Objects as arrows - Category Theory

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Sridhar Ratnakumar

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

Vincent L

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...

Pedro Minicz

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.

Pedro Minicz

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

Sridhar Ratnakumar

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).

Asad Saeeduddin

The notion of treating objects as arrows is called a bicategory

Asad Saeeduddin

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.

Vincent L

what is a hom-category ?