Plethysm Products, Element– and Plus Constructions
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 catego...
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Académie des sciences
2024-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.557/ |
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| author | Kaufmann, Ralph M. Monaco, Michael |
| author_facet | Kaufmann, Ralph M. Monaco, Michael |
| author_sort | Kaufmann, Ralph M. |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-34df0e6b5e4549c6b6807dd34af8cbdc |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-34df0e6b5e4549c6b6807dd34af8cbdc2025-02-07T11:20:51ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-05-01362G435741110.5802/crmath.55710.5802/crmath.557Plethysm Products, Element– and Plus ConstructionsKaufmann, Ralph M.0Monaco, Michael1Purdue University Department of Mathematics, and Department of Physics & Astronomy, West Lafayette, IN 47907, USAPurdue University Department of Mathematics, West Lafayette, IN 47907, USAMotivated 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.557/ |
| spellingShingle | Kaufmann, Ralph M. Monaco, Michael Plethysm Products, Element– and Plus Constructions Comptes Rendus. Mathématique |
| title | Plethysm Products, Element– and Plus Constructions |
| title_full | Plethysm Products, Element– and Plus Constructions |
| title_fullStr | Plethysm Products, Element– and Plus Constructions |
| title_full_unstemmed | Plethysm Products, Element– and Plus Constructions |
| title_short | Plethysm Products, Element– and Plus Constructions |
| title_sort | plethysm products element and plus constructions |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.557/ |
| work_keys_str_mv | AT kaufmannralphm plethysmproductselementandplusconstructions AT monacomichael plethysmproductselementandplusconstructions |