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I am working on formalizing software design using category theory.

However the most natural way for me to express what I want is with a Category where multiple morphisms can join into a single morphism.

Thinking diagrammatically, what I am talking about is multiple arrows (from different objects) merging into a single arrow before reaching an object. Kinda like an inverse arrow split.

Please note that I am not talking about Categories where objects can be (say) concatenations of multiple types. It also does not make sense (for what I am working on) to use currying.

So my question is: has this type of structure been researched? If yes what would be a good reference?

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This structure is known as a multicategory. There are many references; e.g., see Wikipedia for a basic introduction, and read the nlab's page http://ncatlab.org:8080/nlab/show/multicategory as well. One very nice reference is the book Higher Operads, Higher Categories, by Leinster (see page 35). One nice introduction is also the Catsters' video playlist.

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  • $\begingroup$ Perfect! Exactly what I was looking for. Thanks! $\endgroup$ – mbrodersen Dec 15 '14 at 6:03
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The difficulty of composing with a multiset of multiarrows is that (unless the multiset of domains is actually a set), there will in general be more than one way to do that composition.

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