# Are categories, multicategories, and polycategories equally expressive?

Clearly any ordinary category can be interpreted in a multi or polycategory without change (by restricting multimorphisms to be unary). Can every multi or polycategory be interpreted or recast in an ordinary category? Is there any reason other than convenience or simplicity to work only in categories rather than multi or polycategories?

In an ordinary category, a poset is any category where the hom-set of any pair of objects contains only 1 morphism. This could carry over unchanged in a multicategory, however, it seems like you could also define a poset as a binary morphism from a pair of objects to the terminal object (which we could interpret as the truth value).

It seems like the power of category theory comes from enforced compositionalitity of morphisms and the encoding irrelevance of objects (ie we don't need to care what the objects are); and multi and polycategories would seem to still maintain those properties.

edit: also curious how PROPs fit in, they are sort of like polycategories but composition rule is different

• I'm familiar with categories and multicategories. What is a polycategory?? Commented Jan 16, 2018 at 18:37
• ncatlab.org/nlab/show/polycategory Commented Jan 16, 2018 at 18:37
• Multicategory = list of objects for domain, one object for codomain. Polycategory = list of objects for domain, list of objects also for codomain. Commented Jan 16, 2018 at 18:39
• If a category has finite products, then the multicategory and the category are equivalent, right? A morphism from the list $(a,b)$ (which is a pair of morphisms) is the same as a morphism from $a\times b$ (or some other tensor product). But if the category does not have the right limits, then these should be inequivalent, right? Commented Jan 16, 2018 at 18:44

## 1 Answer

Clearly any ordinary category can be interpreted in a multi or polycategory without change.

I don't understand this claim. Multicategories are a generalization of monoidal categories: if $(C, \otimes)$ is a monoidal category it induces a multicategory where the multimorphisms $(C_1, \dots C_n) \to D$ are morphisms $C_1 \otimes \dots \otimes C_n \to D$. They are a strict generalization, in that there are multicategories which do not arise this way for any monoidal category. But if $C$ is just an ordinary category I'm not aware of any way of turning it into a multicategory.

• If we just modify the axioms of a category such that a morphism's domain can (but may not necessarily) consist of 1 or more objects, then wouldn't any ordinary category be left unchanged by keeping morphisms as 1-ary morphisms. For example, a poset in multicategory theory could be defined as "a poset is a multicategory such that all morphisms are unary and there is only a single morphism between any two objects." In that sense, it seems like a multicategory is a strict generalization of a category, not just a monoidal category. Commented Jan 16, 2018 at 19:49
• @Brandon: okay, right, so you can set the multimorphisms to be empty unless $n = 1$. In that case the answer to your title question is still no: multicategories are much more expressive than categories. Commented Jan 16, 2018 at 19:51