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I'm looking for a generalized notion of category (really of symmetric multicategory) which, roughly speaking, doesn't make a distinction between sources and targets. Each "morphism" in such a category has attached to it a multiset of source/targets, and each pair of morphisms can be composed along any common submultiset of source/targets.

Does anyone know of a reference in the literature / a page on the nLab where generalized notions of categories with "unoriented compositions," e.g. the above notion, are defined and named?

Most formalisms I know for dealing with this sort of thing start with something which is "oriented" in that it distinguishes sources and targets and equip it with extra structure allowing one to turn sources into targets and vice versa. But the example I have in mind doesn't have this property: the "morphisms" are finite graphs together with labels on some of their vertices such that no pair of labeled vertices is connected by an edge, and composition is giving by gluing together graphs along labeled vertices with shared labels. I can't interpret this example as, say, a symmetric monoidal category with duals because there are no cups or caps.

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  • $\begingroup$ I would suggest that you describe your structure more precisely and then maybe ask the "categories" list whether anybody has seen an animal like it before, rather than asking for a list of all animals. There are models of linear logic where the morphisms don't really have sources and targets. $\endgroup$
    – Paul Taylor
    Commented Feb 6, 2014 at 23:06
  • $\begingroup$ What you describe reminds a lot of Ross Street's "computads" $\endgroup$
    – Adam Gal
    Commented Feb 6, 2014 at 23:39

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Yes, see

and

  • David Ayala, Higher categories are sheaves on manifolds, talk at FRG Conference on Topology and Field Theories, U. Notre Dame (2012) (video)

    Nick Rozenblyum, Manifolds, Higher Categories and Topological Field Theories, talk Northwestern University (2012) (pdf slides)

    (nLab entry)

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    $\begingroup$ Aha! I think the nLab entry that best captures what I was going for is ncatlab.org/nlab/show/hyperstructure. Thanks for the pointers! Eagerly awaiting Ayala's paper. $\endgroup$
    – Qiaochu Yuan
    Commented Feb 6, 2014 at 23:47
  • $\begingroup$ Yeah, I was thinking about linking to "hyperstructure". The trouble is that while that is clearly the idea, it is also not much more than an idea, unfortunately. And yes, myself I also kept and keep awaiting news from David Ayala and Nick Rozenblyum on their approach. I checked again before posting here and couldn't find a preprint. If anything exists, I'd be grateful for pointers. $\endgroup$
    – Urs Schreiber
    Commented Feb 6, 2014 at 23:50

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