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$\def\Fin{\text{Fin}_*} \def\Set{\text{Set}_*} \def\dd{\mathop{\diamond_\land}}$

The present question is intimately related to another question.

Let $\Fin$ be the category of pointed sets.

The category $[\Fin,\Set]$ of $\Set$-enriched functors has a convolution monoidal product when both categories are endowed with the smash product operation.

This means that we can convolve two parallel functors $F,G : \Fin \to \Set$ to $$ F*G := n\mapsto \int^{pq}Fp\land Gq\land \Fin(p\land q,n) $$ One would be tempted to define, starting from this, a substitution product following Kelly's "On the operads of PJ May": the iterated convolution $$ F^{\ast m} \cong \int^{p_1,...,p_m} Fp_1\land \dots \land Fp_m \land \Fin(p_1\land\dots\land p_m,\_) $$ should define a substitution product $$ F \dd G := n\mapsto \int^m Fm\land G^{\ast m}n $$ However, this is not an associative operation: in Kelly's proof, the fact that convolution and substitution interact in the following way is crucial: $$ (F\dd G)^{\ast m} \cong F^{\ast m}\dd G\tag{$\star$} $$ Because once $(\star)$ is proved, it follows that $$ \begin{align*} (F\dd G)\dd H & = \int^n (F\dd G)n\land H^{*n} \\ & \cong \int^{nm} Fm\land G^{*m}n\land H^{*n} \\ F \dd (G \dd H) & = \int^m Fm \land (G\dd H)^{*m} \\ & \cong \int^m Fm \land (G^{*m}\dd H) \\ & \cong \int^{nm} Fm\land G^{*m}n\land H^{*n}. \end{align*} $$

Let's try to prove $(\star)$ then, adopting a slightly improper notation in order to save space: $\underline p$ is a tuple of the appropriate length, and we write $\int^{\underline{p}} F\underline p \land \Fin(\underline p^\land,\_)$ for the integral defining $F^{\ast m}$: $$ \begin{align*} (F\diamond_\land G)^{\ast m} & =\int^{\underline{k}} \Fin(\underline{k}^\land, \_)\land (F\diamond_\land G)(\underline{k}) \\ & \cong \int^{\underline{k}, \underline{p}} \Fin(\underline{k}^\land, \_) \land \left(Fp_1 \land G^{\ast p_1}k_1\right) \land \cdots \land \left(Fp_m \land G^{\ast p_m}k_m\right) \\ & \cong \int^{\underline{p}} Fp_1\land\cdots\land Fp_m \land (G^{\ast p_1} \ast\cdots\ast G^{\ast p_m}) \\ & \cong \int^{\underline{p}} Fp_1\land\cdots\land Fp_m \land G^{\ast \sum p_i} \\ & \cong \int^{\underline{p}} Fp_1\land\cdots\land Fp_m \land \int^r G^{\ast r}\land \Fin(\sum p_i,r) \end{align*} $$ whre I expanded the definition, rearranged terms, compacted again the definition and finally used the fact that $(\_)^{\ast \_} : [\Fin,\Set]\times \Fin^\text{op} \to [\Fin,\Set]$ is a bifunctor, so that by ninja Yoneda $G^{\ast k} \cong \int^r G^{\ast r}\land \Fin(k,r)$.

Now I'm stuck, because $\Fin(\sum p_i,r) \not\cong \Fin(\bigwedge p_i,r) $; the monoidal structure given by coproduct is different from the one given by smash product; the first categorifies the monoid structure $(\mathbb N, +)$, whereas the second categorifies the monoid structure $(\mathbb N, \circ)$, where $p\circ q := pq-p-q+1$ (if $p,q$ are finite sets with $p,q$ elements; or counting in another way, if $p:= \{0,...,p\}$ then $p\circ q=pq$).

If I had obtained $\Fin(\bigwedge p_i,r)$ instead of $\Fin(\sum p_i,r)$, all would have been ok, because now $$ \begin{align*} & \cong \int^{\underline{p}} Fp_1\land\cdots\land Fp_m \land \int^r G^{\ast r}\land \Fin\big(\bigwedge p_m,r\big) \\ & \cong \int^{\underline{p},r} \Fin(\underline{p}^\land,r) \land Fp_1\land\cdots\land Fp_m \land G^{\ast r} \\ & \cong \int^r F^{\ast m} r \land G^{\ast r} \\ & \cong F^{\ast m} \diamond_\land G \end{align*} $$

It seems then that there is no way to define an operad-like monoidal structure on $[\Fin,\Set]$, neither starting from the framework of my previous question, nor in a purely $\Set$-enriched one, i.e. taking the smash-product monoidal structure on both domain and codomain of $[\Fin,\Set]$.

Why? What went wrong?

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In the same vein as my response to your other question, if pointed finite sets are an eleutheric system of arities, Lawvere theories over that system of arities will be equivalent to monads in a certain monoidal category. This is in section 11 of Lucyshyn-Wright here.

Edit: I briefly outlined how the relationship between eleutheric systems of arities and these monads works here. I haven't spent a huge amount of time working out this part of the paper, so I don't want to risk giving a bad answer. Sections 9,10,11 of Lucyshyn-wright's paper give a very detailed construction of this correspondence.

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    $\begingroup$ Yes, this is a very helpful comment; I was about to rediscover eleutheric arities. I think I'm going to contact Lucyshyn-Wright... $\endgroup$
    – fosco
    Commented Jun 2, 2020 at 15:27
  • $\begingroup$ Judging by the comment, your answer perfectly suffices for the OP. However for an average user here - I believe it is highly cryptic. Could you please elaborate just a little bit? $\endgroup$ Commented Jun 2, 2020 at 15:39
  • $\begingroup$ To be honest, I haven't spent enough time with the profunctor/theory correspondence to give a clean high-level overview. I did give a more thorough answer to what an eleutheric system of arities is. $\endgroup$ Commented Jun 2, 2020 at 16:15

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