Plethysm Products, Element and Plus Constructions

TL;DR

This paper develops a layered theory of monoidal plethysm products for bimodules over categories, linking them to known constructions like Baez-Dolan plus, and introduces new element-based realizations and criteria for operad-like structures.

Contribution

It introduces a comprehensive framework for plethysm products across multiple levels, connecting them to Grothendieck constructions and generalizing known operad constructions.

Findings

Established a new element construction compatible with monoidal structures.

Proved a commutativity result between element and plus constructions.

Provided a criterion for defining operad-like structures as plethysm monoids.

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.