Heaps of modules: categorical aspects

Breaz, Simion; Brzezinski, Tomasz; Rybołowicz, Bernard; Saracco, Paolo

doi:10.1017/fms.2025.10109

Journal article 137 views 23 downloads

Heaps of modules: categorical aspects

Simion Breaz

, Tomasz Brzezinski

, Bernard Rybołowicz

, Paolo Saracco

Forum of Mathematics, Sigma, Volume: 13, Start page: e166

Swansea University Author: Tomasz Brzezinski

PDF | Version of Record

© The Author(s), 2025. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (CC BY).
Download (426.58KB)

Check full text

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...

Full description

Published in:	Forum of Mathematics, Sigma
ISSN:	2050-5094
Published:	Cambridge University Press (CUP) 2025
Online Access:	Check full text
URI:	https://cronfa.swan.ac.uk/Record/cronfa70383

first_indexed	2025-09-18T10:10:23Z
last_indexed	2025-10-25T06:47:17Z
id	cronfa70383
recordtype	SURis
fullrecord	<?xml version="1.0"?><rfc1807><datestamp>2025-10-24T15:42:51.9357651</datestamp><bib-version>v2</bib-version><id>70383</id><entry>2025-09-18</entry><title>Heaps of modules: categorical aspects</title><swanseaauthors><author><sid>30466d840b59627325596fbbb2c82754</sid><ORCID>0000-0001-6270-3439</ORCID><firstname>Tomasz</firstname><surname>Brzezinski</surname><name>Tomasz Brzezinski</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2025-09-18</date><deptcode>MACS</deptcode><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.</abstract><type>Journal Article</type><journal>Forum of Mathematics, Sigma</journal><volume>13</volume><journalNumber/><paginationStart>e166</paginationStart><paginationEnd/><publisher>Cambridge University Press (CUP)</publisher><placeOfPublication/><isbnPrint/><isbnElectronic/><issnPrint/><issnElectronic>2050-5094</issnElectronic><keywords/><publishedDay>6</publishedDay><publishedMonth>10</publishedMonth><publishedYear>2025</publishedYear><publishedDate>2025-10-06</publishedDate><doi>10.1017/fms.2025.10109</doi><url/><notes/><college>COLLEGE NANME</college><department>Mathematics and Computer Science School</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>MACS</DepartmentCode><institution>Swansea University</institution><apcterm>SU Library paid the OA fee (TA Institutional Deal)</apcterm><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.</funders><projectreference/><lastEdited>2025-10-24T15:42:51.9357651</lastEdited><Created>2025-09-18T11:07:29.1013852</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>Simion</firstname><surname>Breaz</surname><orcid>0000-0002-8506-4526</orcid><order>1</order></author><author><firstname>Tomasz</firstname><surname>Brzezinski</surname><orcid>0000-0001-6270-3439</orcid><order>2</order></author><author><firstname>Bernard</firstname><surname>Rybołowicz</surname><orcid>0000-0002-2894-8288</orcid><order>3</order></author><author><firstname>Paolo</firstname><surname>Saracco</surname><orcid>0000-0001-5693-7722</orcid><order>4</order></author></authors><documents><document><filename>70383__35477__928984bc04cc487daf593a72e87a91cd.pdf</filename><originalFilename>70383.VOR.pdf</originalFilename><uploaded>2025-10-24T15:39:01.9656591</uploaded><type>Output</type><contentLength>436814</contentLength><contentType>application/pdf</contentType><version>Version of Record</version><cronfaStatus>true</cronfaStatus><documentNotes>© The Author(s), 2025. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (CC BY).</documentNotes><copyrightCorrect>true</copyrightCorrect><language>eng</language><licence>https://creativecommons.org/licenses/by/4.0</licence></document></documents><OutputDurs/></rfc1807>
spelling	2025-10-24T15:42:51.9357651 v2 70383 2025-09-18 Heaps of modules: categorical aspects 30466d840b59627325596fbbb2c82754 0000-0001-6270-3439 Tomasz Brzezinski Tomasz Brzezinski true false 2025-09-18 MACS 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. Journal Article Forum of Mathematics, Sigma 13 e166 Cambridge University Press (CUP) 2050-5094 6 10 2025 2025-10-06 10.1017/fms.2025.10109 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) 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. 2025-10-24T15:42:51.9357651 2025-09-18T11:07:29.1013852 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Simion Breaz 0000-0002-8506-4526 1 Tomasz Brzezinski 0000-0001-6270-3439 2 Bernard Rybołowicz 0000-0002-2894-8288 3 Paolo Saracco 0000-0001-5693-7722 4 70383__35477__928984bc04cc487daf593a72e87a91cd.pdf 70383.VOR.pdf 2025-10-24T15:39:01.9656591 Output 436814 application/pdf Version of Record true © The Author(s), 2025. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (CC BY). true eng https://creativecommons.org/licenses/by/4.0
title	Heaps of modules: categorical aspects
spellingShingle	Heaps of modules: categorical aspects Tomasz Brzezinski
title_short	Heaps of modules: categorical aspects
title_full	Heaps of modules: categorical aspects
title_fullStr	Heaps of modules: categorical aspects
title_full_unstemmed	Heaps of modules: categorical aspects
title_sort	Heaps of modules: categorical aspects
author_id_str_mv	30466d840b59627325596fbbb2c82754
author_id_fullname_str_mv	30466d840b59627325596fbbb2c82754_***_Tomasz Brzezinski
author	Tomasz Brzezinski
author2	Simion Breaz Tomasz Brzezinski Bernard Rybołowicz Paolo Saracco
format	Journal article
container_title	Forum of Mathematics, Sigma
container_volume	13
container_start_page	e166
publishDate	2025
institution	Swansea University
issn	2050-5094
doi_str_mv	10.1017/fms.2025.10109
publisher	Cambridge University Press (CUP)
college_str	Faculty of Science and Engineering
hierarchytype
hierarchy_top_id	facultyofscienceandengineering
hierarchy_top_title	Faculty of Science and Engineering
hierarchy_parent_id	facultyofscienceandengineering
hierarchy_parent_title	Faculty of Science and Engineering
department_str	School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str	1
active_str	0
description	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.
published_date	2025-10-06T05:26:28Z
_version_	1851641339773452288
score	11.089884

Heaps of modules: categorical aspects

Similar Items