A note on triangulated monads and categories of module spectra

Ivo Dell'Ambrogio; Beren Sanders

doi:10.1016/j.crma.2018.06.007

Homological algebra/Topology

A note on triangulated monads and categories of module spectra

Presented by: the Editorial Board

Ivo Dell'Ambrogio ¹; Beren Sanders ²

¹ Laboratoire de mathématiques Paul-Painlevé, Université de Lille-1, Cité scientifique, Bât. M2, 59665 Villeneuve-d'Ascq cedex, France
² Laboratory for Topology and Neuroscience, École polytechnique fédérale de Lausanne, Bât. MA, Station 8, CH-1015 Lausanne, Switzerland

Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 839-842.

Abstract
Résumé

Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg–Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is ‘essentially monadic’, i.e. becomes monadic after performing the two evident necessary operations of taking the Verdier quotient by the kernel of the right adjoint and idempotent completion. In this sense, the monad itself is ‘intrinsically monadic’. It follows that for any highly structured ring spectrum, its category of homotopy (aka naïve) modules is triangulated if and only if it is equivalent to its category of highly structured (aka strict) modules.

On considère une monade sur une catégorie triangulée, idempotente-complète, avec la propriété que la catégorie de Eilenberg–Moore des modules hérite d'une triangulation. Nous montrons que toute autre adjonction triangulée réalisant cette monade est essentiellement « monadique », c'est-à-dire qu'elle devient monadique après avoir pris le quotient de Verdier par le noyau de l'adjoint à droite et la complétion idempotente. En ce sens, la monade elle-même est « intrinsèquement monadique ». Il s'ensuit que, pour tout spectre en anneaux hautement structuré, sa catégorie des modules d'homotopie (dits aussi naïfs) est triangulée si et seulement si elle est équivalente à sa catégorie des modules hautement structurés (dits aussi stricts).

Received: 2018-06-04
Accepted: 2018-06-28
Published online: 2018-07-06

DOI: 10.1016/j.crma.2018.06.007

Author's affiliations:

Ivo Dell'Ambrogio ¹; Beren Sanders ²

¹ Laboratoire de mathématiques Paul-Painlevé, Université de Lille-1, Cité scientifique, Bât. M2, 59665 Villeneuve-d'Ascq cedex, France
² Laboratory for Topology and Neuroscience, École polytechnique fédérale de Lausanne, Bât. MA, Station 8, CH-1015 Lausanne, Switzerland

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Ivo Dell'Ambrogio; Beren Sanders. A note on triangulated monads and categories of module spectra. Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 839-842. doi : 10.1016/j.crma.2018.06.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.06.007/

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