I'm just beginning to learn category theory, and this question popped up in my mind, which Googling has not been able to resolve (probably because I am not searching the right terms).
A functor between categories $C,C'$ is a map $T$ that associates each object $c\in C$ with an object $Tc\in C'$, and each morphism $\varphi$ in $C$ with a morphism $T\varphi$ in $C'$, satisfying certain niceness properties (takes identity morphism to the identity, etc.)
What about a map $T$ from $C$ to $C'$ that takes objects in $C$ to morphisms in $C'$, and morphisms in $C$ to "morphisms between morphisms" in $C'$ instead, satisfying similar "niceness" properties? Do these have a name?
This is in some sense like a "meta" version of functors, and I imagine they have been studied before. I imagine that higher category theory might be relevant, but I unfortunately do not quite understand the Wikipedia page well enough to determine whether or not it's related.
Unless, perhaps, these maps are not well-defined, or not interesting? If so, I am very intrigued to find out why. Indeed my question above is rather vague as I do not really know how to make the notion of transformations between morphisms rigorous. (Maybe it is captured by the notion of a natural transformations? Or, maybe not.)
Any explanations or directions on where to find out more, or the right terms to search, would be greatly appreciated!
Best Answer
For any category $D$ there is its category of arrows $Arr(D)$. Its objects are the morphisms in $D$, and its morphisms are commuting squares of morphisms in $D$. A functor $C\to Arr(D)$ thus takes the objects of $C$ to morphisms in $D$, and morphisms in $C$ to morphisms between morphisms (at least in a sense). This does not require any higher dimensional considerations.