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In category theory every morphism has two arguments (called source and destination). Are there known generalizations of theory with every morphism having an arbitrary (possibly infinite) number of arguments? (Two morphisms are not required to have the same number of arguments.)

Such structures appear in my research.

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I think I encountered this recently, but I've forgotten the name. But ultimately it doesn't matter if your categories are nice enough: for example, if your category has all products, or if your category is equipped with a symmetric monoidal structure. –  Zhen Lin Feb 24 '12 at 19:44
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Aha, this just came up in seminar today: multicategories. –  Zhen Lin Feb 28 '12 at 18:12
    
@Zhen Lin: "A multicategory is like a category, except that one allows multiple inputs and a single output." No, I need a structure with inputs and outputs indistinguishable (every input may serve as output and vice versa). –  porton Feb 28 '12 at 21:03
    
@porton I don't get where is the problem with Zhen Lin answer, every output of an arrow in a multicategory may serve as (part of) the input of another arrow. Note that in a category target and source of an arrow are well distinct objects, infact they tell you what you can compose on the left(respectively on the right) with you arrow. –  Giorgio Mossa May 19 '12 at 13:41

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So, profunctors do some of what you would like. A profunctor from $\mathcal{C}$ to $\mathcal{D}$ is a functor $F:\mathcal{C}\times\mathcal{D}^{op}\rightarrow sets$.This allows for partially defined functors from $\mathcal{C}$ to $\mathcal{D}$. (see the nlab entry). If you want to vary the number of arguments, simply let $\mathcal{C}$ be $X\coprod X\times X\coprod X^3\coprod...\coprod X^{\infty}$

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