This is a conceptual question about category theory, more than a technical one. It has no practical purpose as far as I know, but it's been bugging me for ages.
The short version: a function can be seen as a special case of a relation, namely one in which every object in the domain is related to exactly one object in the codomain. Functors don't seem to be special cases of profunctors in the same sense. Is there some other, somewhat analogous, sense in which a functor can be seen as a special case of a profunctor?
The long version:
A profunctor can be seen as a particular type of functor: a profunctor $\phi\colon A\nrightarrow B$ is a functor $\phi\colon A^\text{op}\times B \to \mathbf{Set}$. (There seem to be multiple conventions about the order of $A$ and $B$ and where to put the ${}^\text{op}$. I'm using this one.) For objects $a\in\mathrm{Ob}(A), b\in\mathrm{Ob}(B)$, one can think of $\phi(a,b)$ as something like a hom-set between $a$ and $b$, even though they're in different categories. The elements of $\phi(a,b)$ are sometimes called "heteromorphisms." The functoriality of $\phi$ essentially means that these heteromorphisms have to compose with the (homo)morphisms in the categories $A$ and $B$.
This leads to the view that profunctors are to functors as relations are to functions. We can think of a relation $R$ between sets $S$ and $T$ as a function $R:S\times T\to \{0,1\}$, which corresponds to the definition of a profunctor as a functor, and the definition of a profunctor can be seen as a categorification of this.
We can also think of $R$ as just a subset of the Cartesian product $S\times T$. The picture of profunctors in terms of heteromorphisms can be seen as a categorification of this view. From this view, we can also see a function as a special case of a relation: it's a relation in which every object in $S$ is related to exactly one object in $T$.
However, at this point the profunctors$\leftrightarrow$relations analogy seems to break down. It would be very nice if we could say, analogously, that a functor $F\colon A\to B$ is a profunctor in which every object of $A$ is related to exactly one object of $B$. However, it seems this can't work. Consider the following example:
If we want to express a functor that maps $a\in\mathrm{Ob}(A)$ to $b_1\in \mathrm{Ob}(B)$ then we should draw a heteromorphism $h$ from $a$ to $b_1$. But since heteromorphisms must compose with morphisms we are forced to also add a heteromorphism $h{;}f$, contradicting the idea that $a$ should be related to exactly one object in $B$.
The question is, is there some other way in which a functor can be seen as a special case of a profunctor, analogous to the way in which a function can be seen as a special case of a relation? If someone for some reason knew about profunctors (via the heteromorphism definiton) but didn't know what a functor was, could you given them a "categorical" definition of a functor in terms of profunctors?
from Hot Weekly Questions - Mathematics Stack Exchange
Nathaniel
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