Grothendieck categories of enriched functors
type: Item
instance of: scholarly article
title: Grothendieck categories of enriched functors
zbMATH Open document ID: 1333.18014
author: Hassan Al Hwaeer
author: Grigory Garkusha
published in: Journal of Algebra
publication date: 2016-01-29T00:00:00Z
review text: Let \(\mathcal V\) be a closed symmetric monoidal category, \(\mathcal C\) a \(\mathcal V\)-enriched category. The authors study localizations in the category of enriched functors \([\mathcal C,\mathcal V]\), and exhibits an identification between complexes of generalized modules \(Ch(\mathcal C_R)\) and enriched functor \([\text{mod} R,Ch(\text{Mod} R)]\). The notion of ``unital algebraic almost stable homotopy category'' is defined, providing a strict expansion of the class of ``unital algebraic stable homotopy category'' of Hovey-Palmieri-Strickland, by showing that the derived category of generalized modules \(\mathcal D(\mathcal C_R)\) over a commutative ring \(R\) is a unital algebraic almost stable homotopy category that is not a unital algebraic stable homotopy category. In particular, it is shown that \(\text{Ch}(\mathcal C_R)\) is a closed symmetric monoidal category and the derived category \(D(\mathcal C_R)\) is closed symmetric monoidal, and also a compactly generated triangulated category with a duality on compact objects, but the compact objects are strongly dualizable. This satisfies all axioms of Hovey-Palmieri-Strickland's unital algebraic stable homotopy theory apart from the strongly dualizable property on compact objects.
reviewed by: John Zhang
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 18D20
zbMATH DE Number: 6537412
zbMATH Keywords: Grothendieck categories
zbMATH Keywords: enriched categories
zbMATH Keywords: model categories
zbMATH Keywords: stable homotopy theories
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2015.09.052
OpenAlex ID: W2250071622
Wikidata QID: Q112881621
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DOI: 10.1016/J.JALGEBRA.2015.09.052
description: scientific article; zbMATH DE number 6537412