I've recently started studying multi-categories. I find that they allow multiple objects as the source and multiple morphisms onto a single object.

It seems natural to me to have the opposite construction: allow only for a single object as the source and again multiple morphisms, yet onto multiple objects as the target.

I haven't found any literature so far on this type of multicategory.

Any help would be greatly appreciated!

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    $\begingroup$ You can even have both, and then you'll get what's called a polycategory. $\endgroup$ Sep 22, 2016 at 21:28
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    $\begingroup$ Anyway, I would call your thing a comulticategory, by analogy with cooperads. $\endgroup$ Sep 22, 2016 at 21:36
  • $\begingroup$ Thanks @Qiaochu. Do you know of any literature on this constrution by any chance? $\endgroup$ Sep 22, 2016 at 21:47
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    $\begingroup$ There's an isomorphism of 2-categories between multicategories and comulticategories, so there's no reason to expect people to write about the latter. $\endgroup$ Sep 23, 2016 at 4:24

1 Answer 1


A multicategory is the same thing as a colored operad. In a colored cooperad, you have operations with one input and multiple outputs, but instead of composing morphisms, you cocompose (= decompose) them, in the spirit of a cocategory. This is not what you're asking about. IMO that would be called a multicocategory, if one really wanted to stick to the "multicategory" scheme of naming things.

What you're describing, as far as I know, doesn't have a name, so you can make up whichever you want. They are a special case of polycategories (which are themselves special cases of colored PROPs); the difference there is that morphism can have multiple inputs and outputs.

If you want my 2 cents, please don't invent a name obtained by sticking a prefix meaning "multiple" or a variation thereof in front of the word "category". We got too many of these names already, and they're not really descriptive.

If you want to call them "comulticategories", be also aware that what you're considering is not the same thing as a colored cooperad (whereas multicategories are colored operads). Besides when one sticks the prefix "co" in front of the name of an algebraic structure, one generally expects to get a coalgebraic structure...

  • $\begingroup$ Great stuff! Many thanks @Najib for your detailed answer. I will just define this structure and temporarily call it "Dagger Multicategory" and later will think about a better name. $\endgroup$ Sep 23, 2016 at 10:39

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