Multicategories with out-arities Basically, my question is:

Why the emphasis on domains in the notion of multicategory?


I will now give the formal framework to state it correctly.
Passing from categories to multicategories can be done quite simply by looking at the internal definition of a category: it is the data of two sets $C_1,C_0$ and maps $s,t \colon C_1 \rightrightarrows C_0$ and $i \colon C_0 \to C_1$ such that $si = ti = \mathrm{id}_{C_0}$ together with a composition function $m \colon C_1 \times_{s,t} C_1 \to C_1$ satisfying some axioms (for associativity of the composition and neutrality of the identities. If we want multicategories, we want morphisms with multiple sources, so we just change the definition so that the source function $s$ take that into account: a multicategory is the data of
$$ C_0^\ast \stackrel s \leftarrow C_1 \stackrel t \to C_0$$
(where $A^\ast$ is the set of finite words on a set $A$) and a map $i \colon C_0 \to C_1$ such that $ti = \mathrm{id}_{C_0}$ and $si = (C_0 \hookrightarrow C_0^\ast)$ together with a composition $m \colon C_1^\ast \times_{t^\ast,s} C_1 \to C_0$ satisfying some axioms.
Now, taking a really good look at what we did lead us to greater generality. Consider a category $\mathcal E$ with pullbacks and a cartesian monad $\mathbf T = (T,\mu,\eta)$ on $\mathcal E$. For $O$ an object of $\mathcal E$, define the category $\mathrm{Gr_{\mathbf T}}(O)$ of $\mathbf T$-graphs over $O$ to be the full subcategory of spans of $\mathcal E$ generated by those of the form
$$ TO \stackrel s \leftarrow M \stackrel t \to O.$$
Then $\mathrm{Gr_{\mathbf T}}(O)$ is monoidal: the tensor product of $TO \stackrel s \leftarrow M \stackrel t \to O$ and $TO \stackrel {s'} \leftarrow M' \stackrel {t'} \to O$ is defined as follow
$$\require{AMScd}
\begin{CD}
M'\otimes M @>>> M @>t>> O \\
@VVV \lrcorner @VVsV \\
TM' @>>Tt'> TO \\
@VTs'VV \\
TTO \\
@V\mu VV \\
TO
\end{CD}
$$
where the square is cartesian. Now a $\mathbf T$-category is just a monoid in that monoidal category. Take $\mathcal E = \mathsf{Set}$ and either $T = \mathrm{id}_{\mathsf{Set}}$ or $T = (A \mapsto A^\ast)$ to get the classical notion of (small) category and multicategory.
But why is $T$ acting on the source side? It completely breaks some useful classical tools, like the opposite category functor. My two cents would be to define $\mathbf {T,S}$-categories as follow. Like before we have a category $\mathcal E$ with pullbacks, now equipped with two cartesian monads $\mathbf T= (T,\mu_t,\eta_t)$ and $\mathbf S=(S,\mu_s,\eta_S)$ together with a distributive law $\lambda \colon TS \to ST$. Then for any $O \in \mathcal E$ define $\mathrm{Gr}_{\mathbf {T,S,\lambda}}(O)$ to be the full subcategory category of spans on the objects of the form
$$ TO \stackrel s \leftarrow M \stackrel t \to SO. $$
One can again make it monoidal: the tensor product of $TO \stackrel s \leftarrow M \stackrel t \to SO$ and $TO \stackrel {s'} \leftarrow M' \stackrel {t'} \to SO$ is given by
$$\require{AMScd}
\begin{CD}
M'\otimes M @>>> \to @>>> SM @>St>> SSO @>\mu_S>> SO \\
@VVV \lrcorner @. @VVSsV \\
TM' @>>Tt'> TSO @>>\lambda_O> STO \\
@VTs'VV \\
TTO \\
@V\mu VV \\
TO
\end{CD}
$$
and define a $\mathbf{T,S,\lambda}$-category as a monoid in such a category.
I googled around and did not find any reference for this last notion. Is there any? What is the usual name for this notion? (Or is my notion completely wrong for some reason?)
 A: Koslowski, in his article "A monadic approach to polycategories", develops something similar to what you describe (it could even be the exact same thing, actually, I don't know all the details). Given two monads $S$ and $T$ and a distribution relation between the two (a span $TS \overset{\omega}\Leftrightarrow ST$), he builds so-called $\omega$-categories, where the source of the morphisms are $T$-shaped, and the targets are $S$-shaped (in the same sense as "the source of morphisms in a $T$-category are $T$-shaped).
Tom Leinster in his book Higher Operads, Higher Categories mentions $T$-PROs on page 168 (just below proposition 6.6.12), where $T$ is a monad, as a generalization of $T$-categories (in the same way that PROs generalize operads).
The difference seems to lie in the fact that in a polycategory, composition can only occur along one object at a time, whereas in a $T$-PRO, you can compose along multiple objects. So say $T=S$ is the free monoid functor, and $a,b,c$ are some colors. Then if you have morphisms $f : a \to (b,b)$ and $g : (b,b) \to c$, in a $(T,T)$-polycategory you can only compose one of the output of $f$ to one of the inputs of $g$ (and so get a morphism $a \to (b,c)$), while in a $T$-PRO you would be able to plug in both outputs of $f$ into the inputs of $g$ (and so get a morphism $a \to c$).


*

*Tom Leinster. Higher operads, higher categories. Vol. 298. London Mathemati-
cal Society Lecture Note Series. Cambridge University Press, Cambridge, 2004,
pp. xiv+433. isbn: 0-521-53215-9. doi: 10.1017/CBO9780511525896.

*Jürgen Koslowski. “A monadic approach to polycategories”. In: Theory Appl. Categ. 14 (2005), No. 7, 125–156. issn: 1201-561X.
