TITLE: Dioperads vs polycategories
QUESTION [11 upvotes]: 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?

REPLY [6 votes]: 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.