Pretriangulated 2-representations via dg algebra 1-morphisms

Robert Laugwitz; Vanessa Miemietz

doi:10.4171/DM/1029

Pretriangulated 2-representations via dg algebra 1-morphisms

Robert Laugwitz
, Vanessa Miemietz

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Abstract

This paper develops a theory of pretriangulated 2-representations of dg 2-categories. We characterize cyclic pretriangulated 2-representations, under certain compactness assumptions, in terms of dg modules over dg algebra 1-morphisms internal to associated dg 2-categories of compact objects. Further, we investigate the Morita theory and quasi-equivalences for such dg 2-representations. We relate this theory to various classes of examples of dg categorifications from the literature.

Original language	English
Pages (from-to)	1271–1323
Number of pages	53
Journal	Documenta Mathematica
Volume	30
Issue number	6
DOIs	https://doi.org/10.4171/DM/1029
Publication status	Published - 29 Aug 2025

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10.4171/DM/1029Licence: CC BY

10.4171-dm-1029Final published version, 750 KBLicence: CC BY

2-representation theory and categorification
Miemietz, V. (Principal Investigator)
Engineering and Physical Sciences Research Council
2/04/19 → 31/03/23
Project: Research

Cite this

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abstract = "This paper develops a theory of pretriangulated 2-representations of dg 2-categories. We characterize cyclic pretriangulated 2-representations, under certain compactness assumptions, in terms of dg modules over dg algebra 1-morphisms internal to associated dg 2-categories of compact objects. Further, we investigate the Morita theory and quasi-equivalences for such dg 2-representations. We relate this theory to various classes of examples of dg categorifications from the literature.",

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N2 - This paper develops a theory of pretriangulated 2-representations of dg 2-categories. We characterize cyclic pretriangulated 2-representations, under certain compactness assumptions, in terms of dg modules over dg algebra 1-morphisms internal to associated dg 2-categories of compact objects. Further, we investigate the Morita theory and quasi-equivalences for such dg 2-representations. We relate this theory to various classes of examples of dg categorifications from the literature.

AB - This paper develops a theory of pretriangulated 2-representations of dg 2-categories. We characterize cyclic pretriangulated 2-representations, under certain compactness assumptions, in terms of dg modules over dg algebra 1-morphisms internal to associated dg 2-categories of compact objects. Further, we investigate the Morita theory and quasi-equivalences for such dg 2-representations. We relate this theory to various classes of examples of dg categorifications from the literature.

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Pretriangulated 2-representations via dg algebra 1-morphisms

Abstract

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2-representation theory and categorification

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