Category containing its own morphisms as objects

Question

I am trying to construct a language the objects of which are some objects and morphisms between own objects. Not sure even if this construction is a category. Is there any name of this kind of structures?

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** Actually let me add more details: Assume we have two basic objects in category. The objects are $A$ and $B$ and there is a single morphism between them $f: A \rightarrow B$. Then the category should include the $f$ as an object. So we may have another morphism $g: f \rightarrow A$ and in this case $g$ is also an object etc.

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And I am also trying to prove that this structure is not a category. It is also appreciated if one can prove that this is not a category.

Answer 1

I'll be clear that I don't think what you're trying to construct is a category. However, I'll do my best to give a categorical interpretation of what you've put in the question.

Let me be clear that in a category, morphisms are themselves objects only by coincidence. I.e., for a morphism $f: A\to B$, the existence of an object of the category $O$ with $O=f$ mathematically is a coincidence. This property, while true for the standard constructions of some categories, like $\newcommand\Set{\mathbf{Set}}\Set$ is a coincidence in the sense that this property is not preserved under equivalence or even isomorphism of categories. We can always replace a category by an isomorphic category whose set of objects and morphisms are disjoint.

Moreover, without specifying something further, you get very little useful information about the category, since we have no idea what the arrows into or out of the object $O=f$ are, and they don't necessarily have any relation to $f$.

A categorical interpretation

If you want to categorify the observation that in $\Set$ every arrow is itself an object, since we define functions $f:X\to Y$ by their graph $G_f \subseteq X\times Y$, then we can categorify the notion of the graph of a morphism.

If $f:X\to Y$, then I would define a graph of $f$, $G_f$ if it exists to be a subobject $\iota : G_f\hookrightarrow X\times Y$ such that $\pi_1 \iota : G_f\to X$ is an isomorphism and $\pi_2(\pi_1\iota)^{-1} = f$.

Graphs exist in a lot of categories. They exist in toposes, or in categories of algebraic objects, as well as in lots of geometric categories.

Answer 2

What are the "objects of a language"? The entities its first-order quantifiers run over? [That's the usual story!]

Category theory is usually set up with a two-sorted first order language with objects and morphisms as distinct sorts. But it would only be mildly perverse to take the official language of category theory to be single-sorted, with quantifiers running over entities comprising the category's objects and the morphisms (then restricting quantifiers when you want to talk about just the objects, or just the morphisms).

In this single-sorted version, the objects in the single domain of the language will include morphisms between [some of] the domain's own objects.

Presumably, though, you are not interested in this triviality! So I guess the question needs to be refined to give us something clearer to grapple with (so we know whether slice or arrow categories are relevant to real your concerns, for example.)