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Heaps of modules: categorical aspects
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Tomasz Brzezinski ,
Bernard Rybołowicz ,
Paolo Saracco
Forum of Mathematics, Sigma, Volume: 13, Start page: e166
Swansea University Author:
Tomasz Brzezinski
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© The Author(s), 2025. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (CC BY).
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DOI (Published version): 10.1017/fms.2025.10109
Abstract
Connections between heaps of modules and (affine) modules over rings are explored. This leads to explicit, often constructive, descriptions of some categorical constructions and properties that are implicit in universal algebra and algebraic theories. In particular, it is shown that the category of...
| Published in: | Forum of Mathematics, Sigma |
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| ISSN: | 2050-5094 |
| Published: |
Cambridge University Press (CUP)
2025
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa70383 |
| Abstract: |
Connections between heaps of modules and (affine) modules over rings are explored. This leads to explicit, often constructive, descriptions of some categorical constructions and properties that are implicit in universal algebra and algebraic theories. In particular, it is shown that the category of groups with a compatible action of a truss T (also called pointed T-modules) is isomorphic to the category of modules over the ring R(T) universally associated to the truss. This is widely used in the explicit description of free objects. Next, it is proven that the category of heaps of modules over T is isomorphic to the category of affine modules over R(T) and, in order to make the picture complete, that (in the unital case) these are in turn equivalent to a specific subcategory of the slice category of pointed T-modules over R(T). These correspondences and properties are then used to describe explicitly various (co)limits and to compare short exact sequences in the Barr-exact category of heaps of T-modules with short exact sequences as defined previously. |
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| College: |
Faculty of Science and Engineering |
| Funders: |
The research of Simion Breaz is supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI–UEFISCDI, project number PN-III-P4-ID-PCE-2020-0454, within PNCDI III. The research of Tomasz Brzeziński was partially supported by the National Science Centre, Poland, grant no. 2019/35/B/ST1/01115 and is supported by the National Science Centre, Poland, through the WEAVE-UNISONO grant no. 2023/05/Y/ST1/00046. The research of Bernard Rybołowicz was supported by the EPSRC grant EP/V008129/1. This paper was written while Paolo Saracco was a Chargé de Recherches of the Fonds de la Recherche Scientifique - FNRS and a member of the “National Group for Algebraic and Geometric Structures and their Applications” (GNSAGA-INdAM). At the time of acceptance, PS research activity is supported by the Junta de Andalucía, in the framework of the Emergia grant DGP_EMEC_2023_00216. This article is based upon work from COST Action CaLISTA CA21109 supported by COST (European Cooperation in Science and Technology). www.cost.eu. |
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e166 |