{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T03:06:20Z","timestamp":1767927980881,"version":"3.49.0"},"reference-count":35,"publisher":"Cambridge University Press (CUP)","issue":"10","license":[{"start":{"date-parts":[[2023,2,20]],"date-time":"2023-02-20T00:00:00Z","timestamp":1676851200000},"content-version":"unspecified","delay-in-days":111,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here, we define a related structure called a <jats:italic>monoidal reverse differential category<\/jats:italic>, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of quantum computation.<\/jats:p>","DOI":"10.1017\/s096012952200038x","type":"journal-article","created":{"date-parts":[[2023,2,20]],"date-time":"2023-02-20T12:35:59Z","timestamp":1676896559000},"page":"1313-1363","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":3,"title":["Monoidal reverse differential categories"],"prefix":"10.1017","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8742-6263","authenticated-orcid":false,"given":"Geoff","family":"Cruttwell","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan","family":"Gallagher","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Simon Pacaud","family":"Lemay","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dorette","family":"Pronk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2023,2,20]]},"reference":[{"key":"S096012952200038X_ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2007.05.021"},{"key":"S096012952200038X_ref34","doi-asserted-by":"publisher","DOI":"10.1145\/3527634"},{"key":"S096012952200038X_ref10","doi-asserted-by":"crossref","unstructured":"Cruttwell, G. , Gavranovi\u0107, B. , Ghani, N. , Wilson, P. and Zanasi, F. (2022). Categorical foundations of gradient-based learning. In: Proceedings of the 31st European Symposium on Programming.","DOI":"10.1007\/978-3-030-99336-8_1"},{"key":"S096012952200038X_ref16","doi-asserted-by":"crossref","unstructured":"Hadzihasanovic, A. (2015). A diagrammatic axiomatisation for qubit entanglement. In: 2015 30th Annual ACM\/IEEE Symposium on Logic in Computer Science, 573\u2013584.","DOI":"10.1109\/LICS.2015.59"},{"key":"S096012952200038X_ref24","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129520000158"},{"key":"S096012952200038X_ref33","first-page":"1","article-title":"Dagger linear logic for categorical quantum mechanics","volume":"17","author":"Srinivasan","year":"2021","journal-title":"Logical Methods in Computer Science"},{"key":"S096012952200038X_ref28","doi-asserted-by":"publisher","DOI":"10.1109\/LICS.2017.8005064"},{"key":"S096012952200038X_ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s10485-021-09649-7"},{"key":"S096012952200038X_ref8","first-page":"374","article-title":"Linearizing combinators","volume":"38","author":"Cockett","year":"2022","journal-title":"Theory and Applications of Categories"},{"key":"S096012952200038X_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2019.02.014"},{"key":"S096012952200038X_ref35","doi-asserted-by":"crossref","first-page":"3408","DOI":"10.1007\/s10773-008-9772-4","article-title":"A categorical framework for the quantum harmonic oscillator","volume":"47","author":"Vicary","year":"2008","journal-title":"International Journal of Theoretical Physics"},{"key":"S096012952200038X_ref11","first-page":"116","article-title":"Integral and differential structure on the free \n\n\n\n${C^\\infty}$\n\n\n-ring modality","volume":"62","author":"Cruttwell","year":"2021","journal-title":"Cahiers de Topologie et G\u00e9om\u00e9trie Diff\u00e9rentielle Cat\u00e9goriques"},{"key":"S096012952200038X_ref4","first-page":"620","article-title":"Cartesian differential storage categories","volume":"30","author":"Blute","year":"2015","journal-title":"Theory and Applications of Categories"},{"key":"S096012952200038X_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s10485-013-9312-0"},{"key":"S096012952200038X_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s10485-019-09572-y"},{"key":"S096012952200038X_ref7","first-page":"1","article-title":"Integral categories and calculus categories","volume":"29","author":"Cockett","year":"2018","journal-title":"Mathematical Structures in Computer Science"},{"key":"S096012952200038X_ref9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-70583-3_25"},{"key":"S096012952200038X_ref26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-9839-7"},{"key":"S096012952200038X_ref12","unstructured":"Duncan, R. (2006). Types for Quantum Computing. Phd thesis, University of Oxford."},{"key":"S096012952200038X_ref17","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198739623.001.0001"},{"key":"S096012952200038X_ref22","doi-asserted-by":"publisher","DOI":"10.1109\/LICS.2013.36"},{"key":"S096012952200038X_ref29","doi-asserted-by":"publisher","DOI":"10.1145\/2535838.2535879"},{"key":"S096012952200038X_ref13","doi-asserted-by":"publisher","DOI":"10.1109\/LICS52264.2021.9470664"},{"key":"S096012952200038X_ref30","volume-title":"7th workshop on Quantum Physics and Logic (QPL 2010)","author":"Selinger","year":"2010"},{"key":"S096012952200038X_ref27","first-page":"1159","article-title":"Monoidal Grothendieck construction","volume":"35","author":"Moeller","year":"2020","journal-title":"Theory and Applications of Categories"},{"key":"S096012952200038X_ref6","first-page":"1","article-title":"Reverse derivative categories","volume":"152","author":"Cockett","year":"2020","journal-title":"LIPIcs"},{"key":"S096012952200038X_ref3","first-page":"622","article-title":"Cartesian differential categories","volume":"22","author":"Blute","year":"2009","journal-title":"Theory and Applications of Categories"},{"key":"S096012952200038X_ref23","unstructured":"Lemay, J.-S. P. (2019). Why fhilb is not an interesting (co) differential category. In: In the Proceedings of Quantum Programming and Logic (QPL2019)."},{"key":"S096012952200038X_ref32","first-page":"650","article-title":"Framed bicategories and monoidal fibrations","volume":"20","author":"Shulman","year":"2008","journal-title":"Theory and Applications of Categories"},{"key":"S096012952200038X_ref20","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(01)00241-9"},{"key":"S096012952200038X_ref2","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129506005676"},{"key":"S096012952200038X_ref19","doi-asserted-by":"crossref","unstructured":"Hyland, M. and Schalk, A. (1999). Abstract games for linear logic. Electronic Notes in Theoretical Computer Science 29 127\u2013150. CTCS\u201999, Conference on Category Theory and Computer Science.","DOI":"10.1016\/S1571-0661(05)80312-3"},{"key":"S096012952200038X_ref31","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2008.04.022"},{"key":"S096012952200038X_ref21","unstructured":"Jacobs, B. (1999). Categorical Logic and Type Theory, Elsevier."},{"key":"S096012952200038X_ref25","unstructured":"Lemay, J.-S. P. (2021). Coderelictions for free exponential modalities. In: Gadducci, F. and Silva, A. (eds.) 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021), Leibniz International Proceedings in Informatics (LIPIcs), vol. 211, Dagstuhl, Germany, Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik, 19:1\u201319:21."}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S096012952200038X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,12]],"date-time":"2023-04-12T09:40:56Z","timestamp":1681292456000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S096012952200038X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11]]},"references-count":35,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2022,11]]}},"alternative-id":["S096012952200038X"],"URL":"https:\/\/doi.org\/10.1017\/s096012952200038x","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"value":"0960-1295","type":"print"},{"value":"1469-8072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11]]},"assertion":[{"value":"\u00a9 The Author(s), 2023. Published by Cambridge University Press","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}