Objects in category contain no information, they exist simply to serve as the source and target of arrows, that is, they delimit composition. As such, you can formalize categories without ever defining or mentioning objects. I've seen this concept be called the arrow theory of categories or single-sorted definition of a category.

In a graph, one may treat vertices as edges. If C->A->B, then in an alternate graph "A" can be treated as an edge between the two arrows (which are now vertices).

For example, for any two categories A, B, there is a category [A, B] whose objects are functors between the aforementioned categories, and whose morphisms are natural transformations. But these objects (functors) can be perceived as morphisms at a different level, i.e. morphisms in the hom-category [A, B]. In category theory jargon, we are saying that the category of (small) categories is a (strict) bicategory.

What's this notion termed in category theory? The notion of treating objects

asarrows?Can you give some context ? I can see threating Arrow as object, but for the reverse I don't see an easy way to define composition...

Objects in category contain no information, they exist simply to serve as the source and target of arrows, that is, they delimit composition. As such, you can formalize categories without ever defining or mentioning objects. I've seen this concept be called the arrow theory of categories or single-sorted definition of a category.

Just observe that you can identify each object with its identity morphism.

In a graph, one may treat vertices as edges. If C->A->B, then in an alternate graph "A" can be treated as an edge between the two arrows (which are now vertices).

The notion of treating objects as arrows is called a bicategory

For example, for any two categories A, B, there is a category [A, B] whose objects are functors between the aforementioned categories, and whose morphisms are natural transformations. But these objects (functors) can be perceived as morphisms at a different level, i.e. morphisms in the hom-category

`[A, B]`

. In category theory jargon, we are saying that the category of (small) categories is a (strict) bicategory.what is a hom-category ?