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Plethysm Products, Element and Plus Constructions

Ralph M. Kaufmann, Michael Monaco

TL;DR

The paper develops a unified framework for plethysm products on bimodules over categories, organizing three levels (bimodules, relative bimodules, and factorizable bimodules) and linking them through element representations, Kan extensions, and monoidal structures. It connects these categorical constructions to Grothendieck’s element construction and Baez–Dolan’s plus construction, proving a commutativity between element and plus constructions in suitable contexts. A central result shows that when a Feynman category F is the plus construction of a Unique Factorization Category M, the category of strong monoidal functors [F, C]⊗ is equivalent to the category of monoids in the basic element representations, thereby grounding operad-like theories (e.g., operads, properads, hyper modular operads) as plethysm monoids within this framework. The work also develops decorations, enrichments, and various notions of algebras and mergers, providing a flexible toolkit to study operad-like objects across graphical Feynman categories and UFCs. Overall, the results offer a broad, interconnected perspective on plethysm in category theory with strong links to classical operad theory and its higher analogs, enabling new definitions and equivalences for algebras, decorations, and plus constructions.

Abstract

Motivated by viewing categories as bimodule monoids over their isomorphism groupoids, we construct monoidal structures called plethysm products on three levels: that is for bimodules, relative bimodules and factorizable bimodules. For the bimodules, we work in the general setting of actions by categories. We give a comprehensive theory linking these levels to each other as well as to Grothendieck element constructions, indexed enrichments, decorations and algebras. Specializing to groupoid actions leads to applications including the plus construction. In this setting, the third level encompasses the known constructions of Baez-Dolan and its generalizations, as we prove. One new result is that that the plus construction can also be realized an element construction compatible with monoidal structures that we define. This allows us to prove a commutativity between element and plus constructions a special case of which was announced earlier. Specializing the results on the third level yield a criterion, when a definition of operad-like structure as a plethysm monoid -- as exemplified by operads -- is possible.

Plethysm Products, Element and Plus Constructions

TL;DR

The paper develops a unified framework for plethysm products on bimodules over categories, organizing three levels (bimodules, relative bimodules, and factorizable bimodules) and linking them through element representations, Kan extensions, and monoidal structures. It connects these categorical constructions to Grothendieck’s element construction and Baez–Dolan’s plus construction, proving a commutativity between element and plus constructions in suitable contexts. A central result shows that when a Feynman category F is the plus construction of a Unique Factorization Category M, the category of strong monoidal functors [F, C]⊗ is equivalent to the category of monoids in the basic element representations, thereby grounding operad-like theories (e.g., operads, properads, hyper modular operads) as plethysm monoids within this framework. The work also develops decorations, enrichments, and various notions of algebras and mergers, providing a flexible toolkit to study operad-like objects across graphical Feynman categories and UFCs. Overall, the results offer a broad, interconnected perspective on plethysm in category theory with strong links to classical operad theory and its higher analogs, enabling new definitions and equivalences for algebras, decorations, and plus constructions.

Abstract

Motivated by viewing categories as bimodule monoids over their isomorphism groupoids, we construct monoidal structures called plethysm products on three levels: that is for bimodules, relative bimodules and factorizable bimodules. For the bimodules, we work in the general setting of actions by categories. We give a comprehensive theory linking these levels to each other as well as to Grothendieck element constructions, indexed enrichments, decorations and algebras. Specializing to groupoid actions leads to applications including the plus construction. In this setting, the third level encompasses the known constructions of Baez-Dolan and its generalizations, as we prove. One new result is that that the plus construction can also be realized an element construction compatible with monoidal structures that we define. This allows us to prove a commutativity between element and plus constructions a special case of which was announced earlier. Specializing the results on the third level yield a criterion, when a definition of operad-like structure as a plethysm monoid -- as exemplified by operads -- is possible.
Paper Structure (79 sections, 55 theorems, 82 equations, 2 figures, 2 tables)

This paper contains 79 sections, 55 theorems, 82 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction and Overview
  2. Introduction
  3. Results and constructions for bimodules
  4. First level: bimodules
  5. Second level: relative bimodules and element representations
  6. Third level: factorizable bimodules
  7. Decorations and Algebras
  8. Application to classical theory
  9. Connection to other points of view on operad--like objects.
  10. Bimodules
  11. Bimodules and Representations
  12. Group actions
  13. General actions
  14. Bimodules
  15. Plethysm
  16. ...and 64 more sections

Key Result

Proposition 2.2

There is a one-to-one correspondence between $G$-actions and functors $\rho: \underline{G} \longrightarrow {\mathcal{S}}et$.

Figures (2)

  • Figure 1: In (A), the top and bottom rows are objects of $\mathcal{A}^\boxtimes$ with each individual $B$ and $B'$ an object of $\mathcal{A}$. The two middle rows are objects of $\mathcal{A}$ with each individual $b$ and $b'$ an object of ${\mathcal{V}}$. The collection of line segments connecting a $B$-object to several $b$-objects represents an isomorphism from $B$ to the monoidal product of those $b$-objects. The arrows in the middle are morphisms in ${\mathcal{V}}$. These can also be pictured as "brick wall" diagrams (B) like those found in KMoManin.
  • Figure :

Theorems & Definitions (202)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • ...and 192 more