{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T07:19:55Z","timestamp":1740122395288,"version":"3.37.3"},"reference-count":23,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T00:00:00Z","timestamp":1680307200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T00:00:00Z","timestamp":1681084800000},"content-version":"vor","delay-in-days":9,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"University of Bergen"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Appl Categor Struct"],"published-print":{"date-parts":[[2023,4]]},"abstract":"Abstract<\/jats:title>We consider profunctors \"Equation missing\" between posets and introduce their graph<\/jats:italic> and ascent<\/jats:italic>. The profunctors $$\\text {Pro}(P,Q)$$<\/jats:tex-math>\n \n Pro<\/mml:mtext>\n (<\/mml:mo>\n P<\/mml:mi>\n ,<\/mml:mo>\n Q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> form themselves a poset, and we consider a partition $$\\mathcal {I}\\sqcup \\mathcal {F}$$<\/jats:tex-math>\n \n I<\/mml:mi>\n \u2294<\/mml:mo>\n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of this into a down-set $$\\mathcal {I}$$<\/jats:tex-math>\n I<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and up-set $$\\mathcal {F}$$<\/jats:tex-math>\n F<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, called a cut<\/jats:italic>. To elements of $$\\mathcal {F}$$<\/jats:tex-math>\n F<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> we associate their graphs, and to elements of $$\\mathcal {I}$$<\/jats:tex-math>\n I<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> we associate their ascents. Our basic results is that this, suitably refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of $$Q \\times P$$<\/jats:tex-math>\n \n Q<\/mml:mi>\n \u00d7<\/mml:mo>\n P<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Cuts in finite Booleans lattices correspond precisely to finite simplicial complexes. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letterplace ideals for posets. We study $$\\text {Pro}({\\mathbb N}, {\\mathbb N})$$<\/jats:tex-math>\n \n Pro<\/mml:mtext>\n (<\/mml:mo>\n N<\/mml:mi>\n ,<\/mml:mo>\n N<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Such profunctors identify as order preserving maps $$f: {\\mathbb N}\\rightarrow {\\mathbb N}\\cup \\{\\infty \\}$$<\/jats:tex-math>\n \n f<\/mml:mi>\n :<\/mml:mo>\n N<\/mml:mi>\n \u2192<\/mml:mo>\n N<\/mml:mi>\n \u222a<\/mml:mo>\n {<\/mml:mo>\n \u221e<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For our applications when P<\/jats:italic> and Q<\/jats:italic> are infinite, we also introduce a topology on $$\\text {Pro}(P,Q)$$<\/jats:tex-math>\n \n Pro<\/mml:mtext>\n (<\/mml:mo>\n P<\/mml:mi>\n ,<\/mml:mo>\n Q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, in particular on profunctors $$\\text {Pro}({\\mathbb N},{\\mathbb N})$$<\/jats:tex-math>\n \n Pro<\/mml:mtext>\n (<\/mml:mo>\n N<\/mml:mi>\n ,<\/mml:mo>\n N<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10485-023-09711-6","type":"journal-article","created":{"date-parts":[[2023,4,10]],"date-time":"2023-04-10T08:03:03Z","timestamp":1681113783000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Profunctors Between Posets and Alexander Duality"],"prefix":"10.1007","volume":"31","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9796-7530","authenticated-orcid":false,"given":"Gunnar","family":"Fl\u00f8ystad","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,4,10]]},"reference":[{"key":"9711_CR1","doi-asserted-by":"crossref","unstructured":"Baez, J.C.: Isbell duality, Notices of the American Mathematical Society 70(1) 140\u2013141 (2023)","DOI":"10.1090\/noti2685"},{"key":"9711_CR2","unstructured":"B\u00e9nabou, J.: Distributors at work, Lecture notes of a course at TU Darmstadt written by Thomas Streicher 11 (2000). www2.mathematik.tudarmstadt.de\/streicher\/FIBR\/DiWo.pdf"},{"key":"9711_CR3","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525872","volume-title":"Handbook of Categorical Algebra: volume 1, Basic Category Theory","author":"F Borceux","year":"1994","unstructured":"Borceux, F.: Handbook of Categorical Algebra: volume 1, Basic Category Theory, vol. 1. 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