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.

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    $\begingroup$ One of the points of category theory, ordinary and higher, is concentrating everything on morphisms. You can replace abelian groups with sausages, as long as you use the same morphisms you end up with the same category. $\endgroup$ – Fernando Muro May 22 '15 at 18:30
  • $\begingroup$ see my last comment below to understand the motivation of "keeping track of objects inner structure". $\endgroup$ – sure May 22 '15 at 18:46

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)

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  • $\begingroup$ What if there is no terminal object? Moreover, if you take the category of group objects $Grp(\mathcal{C})$ of some category $\mathcal{C}$, I'm not sure to see how does category theory sees that the objects you consider are actually functors. It also seems that it only sees it indirectly through its "elements", but one could also make this appears explicitly right? $\endgroup$ – sure May 22 '15 at 13:25
  • $\begingroup$ Example: take $Grp(Grp)$. The trivial group object is indeed a group internal to Grp, yet all internal homomorphism between the trivial group object and another group object (that is, any natural transformation between two group object functors with codomain Grp) is trivial. Here, you can't discriminate anything without keeping 1-morphism. $\endgroup$ – sure May 22 '15 at 15:13
  • $\begingroup$ You don't always need terminal objects. Every element of a group $G$ induces a unique morphism $\mathbb Z\to G$ for example. $\endgroup$ – Wouter Stekelenburg May 22 '15 at 16:11
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    $\begingroup$ Moreover, you can pick out $\mathbb{Z}$ categorically as the group $G$ whose only idempotents $p: G \to G$ with respect to composition are the identity and the trivial map, and which admits more than one map $G \to H$ for any nontrivial $H$. This was discussed here: mathoverflow.net/questions/194047/… $\endgroup$ – Todd Trimble May 22 '15 at 17:42
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    $\begingroup$ @sure: The point is that what's important is what you can tell with arrows; don't focus too much on recovering the specific concrete representation (although that can be recovered up to isomorphism in a lot of cases). In fact, sometimes the set-theoretic representation is quite wrong -- e.g. in algebraic geometry, the affine line over $\mathbf{F}_2$ used to consist of two points, which is quite insufficient to capture its structure; e.g. it's incapable of distinguishing between $x^2 + x$ and the zero function. Arrows used to be the only way we knew to understand its structure! $\endgroup$ – user13113 May 22 '15 at 22:02

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.

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    $\begingroup$ I'm downvoting because 1) I believe that the concept of "2-floor category" is something that you just invented for the purpose of this answer. 2) I cannot perceive the difference between your 2-floor category and teh well known concept of a (strict) 2-category. 3) I don't see the relationship between all this and your question. $\endgroup$ – André Henriques Jun 7 '15 at 21:41
  • $\begingroup$ 1) I invented it in order to "remember" the objects and 1-arrows of a category, say, in a slice category, or any functor category. 2) In a 2-category, the vertical composition requires that the 1-arrows share the same domain and codomain. In the case of 2-floor category, you don't need that. 3) My question is basically "how can I remember the stratification process resulting in building categories with categories ?" This is important if you want to theorize what is a definition of a theory (think of definition/theory build over other definitions/theories, and this recursively). $\endgroup$ – sure Jun 7 '15 at 22:46
  • $\begingroup$ A typical example of 2-floor category is given by any subcategory of Cat without terminal object where you remember the objects and arrows of in the categories. Feel free to read definitions and examples more carefully before down voting an answer you don't understand next time $\endgroup$ – sure Jun 7 '15 at 22:46
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    $\begingroup$ You keep talking of not having a terminal object. If you explained your actual case of interest it might help us understand why you want this example. $\endgroup$ – David Roberts Jun 8 '15 at 0:05
  • $\begingroup$ You seem to think that having a terminal object is enough to "see" inside the inner structure. This is false, you also need the good amount of arrows with domain $1$. In most syntactic categories (used to define formally what is a theory or a definition), you don't have them. The point of 2-floor categories is to be able to remember the syntactic inner structure of your objects in a single framework. A 1-category is just a graph with composition: if you add, remove, or change something inside, you might lose the interpretation you started from. This is something I don't want to. $\endgroup$ – sure Jun 8 '15 at 8:20

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