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Star-multiplicative graphs in pointed protomodular categories

2010-02-05Journal article Restricted access
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datacite.subject.fosCiências Naturais::Matemáticas
datacite.subject.sdg03:Saúde de Qualidade
datacite.subject.sdg09:Indústria, Inovação e Infraestruturas
datacite.subject.sdg12:Produção e Consumo Sustentáveis
dc.contributor.authorMartins-Ferreira, N.
dc.date.accessioned2025-12-15T17:06:35Z
dc.date.available2025-12-15T17:06:35Z
dc.date.issued2010-02-05
dc.descriptionFonte: http://www.tac.mta.ca/tac/volumes/23/9/23-09.pdf
dc.description.abstractProtomodularity, in the pointed case, is equivalent to the Split Short Five Lemma. It is also well known that this condition implies that every internal category is in fact an internal groupoid. In this work, this is condition (II) and we introduce two other conditions denoted (I) and (III). Under condition (I), every multiplicative graph is an internal category. Under condition (III), every star-multiplicative graph can be extended (uniquely) to a multiplicative graph, a problem raised by G. Janelidze in [10] in the semiabelian context. When the three conditions hold, internal groupoids have a simple description, that, in the semiabelian context, correspond to the notion of internal crossed module, in the sense of [10].eng
dc.description.sponsorshipThis work was done during the Post-Doctoral position held by the author at CMUC, supported by the FCT grant SFRH/BPD/4321/2008.
dc.identifier.citationMartins-Ferreira, N. (2010). Star-multiplicative graphs in pointed protomodular categories. Theory and Applications of Categories, 23(9), 170-198. DOI: https://doi.org/10.70930/tac/iodb71v5.
dc.identifier.doi10.70930/tac/iodb71v5
dc.identifier.issn1201-561X
dc.identifier.urihttp://hdl.handle.net/10400.8/15068
dc.language.isoeng
dc.peerreviewedyes
dc.publisherMount Allison University
dc.rights.uriN/A
dc.subjectInternal category
dc.subjectinternal groupoid
dc.subjectre°exive graph
dc.subjectmultiplicative graph
dc.subjectstar-multiplicative graph
dc.subjectjointly epic pair
dc.subjectadmissible pair
dc.subjectjointly epic split extension
dc.subjectsplit short ¯ve lemma
dc.subjectpointed protomodular
dc.titleStar-multiplicative graphs in pointed protomodular categorieseng
dc.typejournal article
dspace.entity.typePublication
oaire.citation.endPage198
oaire.citation.issue9
oaire.citation.startPage170
oaire.citation.titleTheory and Applications of Categories
oaire.citation.volume23
oaire.versionhttp://purl.org/coar/version/c_970fb48d4fbd8a85
person.familyNameMartins-Ferreira
person.givenNameNelson
person.identifier485301
person.identifier.ciencia-idB115-B65E-24AA
person.identifier.orcid0000-0002-4199-7367
person.identifier.ridN-1699-2013
person.identifier.scopus-author-id24598020700
relation.isAuthorOfPublication52406f6a-2c36-4e9a-9996-d3cc719d46bf
relation.isAuthorOfPublication.latestForDiscovery52406f6a-2c36-4e9a-9996-d3cc719d46bf

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Protomodularity, in the pointed case, is equivalent to the Split Short Five Lemma. It is also well known that this condition implies that every internal category is in fact an internal groupoid. In this work, this is condition (II) and we introduce two other conditions denoted (I) and (III). Under condition (I), every multiplicative graph is an internal category. Under condition (III), every star-multiplicative graph can be extended (uniquely) to a multiplicative graph, a problem raised by G. Janelidze in [10] in the semiabelian context. When the three conditions hold, internal groupoids have a simple description, that, in the semiabelian context, correspond to the notion of internal crossed module, in the sense of [10].
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