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

Jean-Baptiste Vienney

TL;DR

This work characterizes symmetric powers in $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal categories through a bijection between permutation splittings and binomial bimonoids, revealing a canonical bialgebraic framework for $n$-th symmetric powers. It shows that in this setting the biassociativity and bicommutativity axioms can be omitted without loss of structure, yielding a streamlined presentation of binomial bimonoids. The results generalize the classical polynomial/ divided-power constructions (exemplified by $k[x]$ and its graded variants) to a broad, coordinate-free categorical context, and they establish that binomial bimonoid structure is a property determined by the underlying graded object up to isomorphism. The paper also provides concrete instances in categories such as modules over commutative $\mathbb{Q}_{\ge 0}$-algebras, sets with relations, and suplattices, illustrating how permutation splittings induce binomial bimonoids in diverse environments.

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 $\frac{1}{n!}\underset{σ\in S_n}{\sum}σ\colon A^{\otimes n} \rightarrow A^{\otimes n}$. 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.

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

TL;DR

This work characterizes symmetric powers in -linear symmetric monoidal categories through a bijection between permutation splittings and binomial bimonoids, revealing a canonical bialgebraic framework for -th symmetric powers. It shows that in this setting the biassociativity and bicommutativity axioms can be omitted without loss of structure, yielding a streamlined presentation of binomial bimonoids. The results generalize the classical polynomial/ divided-power constructions (exemplified by and its graded variants) to a broad, coordinate-free categorical context, and they establish that binomial bimonoid structure is a property determined by the underlying graded object up to isomorphism. The paper also provides concrete instances in categories such as modules over commutative -algebras, sets with relations, and suplattices, illustrating how permutation splittings induce binomial bimonoids in diverse environments.

Abstract

In any symmetric monoidal category, the -th (co)equalizer symmetric power of an object is the (co)equalizer of all the permutations from to itself. If the symmetric monoidal category is -linear, that is, enriched over -modules, the notions of -th equalizer symmetric power and -th coequalizer symmetric power are equivalent. In this context, the -th symmetric power of can be described as the intermediate object in a splitting of the idempotent . 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 -linear symmetric monoidal category a bijection between operations making a graded object 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 -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 -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 of positive characteristic are non-isomorphic binomial -bialgebras with isomorphic underlying -graded vector spaces.
Paper Structure (42 sections, 33 theorems, 311 equations, 4 figures)

This paper contains 42 sections, 33 theorems, 311 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Symmetric powers in $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal categories.
  3. The example of $(k_n[x])_{n \ge 0}$.
  4. Other examples.
  5. Main results.
  6. Outline of the rest of the paper.
  7. Related works.
  8. Conventions and notations.
  9. Acknowldegments.
  10. String diagrams and the bialgebraic structure of $k[x]$
  11. String diagrams and bicommutative bimonoids
  12. i. String diagrams for symmetric monoidal categories.
  13. ii. String diagrams for additive symmetric monoidal categories.
  14. iii. Commutative monoids.
  15. iv. Cocommutative comonoids.
  16. ...and 27 more sections

Key Result

Proposition 4

Let $(\mathsf{C},\otimes,I)$ be an additively idempotent additive symmetric monoidal category. Then $(\mathsf{C},\otimes,I)$ is a $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal category by defining for all $q \in \mathbb{Q}_{\ge 0}$ and morphisms $f$.

Figures (4)

  • Figure 1: Axioms for a binomial bimonoid
  • Figure 2: Equivalent axioms for a binomial bimonoid
  • Figure 3: Equivalent axioms for a binomial bimonoid in a $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal category
  • Figure 4: String diagram representation of $\psi_\sigma(u,v,w,x)=(\alpha,\beta,\gamma,\delta) \in (\phi'_{a_\sigma,b_\sigma,c_\sigma,d_\sigma})^{-1}(\sigma)$. The permutations $\alpha$ and $\beta$ are obtained from $u,v$ and $w,x$ as depicted. Then, there exist unique $\gamma$ and $\delta$ such that $\phi'_{a_\sigma,b_\sigma,c_\sigma,d_\sigma}(\alpha,\beta,\gamma,\delta)$, which is the permutation represented by the whole diagram, is equal to $\sigma$.

Theorems & Definitions (89)

  • Definition 1
  • Example 2
  • Definition 3
  • Proposition 4
  • proof
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Definition 9
  • ...and 79 more