{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:41:33Z","timestamp":1773250893563,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T00:00:00Z","timestamp":1602547200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T00:00:00Z","timestamp":1602547200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB-TRR 195SFB-TRR 195"],"award-info":[{"award-number":["SFB-TRR 195SFB-TRR 195"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2021,2]]},"abstract":"Abstract<\/jats:title>We discuss Peter Freyd\u2019s universal way of equipping an additive category $$\\mathbf {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with cokernels from a constructive point of view. The so-called Freyd category $$\\mathcal {A}(\\mathbf {P})$$<\/jats:tex-math>\n \n A<\/mml:mi>\n (<\/mml:mo>\n P<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is abelian if and only if $$\\mathbf {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has weak kernels. Moreover, $$\\mathcal {A}(\\mathbf {P})$$<\/jats:tex-math>\n \n A<\/mml:mi>\n (<\/mml:mo>\n P<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has decidable equality for morphisms if and only if we have an algorithm for solving linear systems $$X \\cdot \\alpha = \\beta $$<\/jats:tex-math>\n \n X<\/mml:mi>\n \u00b7<\/mml:mo>\n \u03b1<\/mml:mi>\n =<\/mml:mo>\n \u03b2<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for morphisms $$\\alpha $$<\/jats:tex-math>\n \u03b1<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\beta $$<\/jats:tex-math>\n \u03b2<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$\\mathbf {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We give an example of an additive category with weak kernels and decidable equality for morphisms in which the question whether such a linear system admits a solution is computationally undecidable. Furthermore, we discuss an additional computational structure for $$\\mathbf {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that helps solving linear systems in $$\\mathbf {P}$$<\/jats:tex-math>\n P<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and even in the iterated Freyd category construction $$\\mathcal {A}( \\mathcal {A}(\\mathbf {P})^{\\mathrm {op}} )$$<\/jats:tex-math>\n \n A<\/mml:mi>\n (<\/mml:mo>\n A<\/mml:mi>\n \n \n (<\/mml:mo>\n P<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n op<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, which can be identified with the category of finitely presented covariant functors on $$\\mathcal {A}(\\mathbf {P})$$<\/jats:tex-math>\n \n A<\/mml:mi>\n (<\/mml:mo>\n P<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The upshot of this paper is a constructive approach to finitely presented functors that subsumes and enhances the standard approach to finitely presented modules in computer algebra.<\/jats:p>","DOI":"10.1007\/s10485-020-09612-y","type":"journal-article","created":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T05:02:58Z","timestamp":1602565378000},"page":"171-211","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["A Constructive Approach to Freyd Categories"],"prefix":"10.1007","volume":"29","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9673-0811","authenticated-orcid":false,"given":"Sebastian","family":"Posur","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,10,13]]},"reference":[{"key":"9612_CR1","doi-asserted-by":"crossref","unstructured":"Auslander, M.: Coherent functors. 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