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Composition of distributors or profunctors would be an example. But
composition is only defined up to isomorphism and so one gets a
bicategory. This was done in the second half of the sixties by Benabou.

But the situation can be rectified when redefining distributors from A to B
as cocontinuous functors from Psh(A) to Psh(B).

Thomas

PS I take the opportunity to thank Bob for organising and moderating the
categories list for such a long time.
And thanks to the people who have taken over! I missed it for quite some
time already!

> In a current project we have the following situation. For a category
> we are attempting to define, we know what the objects are, and also
> the morphisms. Unfortunately we do not have an obvious composition
> operation. What we do have is a "colimit" operation, which operates on
> a directed graph labelled by our objects and morphisms, and returns a
> putative colimit object equipped with a family of morphisms in the
> usual way (or fails.)
>
> We then define the composite of morphisms A->B, B->C to be the
> colimit of the diagram A->B->C. We then check that this composition
> operation satisfies the axioms of a category, and that our earlier
> colimit construction is indeed an actual colimit with respect to the
> compositional structure. It seems that everything works fine, and we
> are happy.
>
> My question is whether this has any precedent in the literature. The
> situation as I have described it is a bit simplified, in reality there
> is some higher categorical stuff going on. Personally I'm sure I've
> read similar things in the literature in the past but I can't track
> them down now that I actually need them. The nLab article on
> "composition" has some stuff about this with regard to transfinite
> composition, but we're not trying to do anything transfinite here.
>
> Best wishes,
> Jamie
>
>
>
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