As defined by Gan, a dioperad consists of sets of operations $P(n,m)$ with "$n$ inputs and $m$ outputs", which can be composed by joining one output of one operation to one input of another, giving rise to composition operations

$$ P(n_1,m_1) \times P(n_2,m_2) \to P(n_1+n_2-1,m_1+m_2-1). $$

Dioperads are apparently motivated by the fact that some PROPs are freely generated by dioperads, because they have a generators-and-relations description in which all the relations only involve composites of the sort that exist in a dioperad. In such a case, the dioperad is evidently a simpler object to study than the entire PROP.

As defined by Szabo, a polycategory has a set of objects, and for any $n$-tuple and $m$-tuple of objects $a_1,\dots,a_n$ and $b_1,\dots,b_m$, a set of morphisms $P(a_1,\dots,a_n; b_1,\dots,b_n)$, with composition operations that join one of the targets of one morphism to one of the sources of another. If we write $\Gamma,a,\Delta$ for a list of objects containing $a$ somewhere in the middle, then this composition operation looks like

$$P(\Gamma; \Delta,a,\Psi) \times P(\Xi,a,\Upsilon; \Phi) \to P(\Xi,\Gamma,\Upsilon; \Delta,\Phi,\Psi).$$

Polycategories are motivated by providing a semantics for classical linear logic, whose cut rule looks like this composition rule:

$$ \frac{\Gamma \vdash \Delta,A,\Psi \qquad \Xi,A,\Upsilon \vdash \Phi}{\Xi,\Gamma,\Upsilon \vdash \Delta,\Phi,\Psi} $$

One generally thinks of the source objects as joined by a "multiplicative conjunction" tensor product $\otimes$ and the target objects as joined instead by a "multiplicative disjunction" tensor product $⅋$, and this can be made precise by showing that "representable" polycategories are precisely linearly distributive categories.

It looks to me as though (ignoring questions of enrichment) a dioperad is just a one-object polycategory, in the same way that an ordinary operad is a one-object multicategory — or in other words, a polycategory is a "colored dioperad". Is this true? If so, is it in print anywhere? If not, what is the difference?

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    $\begingroup$ Sure looks that way to me, although I've never heard the word "polycategory" before. If memory serves, Gan's definition explicitly requests that the set $P(n,m)$ be a bimodule for the symmetric groups $S_n$ and $S_m$. Certainly I make that request in my couple papers, and Bruno does as well. I assume Szabo also asks for the ability to permute the inputs and outputs? Of course, there really should be both planar (associative) and symmetric (commutative) versions, and probably also $E_d$ versions for all dimensions $d$... $\endgroup$ – Theo Johnson-Freyd Jun 15 '17 at 23:46
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    $\begingroup$ I'd be willing to put forward the wager that the sentence "Polycategories are colored dioperads" isn't written in the literature. My reason for believing this is that I, as I already said, never heard of polycategories, so I wonder how much literature there is on them (perhaps a lot in linear logic?), and I also don't know of a lot of work on dioperads (Gan's original paper, Bruno's thesis, my couple papers, surely others... but not a lot). Indeed, Bruno showed that the homotopy theories of dioperads and props are really different, and props are usually (but not always) the "correct" choice. $\endgroup$ – Theo Johnson-Freyd Jun 15 '17 at 23:51
  • $\begingroup$ I would of course be happy to be corrected --- literature I am unaware of is literature I can learn from. $\endgroup$ – Theo Johnson-Freyd Jun 15 '17 at 23:52
  • $\begingroup$ Polycategories of course also have a symmetric and non-symmetric version, just like all kinds of multicategory/operad. Szabo's original definition may have been the non-symmetric one, but if that's the only difference then I would regard the answer as being yes. $\endgroup$ – Mike Shulman Jun 19 '17 at 15:43

In Martin Markl's article "Operads and PROPs," just after Def. 64, the dioperad-polycategory connection is briefly mentioned. Markl attributed this observation to Leinster.

In my book with Mark W. Johnson "A Foundation for PROPs, Algebras, and Modules," in Section 11.5 we wrote down the explicit axioms for a colored dioperad. The relevant graphs and monads were written down earlier in Part I, as explained further down in Section 11.5. If you take out the equivariant structure of a C-colored dioperad (Def. 11.18), the remaining structure is a polycategory with object set C.

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