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
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