Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

As mentioned here, there is a trace operation on the monoidal category of profunctors given by taking coends: for any profunctor $F : A\times X \nrightarrow B \times X$, there is a profunctor $Tr^X(F) : A \nrightarrow B$ defined by $$Tr^X(F)(a,b) = \int^x F((a,x),(b,x))$$ I am interested in the sort of dual operation, where we take an end rather than a coend: $$CoTr^X(F)(a,b) = \int_x F((a,x),(b,x))$$ Note that $CoTr^X(F)$ has the same type ($A \nrightarrow B$) as $Tr^X(F)$.

My question is,

To what extent can the operation $CoTr^X$ be seen as a "cotrace"?

I've found a bit of information googling on "cotraces", but nothing very comprehensive. Specifically, I'd like to know the following:

  1. First of all, is there a commonly-accepted definition of "cotrace", and what is its relationship to the trace?
  2. Is there a string-diagrammatic definition?
  3. One way to view the operation $CoTr^X$ is as a limited form of closure for the "external monoidal" structure on profunctors, in the sense that $${\bf Prof}(G, CoTr^X F) = {\bf Prof}(G \times Hom_X, F)$$ holds naturally in $G : A \nrightarrow B$. That is, we can view $CoTr^X(F)$ as "$Hom_X \multimap F$". Is this part of the general definition of cotrace (assuming the answer to (1) is positive), or is it a special feature of this particular operation on Prof?
share|improve this question

1 Answer 1

Your exact set of questions appears to have languished unanswered for some time, but I can offer at least a partial answer.

You appear to have rediscovered the notion of Tambara modules.

The comonad $CoTr^X$ is talked about in some depth in Doubles for Monoidal Categories by Pastro and Street. They also dig into the left adjoint of this construction, which is a monad on $Prof$, such that the "strong" profunctors are just its algebras. They talk a fair bit about point #3 as well, though as $Hom$ is the unit for profunctor composition, they can just fuse it away.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.