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?

• I'm not sure if this is directly related, but in Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures, the authors consider distributive laws of 2-monads over free cocompletion, which is analogous to the setting here. In particular, they show that the 2-monad for "categories with an initial object" does not distribute. Perhaps in Shulman's setting things are different, but this seems a little surprising. Mar 17, 2020 at 17:05
• You should email Nicola or one of the other authors of the paper @varkor has mentioned. I couldn't do better than them :) just one minor remark: strictly speaking, size issues and coherence conditions force you to consider a generalisation of distributive laws: you need pseudo-distr-laws where at least one of the functors is a relative monad! Mar 17, 2020 at 22:21
• Using small presheaves instead of arbitrary presheaves, as Shulman does, is another way to avoid the size issues, allowing you to avoid considering relative pseudomonads. (Though this seems a little less elegant than the relative setting to me.) Mar 17, 2020 at 22:26
• I agree with you. Mar 17, 2020 at 22:34
• Thanks for the reference! The conclusion I take is that there is no such a distributive law if we use presheaves $\mathcal{C}^{\rm op} \to {\rm Set}$. Do you think it could be useful to consider pointed presheaves $\mathcal{C}^{\rm op} \to {\rm Set_*}$ on ${\rm Set}_*$-enriched categories $\mathcal{C}$? I want to understand how reduced operads give monads on ${\rm Set}_*$ (well, actually on ${\rm Top}_*$), so this seems reasonable to look at. What is the reason this is not being considered? Mar 18, 2020 at 12:37

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 $$1$$.

• Thanks for the clarification! I've edited the post to mention I had a wrong interpretation. Do I observe correctly that your method gives in fact a definition for semi-cocartesian operads in any semi-cartesian monoidal category $\mathcal{V}$? The monad $T$ on $\mathcal{V}$-${\rm Prof}$ would have $TA((a_i),(b_j)) = \bigsqcup_{\phi: m \to n} \prod_{i=1}^m A(a_i,b_{\phi(i)})$ and for composition we only need projections. Mar 19, 2020 at 11:31
• @BastiaanCnossen Yes, I believe that's correct. Mar 20, 2020 at 18:08
• Is it correct that every semi-cocartesian operad is reduced? We should have the axiom that if $\phi: m \hookrightarrow n$ is an injective map and $O$ is a semi-cocartesian operad, then for any operation $a \in O(n), b_i \in O(k_i)$, application of $O(\phi)(a) \in O(m)$ to $(b_{\phi(1)}, \dots, b_{\phi(m)})$ is equal to the restriction of $a(b_1, \dots, b_n) \in O(k_1 + \dots + k_n)$ along the injective map $k_{\phi(1)} + \dots + k_{\phi(m)} \hookrightarrow k_1 + \dots + k_n$, from which it follows that each $0$-ary operation $o = 1(o)$ is equal to $O(0 \to1)(1) \in O(0)$. Correct? Apr 15, 2020 at 13:02
• @BastiaanCnossen At first I was convinced by that, but as I think about it some more I don't think that is actually one of the axioms. The axioms of that sort in generalized operads come from profunctor composition, which means they are of the "equivariance" sort: "acting covariantly on $a$ is equivalent to acting contravariantly on $b_i$". Your axiom instead involves an action on both $a$ and $b$ on one side of the equation. May 6, 2020 at 20:20
• I think this is still part of the naturality in $(k_1, \dots, k_n)$. A priori the multiplication of $O$ should consist of maps $O(n) \times \bigsqcup_{\psi: n' \to n} \prod_{i=1}^{n'} O(k_i) \to O(k_1 + \dots + k_{n'})$ that are extranatural in $n \in T1$ but natural in $(k_1, \dots, k_{n'}) \in (T^21)^{\rm op}$. The axiom I wrote corresponds to naturality for the map $(k_1, \dots, k_{n'}) \to (k_{\phi(1)}, \dots, k_{\phi(m)})$ in $T^21^{\rm op}$, where $n' = n$, combined with extranaturality in $n \in T1$ to reduce back to the case $\psi = {\rm id}$. May 7, 2020 at 23:10