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A while ago, we had some brief discussion on potential higher structures in categorical probability theory. We now have a definition of a certain abstract categorical structure which captures these phenomena and also many other examples. See the current working document.
Differently from “ordinary” higher category theory, composition in our “compositories” has the property that composing an $S^1/\mathbb{Z[n$-morphism with an $m$-morphism along a common $k$-morphism face results in an $(n+m-k)$-morphism. We believe that this kind of composition is a natural structure which arises in many situations. Think, for example, of the nerve of a category, in which sequences of composable morphisms can simply be concatenated. Other examples arise from mathematical structures which can be glued along pairwise intersection, even though the general sheaf condition fails; this applies e.g. to the presheaf of metrics on a set.
Compositories may also provide a potential answer to Urs’ question on hyperstructures and higher spans.
Now the questions are:
Thanks for any feedback!
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