Re: Defining composition via colimit

Re: Defining composition via colimit

[JS Lemay]

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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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Re: Defining composition via colimit

[Richard Garner]

Hi Jamie,

That does sound interesting. Did you look at:

- Kelly, G. M., & Le Creurer, I. J. On the monadicity over graphs of
  categories with limits. Cahiers de Topologie et G\'eom\'etrie
  Diff\'erentielle Cat\'egoriques, 1997(38), 179–191.

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:

> 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

Defining composition via colimit

[Jamie Vicary]

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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