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The category of extensions and a characterisation of n-exangulated functors

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Abstract

Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information. We study how functors between categories of extensions relate to those at the level of the original categories. When the additive categories in question are n-exangulated, this leads to a characterisation of n-exangulated functors. Our approach enables us to study n-exangulated categories from a 2-categorical perspective. We introduce n-exangulated natural transformations and characterise them using categories of extensions. Our characterisations allow us to establish a 2-functor between the 2-categories of small n-exangulated categories and small exact categories. A similar result with no smallness assumption is also proved. We employ our theory to produce various examples of n-exangulated functors and natural transformations. Although the motivation for this article stems from representation theory and the study of n-exangulated categories, our results are widely applicable: several require only an additive category equipped with a biadditive functor with no extra assumptions; others can be applied by endowing an additive category with its split n-exangulated structure.

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Acknowledgements

The authors are very grateful to, and thank: Thomas Brüstle for their helpful comments that led to Proposition 3.2 in its present form; Hiroyuki Nakaoka for their time and a discussion that led to the development of Sect. 4; and Peter Jørgensen for a useful discussion relating to Example 5.6. We also thank the referee for their careful reading of the paper and their comments.   The first author is grateful to have been supported during part of this work by the Alexander von Humboldt Foundation in the framework of an Alexander von Humboldt Professorship endowed by the German Federal Ministry of Education. Parts of this work were carried out while the second author participated in the Junior Trimester Program “New Trends in Representation Theory” at the Hausdorff Research Institute for Mathematics in Bonn. She would like to thank the Institute for excellent working conditions. The third author is grateful to have been supported by Norwegian Research Council project 301375, “Applications of reduction techniques and computations in representation theory”. The fourth author gratefully acknowledges support from: the Danish National Research Foundation (grant DNRF156); the Independent Research Fund Denmark (grant 1026-00050B); the Aarhus University Research Foundation (grant AUFF-F-2020-7-16); the Engineering and Physical Sciences Research Council (grant EP/P016014/1); and the London Mathematical Society with support from Heilbronn Institute for Mathematical Research (grant ECF-1920-57).

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  1. Department of Mathematics, Aarhus University, 8000, Aarhus C, Denmark

    Raphael Bennett-Tennenhaus & Amit Shah

  2. Department of Mathematical Sciences, NTNU, 7491, Trondheim, Norway

    Johanne Haugland & Mads Hustad Sandøy

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  1. Raphael Bennett-Tennenhaus
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Bennett-Tennenhaus, R., Haugland, J., Sandøy, M.H. et al. The category of extensions and a characterisation of n-exangulated functors. Math. Z. 305, 44 (2023). https://doi.org/10.1007/s00209-023-03341-3

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