While we are talking # about enriched category theory, I have a related question. I am looking for the goood way to think of the following kind of situation:

Suppose $V$ a closed symmetric monoidal category with coproducts that are respected by the tensor product, $(a \coprod b) \otimes c \simeq (a \otimes c) \coprod (b \otimes c)$.

Let $\mathbf{B} V$ be the corresponding bicategory and consider for any ordinary category $C$ a lax functor

$F : C \to \mathbf{B} V
\,.$

This is something close to a $V$-enriched category: for each $(a \stackrel{f}{\to} b) \in Mor(C)$ there is $F(a,f,b) \in Ob(V)$ and composition operations

$F(a,f,b)\otimes F(b,g,c) \to F(a,g\circ f, c)$

etc.

Okay, now the observation that I am looking for comments on (nothing profound, but anyway):

Let $1 : C \to \mathbf{B} V$ be the constant functor.

Then lax transformations

$\eta : F \Rightarrow 1$

consist of a collection of morphisms in $V$

$\eta_f : F(a,f,b) \otimes \eta_b \to \eta_a$

satisfying some condition. The hom-adjunct of these (I am assuming $V$ to be closed monoidal, recall) is

$\bar \eta_f : F(a,f,b) \to [\eta_b,\eta_a]$

and the conditions say that this can be read an an “enriched functor”

$\bar \eta: Graph(F) \to V
\,,$

(When $C$ is a codiscrete category this is literally an enriched functor of $V$-enriched categories.)

So $\bar \eta$ is a module over $Graph(F)$.

When this arises in practice, and when $V$ has a 0-object, people like to rephrase this by forming the objects in $V$

$A :=
\coprod_{(a \stackrel{f}{\to}b) \in Mor(C)} F(a,f,b)$

and

$K := \coprod_{a \in Obj(V)} \eta_a$

and consider $A$ as an algebra internal to $V$ with product in components

$F(a,f,b)\otimes F(c,g,d)
\to
\left \lbrace
\array{
F(a,g\circ f,c) & if b = c
\\
0 & otherwise
}
\right.$

the composition if defined and 0 otherwise. Similarly $K$ becomes a module over $A$

$\bar \eta : A \otimes K \to K
\,.$

The famous example that I am thinking of are linear groupoid representations as they are considered in the context of Drinfeld doubles.

In that case $C$ is some finite groupoid, $V = Vect$, $F : C \to \mathbf{B}Vect$ sends all morphisms to the tensor unit and on 2-cells is a groupoid 2-cocycle $\alpha$.

Then a transformation $\eta : F \to 1$ is an $\alpha$-twisted linear representation of $C$.

But instead of saying it this way, people like to form the $\alpha$-twisted groupoid algebra of $C$ and say that $\eta$ is a module over that. This being a special case of the general construction I just tried to describe.

I am thinking there should be some standard enriched-category theoretic way to think of this passage from lax transformations to modules over algebras. Probably I shouldn’t post this here but think about this a bit more myself. But anyway.

## Re: nLab - More General Discussion

I am inclined to follow the entry on distributor and say $Set Mod$ instead of $Dist$ or $Prof$.

I think, generally, if one can help it is useful not to name categories after their morphisms instead of after their objects.

So one should ask: which role do locally small categories play if I allow morphisms between them to be “distributors”=”profunctors”=”bimodules”?

The answer is: in that case they behave like (bases for) modules. So the 2-category they form should be called a 2-category of modules. Of $Set$-modules in the case of locally small categories. Of $V$-modules in the general case.

By the way, there are entries on [[profunctor]] and [[bimodule]], too. At the moment [[profunctor]] just redirects, while [[bimodule]] mentions information pretty much overlapping with [[distributor]].