A bialgebraic characterization of symmetric powers in $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal categories

Abstract

In any symmetric monoidal category, the $n$-th (co)equalizer symmetric power of an object $A$ is the (co)equalizer of all the permutations from $A^{\otimes n}$ to itself. If the symmetric monoidal category is $\mathbb{Q}_{\ge 0}$-linear, that is, enriched over $\mathbb{Q}_{\ge 0}$-modules, the notions of $n$-th equalizer symmetric power and $n$-th coequalizer symmetric power are equivalent. In this context, the $n$-th symmetric power of $A$ can be described as the intermediate object $A_n$ in a splitting of the idempotent $\frac{1}{n!}\underset{\sigma \in S_n}{\sum}\sigma\colon A^{\otimes n} \rightarrow A^{\otimes n}$. We define a permutation splitting as a countable family of such splittings. The main goal of this paper is to prove two theorems. The first theorem exhibits in any $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal category a bijection between operations making a graded object $(A_n)_{n \ge 0}$ into a permutation splitting and operations making this graded object into a bialgebraic structure that we call a binomial bimonoid. Binomial bimonoids can be defined in any additive symmetric monoidal category. The second theorem shows that, in any $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal category, the biassociativity and bicommutativity axioms may be omitted from the definition of a binomial bimonoid. We then show that being a binomial bimonoid in a $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal category is a property: two binomial bimonoids are isomorphic whenever their underlying graded objects are isomorphic. This result does not extend to arbitrary additive symmetric monoidal categories since both the one-variable polynomial algebra and the one-variable divided power polynomial algebra over a field $k$ of positive characteristic are non-isomorphic binomial $k$-bialgebras with isomorphic underlying $\mathbb{N}$-graded vector spaces.