**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?

notdistribute. Perhaps in Shulman's setting things are different, but this seems a little surprising. $\endgroup$ – varkor Mar 17 at 17:05relativemonad! $\endgroup$ – Fosco Mar 17 at 22:21