{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T00:37:23Z","timestamp":1747183043318,"version":"3.40.5"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"7","license":[{"start":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T00:00:00Z","timestamp":1603843200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2021,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given a monoidal category <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline1.png\"\/><jats:tex-math>$\\mathcal C$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> with an object <jats:italic>J<\/jats:italic>, we construct a monoidal category <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline2.png\"\/><jats:tex-math>$\\mathcal C[{J^ \\vee }]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> by freely adjoining a right dual <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline3.png\"\/><jats:tex-math>${J^ \\vee }$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> to <jats:italic>J<\/jats:italic>. We show that the canonical strong monoidal functor <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline4.png\"\/><jats:tex-math>$\\Omega :\\mathcal C \\to \\mathcal C[{J^ \\vee }]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> provides the unit for a biadjunction with the forgetful 2-functor from the 2-category of monoidal categories with a distinguished dual pair to the 2-category of monoidal categories with a distinguished object. We show that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline5.png\"\/><jats:tex-math>$\\Omega :\\mathcal C \\to \\mathcal C[{J^ \\vee }]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is fully faithful and provide coend formulas for homs of the form <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline6.png\"\/><jats:tex-math>$\\mathcal C[{J^ \\vee }](U,\\,\\Omega A)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline7.png\"\/><jats:tex-math>$\\mathcal C[{J^ \\vee }](\\Omega A,U)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline8.png\"\/><jats:tex-math>$A \\in \\mathcal C$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline9.png\"\/><jats:tex-math>$U \\in \\mathcal C[{J^ \\vee }]$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. If <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline10.png\"\/><jats:tex-math>${\\rm{N}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> denotes the free strict monoidal category on a single generating object 1, then <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline11.png\"\/><jats:tex-math>${\\rm{N[}}{{\\rm{1}}^ \\vee }{\\rm{]}}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the free monoidal category Dpr containing a dual pair \u2013 \u02e7 + of objects. As we have the monoidal pseudopushout <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129520000274_inline12.png\"\/><jats:tex-math>$\\mathcal C[{J^ \\vee }] \\simeq {\\rm{Dpr}}{{\\rm{ + }}_{\\rm{N}}}\\mathcal C$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, it is of interest to have an explicit model of Dpr: we provide both geometric and combinatorial models. We show that the (algebraist\u2019s) simplicial category \u0394 is a monoidal full subcategory of Dpr and explain the relationship with the free 2-category Adj containing an adjunction. We describe a generalization of Dpr which includes, for example, a combinatorial model Dseq for the free monoidal category containing a duality sequence <jats:italic>X<\/jats:italic><jats:sub>0<\/jats:sub> \u02e7 <jats:italic>X<\/jats:italic><jats:sub>1<\/jats:sub> \u02e7 <jats:italic>X<\/jats:italic><jats:sub>2<\/jats:sub> \u02e7 \u2026 of objects. Actually, Dpr is a monoidal full subcategory of Dseq.<\/jats:p>","DOI":"10.1017\/s0960129520000274","type":"journal-article","created":{"date-parts":[[2020,10,28]],"date-time":"2020-10-28T04:56:57Z","timestamp":1603861017000},"page":"748-768","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Freely adjoining monoidal duals"],"prefix":"10.1017","volume":"31","author":[{"given":"Kevin","family":"Coulembier","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2548-3005","authenticated-orcid":false,"given":"Ross","family":"Street","sequence":"additional","affiliation":[]},{"given":"Michel","family":"van den Bergh","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,10,28]]},"reference":[{"unstructured":"Delpeuch, A. (2019). Autonomization of monoidal categories, 25 pp. see arXiv:1411.3827v3.","key":"S0960129520000274_ref5"},{"doi-asserted-by":"publisher","key":"S0960129520000274_ref16","DOI":"10.1016\/0022-4049(92)00039-T"},{"key":"S0960129520000274_ref14","volume-title":"Lecture Notes in Mathematics","volume":"265","author":"Saavedra Rivano","year":"1972"},{"doi-asserted-by":"publisher","key":"S0960129520000274_ref9","DOI":"10.1090\/S0002-9947-1958-0131451-0"},{"key":"S0960129520000274_ref4","first-page":"733","article-title":"Note on Frobenius monoidal functors","volume":"14","author":"Day","year":"2008","journal-title":"The New York Journal of Mathematics"},{"key":"S0960129520000274_ref15","first-page":"81","article-title":"The free adjunction","volume":"27","author":"Schanuel","year":"1986","journal-title":"Cahiers de Topologie et G\u00e9om\u00e9trie Diff\u00e9rentielle Cat\u00e9goriques"},{"key":"S0960129520000274_ref17","first-page":"111","article-title":"Fibrations in bicategories","volume":"21","author":"Street","year":"1980","journal-title":"Cahiers de Topologie et G\u00e9om\u00e9trie Diff\u00e9rentielle"},{"doi-asserted-by":"publisher","key":"S0960129520000274_ref6","DOI":"10.1007\/978-3-642-99902-4_22"},{"unstructured":"Joyal, A. and Street, R. (1988). Planar diagrams and tensor algebra (handwritten notes); see http:\/\/web.science.mq.edu.au\/~street\/PlanarDiags.pdf.","key":"S0960129520000274_ref7"},{"key":"S0960129520000274_ref19","first-page":"251","article-title":"Relations between the \u2018percolation\u2019 and \u2018colouring\u2019 problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the \u2018percolation\u2019 problem","volume":"322","author":"Temperley","year":"1971","journal-title":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences"},{"key":"S0960129520000274_ref18","first-page":"699","article-title":"Braids among the groups","volume":"63","author":"Street","year":"1998","journal-title":"Seminarberichte aus dem Fachbereich Mathematik"},{"key":"S0960129520000274_ref13","volume-title":"Graduate Texts in Mathematics","volume":"5","author":"Lane","year":"1971"},{"key":"S0960129520000274_ref20","first-page":"705","volume-title":"Contemporary Mathematics","volume":"78","author":"Yetter","year":"1988"},{"doi-asserted-by":"publisher","key":"S0960129520000274_ref11","DOI":"10.1016\/j.aim.2014.03.003"},{"key":"S0960129520000274_ref10","first-page":"66","volume-title":"Lecture Notes in Mathematics","volume":"281","author":"Kelly","year":"1972"},{"doi-asserted-by":"publisher","key":"S0960129520000274_ref8","DOI":"10.1016\/0001-8708(91)90003-P"},{"key":"S0960129520000274_ref1","first-page":"3","article-title":"Adjonctions et monades au niveau des 2-cat\u00e9gories","volume":"15","author":"Auderset","year":"1974","journal-title":"Cahiers de Topologie et G\u00e9om\u00e9trie Diff\u00e9rentielle Cat\u00e9goriques"},{"doi-asserted-by":"publisher","key":"S0960129520000274_ref2","DOI":"10.1006\/aima.1998.1724"},{"unstructured":"B\u00e9nabou, J. (1973). Les distributeurs, Univ. Catholique de Louvain, S\u00e9minaires de Math. Pure, Rapport No. 33.","key":"S0960129520000274_ref3"},{"key":"S0960129520000274_ref12","first-page":"135","article-title":"Metric spaces, generalized logic and closed categories","volume":"53","author":"(Bill) Lawvere","year":"2002","journal-title":"Rendiconti del Seminario Matematico e Fisico di Milano"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129520000274","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,28]],"date-time":"2022-02-28T12:57:55Z","timestamp":1646053075000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129520000274\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,10,28]]},"references-count":20,"journal-issue":{"issue":"7","published-print":{"date-parts":[[2021,8]]}},"alternative-id":["S0960129520000274"],"URL":"https:\/\/doi.org\/10.1017\/s0960129520000274","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"type":"print","value":"0960-1295"},{"type":"electronic","value":"1469-8072"}],"subject":[],"published":{"date-parts":[[2020,10,28]]},"assertion":[{"value":"\u00a9 The Author(s), 2020. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}}]}}