# Substitution structure on pointed sets

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

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?