On the Distributive Law in Cartesian Multicategories

Cartesian operads can be seen as a way to encode algebraic theories; though they are essentially equivalent to Lawvere theories, the perspective is different. Cartesian multicategories (or cartesian colored operads) correspond to the many sorted theories. In this talk we present cartesian multicategories as algebras for a natural monad $\mathrm{𝐜𝐚𝐫𝐭}$ on symmetric multicategories, and show how this notion is in several respects conceptually advantageous. This analysis is carried out in the framework of unbiased symmetric multicategories [1] where, as is natural in the symmetric context, sequences are replaced by families.

In fact, unbiased symmetric multicategories are simply sum-preserving double functors $𝕄\to \mathbb{P}\mathrm{b}$ (to the double category of pullback squares in finite sets) which are discrete fibrations in the tight direction. The loose arrows of $𝕄$ should be thought of as families of arrows in the multicategory, indexed by the underlying functions in $\mathbb{P}\mathrm{b}$, while tight arrows and cells give reindexing. The classical axioms are in this way naturally embodied by the double categorical structure. This approach, which removes the awkwardness arising from an unnatural skeletal indexing, turns out to be rather effective and opens up new perspectives. In particular, it allows for a base sensitive study of multicategories and renders transparent the links with Joyal’s species.

For instance, plain multicategories are sum-preserving double discrete fibrations $𝕄\to 𝕋\mathrm{ot}$, where $𝕋\mathrm{ot}$ is (the double categorical form of) the multiplicative species of total orders. Furthermore, unbiased symmetric monoidal categories arise when $𝕄\to \mathbb{P}\mathrm{b}$ is, in the loose direction, an opfibration, while the algebras for a symmetric multicategory $𝕄\to \mathbb{P}\mathrm{b}$ are morphisms $𝔸\to 𝕄$ which are discrete opfibrations, and so are themselves symmetric multicategories.

Symmetric operads can be seen as multiplicative species of structures and cartesian operads are obtained by endowing them also with a sum, such that multiplication-composition distributes over sums. In fact, we define cartesian multicategories as the algebras for the monad which takes a symmetric multicategory $𝕄$ to the multicategory $\mathrm{𝐜𝐚𝐫𝐭}𝕄$ whose loose arrows are spans (actually, "enhanced" spans [1]) formed by a tight arrow $f$ and a loose arrow $\alpha$ in $𝕄$; these are composed in the usual way for spans, except that we use $𝕄$-cells in place of pullbacks. Thus, a cartesian structure $\sum :\mathrm{𝐜𝐚𝐫𝐭}𝕄\to 𝕄$ provides a way to evaluate spans, giving the "sum" ${\sum }_f\alpha$ of $\alpha$ along $f$. The functoriality of $\sum$ on loose arrows says that the sum of a composition of spans is the same as the composition of their sums: ${\sum }_g\beta {\sum }_f\alpha ={\sum }_{fh}\beta \gamma$, Since cells in $𝕄$ (like $\mathrm{@}$) are defined by reindexing (which, for non-injective functions, duplicate elements), this can be indeed seen as a kind of generalized distributive law, holding in any cartesian multicategory.

If $R$ is a monoid, we have the "cocartesian" operad ${ℝ}_{▷}$ associated to the multiplicative species $R^I$ of labellings in $R$: a structure over a set $I$ is a family $\alpha :I\to R$. Cartesian structures on ${ℝ}_{▷}$ correspond to rig structures on $R$, and the distributive law therein is of course the usual one.

On the other hand, for any set $S$, the species $S^{S^I}$ gives the endomorphism operad $𝔼\mathrm{nd}(S)$ which is also cartesian: sums are given by duplication-deletion of variables in the multivariate functions $\alpha :S^I\to S$, and the distributive law says, for instance, that the two ways of calculating $\beta (\alpha (x);\alpha (x))$ are equivalent: we can either duplicate the value $\alpha (x)$ and evaluate $\beta$ at it, or duplicate the function $\alpha$ itself, compose it with $\beta$ and evaluate it at the duplicate of $x$. Cartesian morphisms ${ℝ}_{▷}\to 𝔼\mathrm{nd}(S)$ amount to modules $S$ on $R$, and can themselves be considered as cartesian multicategories over ${ℝ}_{▷}$; their loose arrows over ${(r_i)}_{i\in I}$ are "combinations" ${(r_i{s}_i)}_{i\in I}$ and sums reduce them on equal $s_i$.