As defined by [Gan](https://arxiv.org/abs/math/0201074), 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](https://ncatlab.org/nlab/show/PROP) 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](http://www.tandfonline.com/doi/abs/10.1080/00927877508822067), a [polycategory](https://ncatlab.org/nlab/show/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](https://ncatlab.org/nlab/show/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](https://ncatlab.org/nlab/show/linearly+distributive+category).


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?



Dioperads vs polycategories
In Martin Markl's article "[Operads and PROPs](https://arxiv.org/abs/math/0601129)," 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](http://bookstore.ams.org/surv-203/)," 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. 


