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?

• 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 $d$... – Theo Johnson-Freyd Jun 15 '17 at 23:46
• 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. – Theo Johnson-Freyd Jun 15 '17 at 23:51
• I would of course be happy to be corrected --- literature I am unaware of is literature I can learn from. – Theo Johnson-Freyd Jun 15 '17 at 23:52
• 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. – Mike Shulman Jun 19 '17 at 15:43