-cells are functors, 2-cells are natural transformations. As an abstract 2-category, Cat doesn't "see" the inner structure of 0-cells and treat them as pure syntax. It is as if there was some "forgetful inner structure functor" that was applied to a higher dimensional analog on object. The same happens when one construct a category where objects are diagrams of another category. Indeed, let $\mathcal{C}$ be an abstract category. Let $Grp(\mathcal{C})$ be the category where objects are group objects of $\mathcal{C}$ and arrows are internal homomorphism. In this example, one is once again forgetting the inner structure of group object and see them as 0-cell (i.e., point on the underlying graph).

More formally, one can call $2*$-category a "collection" of categories + functors between them (in order to mimic a subcategory of Cat where one remembers the inner structure of objects). A $2*$-functor would take as input objects, arrows, and functors of a $2*$-category, and maps them respectively to objects, arrows, and functors with obvious axioms (preservation of composition of functors + unit). The "forgetful inner structure functor" is then the $2*$-functor injective on functors, sending all objects of a given $2*$-category to a constant object of the same category, all arrows to the identity, but keeping tracks of the functors.

PS: my formalization is just done here to clarify the idea, its obviously not the way to go because it doesn't encompass the example of $Grp(\mathcal{C})$. Actually, it would be an even higher dimensional analog. A group object in $\mathcal{C}$ is already a $2*$-category: it is a functor from the sketch of a group object to $\mathcal{C}$, and arrows between them is given by natural transformations between such functors. So basically, $Grp(\mathcal{C})$ is a pretty good example of mix between higher-dimensional category theory both on objects and arrows.

Answer 1

Cat does see the inner category structure of its 0-cells. In fact, it sees it in a very direct manner: any small category $C$ is isomorphic to the hom-category $\hom_{\mathbf{Cat}}(\mathbf{1}, C)$, where $\mathbf{1}$ is the terminal category.

This is typical of how category theory thinks of structure: structure is what you see with arrows. If you construct a category and find that there is some aspect of your objects that you can't see with arrows, then that aspect isn't part of the structure you're studying. (or alternatively, you constructed the wrong category)

Answer 2

I'm posting an answer to my own thread in order to show that the concept I was looking for is actually easily formalizable and useful.

A $2$-floor category is a collection of objects $A$, $B$, $C$, $\ldots$ called $0$-arrows and written $\mathcal{C}_0$, a collection of $1$-arrows $f$, $g$, $h$, $\ldots$ between objects written $\mathcal{C}_1$, and a collection of $2$-arrows $F$, $G$, $H$, $\ldots$ between $1$-arrows written $\mathcal{C}_2$. This comes with two kinds of source and target maps, $s_1, t_1: \mathcal{C}_1 \rightarrow \mathcal{C}_0$ and $s_2, t_2: \mathcal{C}_2 \rightarrow \mathcal{C}_1$, together with two compositions defined as the pullback of source and target maps $\circ_1 : \mathcal{C}_1 \times_{\mathcal{C}_0} \mathcal{C}_1 \rightarrow \mathcal{C}_0$ and $\circ_2: \mathcal{C}_2 \times_{\mathcal{C}_1} \mathcal{C}_2 \rightarrow \mathcal{C}_1$ such that:

1) The $0$-arrows and the $1$-arrows together with $\circ_1$, $(\mathcal{C}_0, \mathcal{C}_1, \circ_1, s_1, t_1)$, form a category. Hence, $\circ_1$ is associative, unital, and is to be seen as horizontal composition of $1$-arrows. We call $Id_A$ the identity arrow at $A$.

2) The $1$-arrows and the $2$-arrows together with $\circ_2$, $(\mathcal{C}_1, \mathcal{C}_2, \circ_2, s_2, t_2))$ form a category. Hence, $\circ_2$ is associative, unital and is to be seen as vertical composition of $2$-arrows. We call $Id_f$ the identity arrow at $f$.

3) We also introduce a functorial horizontal composition of $2$-arrows $\circ_h$ defined for 2-arrows $F$ and $G$ such that $t_1 s_2 F = s_1 s_2 G$, $t_1 t_2 F = s_1 t_2 G$, and is such that it preserves source and target maps. That is, $s_2 \circ_h(F,G) = s_1 G \circ_1 s_1 F $, $t_2 \circ_h(F,G) = t_1 G \circ_1 t_1 F$. Let $F$, $F'$, $G$, $G'$ be four $2$-arrows such that $F' \circ_2 F$, $G' \circ_2 G$ and $\circ_h( (F',G') \circ (F,G))$ makes sense. By functoriality, we have

$\circ_h( (F',G') \circ (F,G)) = \circ_h( F' \circ_2 F, G' \circ_2 G) = \circ_h(F',G') \circ_2 \circ_h (F,G),$

that is in usual notations $(G' \circ_h F')\circ_2 (G \circ_h F) = (G' \circ_2 G) \circ_h (F' \circ_2 F)$ an interchange laws for $2$-arrows.

Example: Let $\mathcal{C}$ and $\mathcal{D}$ be two different categories and $F: \mathcal{C} \rightarrow \mathcal{D}$ a functor. If we let $\mathcal{C}_0 = Obj(\mathcal{C}) \cup Obj(\mathcal{D})$, $\mathcal{C}_1 = Arr(\mathcal{C}) \cup Arr(\mathcal{D})$ and add $2$-arrows $F_f: f \rightarrow F(f)$ we obtain a $2$-category where the horizontal composition is defined as $F_f \circ_h F_g = F_{f \circ g}$ whenever $F(f \circ g)$ makes sense.

Let $Indisc(3)$ be the category generated by $3$ objects $A$, $B$, $C$ and three arrows$ f: A \rightarrow B$, $g: B \rightarrow C$, and $h: C\rightarrow A$ with relations $h \circ g \circ f = Id$. Any endofunctor induces a 2-floor category with three objects.

Any category that is build "over" another category (say that its objets are some subgraph of it) is a 2-floor category. This concept is interesting when one wants to remember the construction process of categories over categories, because it allows to remember everything.