Plethysm Products, Element– and Plus Constructions

Kaufmann, Ralph M.; Monaco, Michael

Comptes Rendus. Mathématique, Tome 362 (2024) no. G4, pp. 357-411

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 the plus construction can also be realized as 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 yields a criterion for when a definition of operad–like structure as a plethysm monoid —as exemplified by operads— is possible.

En considérant les catégories comme des monoïdes bimodules sur leurs groupoïdes d’isomorphisme, nous construisons des structures monoïdales appelées produits de pléthysme à trois niveaux : c’est-à-dire pour les bimodules, les bimodules relatifs et les bimodules factorisables.

Pour les bimodules, nous travaillons dans le cadre général des actions par catégories. Nous donnons une théorie complète reliant ces niveaux entre eux ainsi qu’aux constructions d’éléments de Grothendieck, aux enrichissements indexés, décorations et algèbres.

La spécialisation dans les actions de groupoïdes conduit à des applications telles que la construction plus.

Dans ce cadre, le troisième niveau englobe les constructions connues de Baez–Dolan et ses généralisations, comme nous le prouvons. Un nouveau résultat est que la construction plus peut aussi être réalisée comme une construction d’éléments compatible avec les structures monoïdales que nous définissons. Cela nous permet de prouver une commutativité entre les constructions d’éléments et les constructions plus, dont un cas particulier a été annoncé précédemment. En spécialisant les résultats du troisième niveau, nous obtenons un critère pour savoir quand une définition d’une structure de type opérade en tant que monoïde pléthysmique (comme illustré par les opérades) est possible.

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