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Email Gateway (Nov 02 2023 at 19:06):Dear all,
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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Email Gateway (Nov 02 2023 at 23:35):Hi Jamie,
That does sound interesting. Did you look at:
This paper, among other things, shows that categories with limits
indexed by finite acyclic directed graphs are monadic over directed
graphs. It seems like you could be describing an algebra structure for
this monad (which is a fortiori a category). The paper gives a fairly
explicit presentation of this monad by generators and relations, so you
should be able to check fairly easily if that is indeed what is going
on.
Richard
Jamie Vicary <jamievicary@gmail.com> writes:
Email Gateway (Nov 03 2023 at 10:29):[[The following message is sent on behalf of streicher@mathematik.tu-darmstadt.d -- for whatever reason it did not seem to have been sent/approved properly to the mailing list, apologies if you receive multiple copies]]
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!