Semi-cocartesian operads

Question

Context: In this interesting blog post, Mike Shulman indicates an approach for defining generalized types of operads. If I interpret the details correctly, (edit: which I apparently did not,) the idea is to consider a monad $T: {\rm \bf CAT} \to {\rm \bf CAT}$ on the category of locally small categories equipped with a distributive law $TP \implies PT$ from the monad $T$ to the small presheaves monad $P$ with \begin{align} P(\mathcal{C}) = \{\text{small presheaves } F: \mathcal{C}^{\rm op} \to {\rm Set} \} \subseteq {\rm Set}^{\mathcal{C}^{\rm op}}. \end{align} (See also this nLab article by Todd Trimble, or chapter 6 of the book Coend calculus by Fosco Loregian.) The presheaf category $P(T(1)) = {\rm Set}^{T(1)^{\rm op}}$ then admits a canonical monoidal structure, often called the `substitution product', and $T$-operads are defined as monoid objects in this category. Well-known examples are symmetric operads, non-symmetric operads and cartesian operads (Lawvere theories), which correspond to the monads on ${\rm \bf CAT}$ that characterize symmetric monoidal categories, monoidal categories and cartesian categories respectively.

Shulman then describes another generalized type of operads: semi-cocartesian operads. It uses the monad $T_{\rm sccs}$ which characterizes semi-cocartesian symmetric monoidal categories: symmetric monoidal categories whose monoidal unit is the initial object. He argues why any reduced operad $\mathcal{O}$ (i.e. $\mathcal{O}(0)$ is the final object) is naturally semi-cocartesian, and that seeing $\mathcal{O}$ as such gives a natural explanation for the basepoint identifications in the monad used by May in his work on operads.

Question: What is the necessary distributive law $T_{\rm sccs} P \implies PT_{\rm sccs}$?

Unfortunately, Shulman doesn't describe the distributive law that we need to make sense of semi-cocartesian operads, and I am not able to reproduce it. The problem I have with defining it is that ${\rm Set}$ is not semi-cocartesian. I guess that the approach should be adapted a little, for example by replacing presheaves by pointed presheaves with values in ${\rm Set}_*$, but then I am not sure how to recover the reduced operads of May as an example. Can someone help me out here?

Answer 1

I'd like to emphasize that nowhere in the linked blog post did I talk about distributive laws. It's true that some people like to define generalized multicategories using distributive laws over $P$, but that's not my preferred framework. My preferred framework is the one I linked to in the post that Geoff Cruttwell and I wrote about here, where instead of monads with a distributive law over $P$ we simply talk about monads on $\rm Prof$ --- and not the bicategory $\rm Prof$, but the double category $\rm Prof$. Having a distributive law over $P$ is one way to lift a monad to $\rm Prof$, because $\rm Prof$ is (up to size considerations) the Kleisli bicategory of $P$. But it's not the only way, so it's useful to work in a general framework that doesn't assume the lifting is obtained in that way. (It also frees us from size worries.) Using double categories (and more generally virtual double categories) also gives us the freedom to talk about monads whose underlying functor is lax. And finally, there are important examples of generalized multicategories arising from monads on double categories that are not the Kleisli bicategory of anything.

I'm not sure exactly how much of this extra freedom gets used in this example, but I am pretty sure that this example works fine if you forget about distributive laws and just think about monads on the double category $\rm Prof$. Nearly all monads of this sort extend immediately to monads on $\rm Prof$ by thinking of a profunctor as a collection of "heterogeneous homsets" and acting on them in the same way that you act on "homogeneous" homsets inside a single category.

In this particular case, for a category $A$ the objects of $T A$ are finite lists of objects of $A$, and the morphisms of $T A$ from $(a_i)_{1\le i \le m}$ to $(b_j)_{1\le j \le n}$ are injective functions $\phi : m\to n$ together with morphisms $a_i \to b_{\phi(i)}$. So we can define it exactly the same way on a profunctor $H : A \nrightarrow B$: for $(a_i)_{1\le i \le m} \in T A$ and $(b_j)_{1\le j \le n}\in T B$, an element of $T H((a_i),b_j))$ is an injective function $\phi : m\to n$ together with elements of $H(a_i, b_{\phi(i)})$.

Now you can build the horizontal-Kleisli (virtual) double category of this monad $T$ on $\rm Prof$. A semi-cocartesian operad is then a (horizontal) monoid in that h-Kleisli double category on the object $1$.