{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T00:35:08Z","timestamp":1767918908460,"version":"3.49.0"},"reference-count":18,"publisher":"Association for Computing Machinery (ACM)","issue":"POPL","content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["Proc. ACM Program. Lang."],"published-print":{"date-parts":[[2026,1,8]]},"abstract":"<jats:p>In many situations one encounters an entity that resembles a monoid. It consists of a carrier and two operations that resemble a unit and a multiplication, subject to three equations that resemble associativity and left and right unital laws. The question then arises whether this entity is, in fact, a monoid in a suitable sense.<\/jats:p>\n                  <jats:p>Category theorists have answered this question by providing a notion of monoid in a monoidal category, or more generally in a multicategory. While these encompass many examples, there remain cases which do not fit into these frameworks, such as the notion of relative monad and the modelling of call-by-push-value sequencing. In each of these examples, the leftmost and\/or the rightmost factor of a multiplication or associativity law seems to be distinguished.<\/jats:p>\n                  <jats:p>To include such examples, we generalize the multicategorical framework in two stages.<\/jats:p>\n                  <jats:p>Firstly, we move to the framework of a left-skew multicategory (due to Bourke and Lack), which generalizes both multicategory and left-skew monoidal category. The notion of monoid in this framework encompasses examples where only the leftmost factor is distinguished, such as the notion of relative monad.<\/jats:p>\n                  <jats:p>Secondly, we consider monoids in the novel framework of a bi-skew multicategory. This encompasses examples where both the leftmost and the rightmost factor are distinguished, such as the notion of a category on a span, and the modelling of call-by-push-value sequencing.<\/jats:p>\n                  <jats:p>In the bi-skew framework (which is the most general), we give a coherence result saying that a monoid corresponds to an unbiased monoid, i.e. a map from the terminal bi-skew multicategory.<\/jats:p>","DOI":"10.1145\/3776727","type":"journal-article","created":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T18:59:43Z","timestamp":1767898783000},"page":"2468-2493","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["What Is a Monoid?"],"prefix":"10.1145","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0864-1876","authenticated-orcid":false,"given":"Paul Blain","family":"Levy","sequence":"first","affiliation":[{"name":"University of Birmingham, Birmingham, United Kingdom"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0277-8217","authenticated-orcid":false,"given":"Morgan","family":"Rogers","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Sorbonne Paris Nord, Villetaneuse, France"}]}],"member":"320","published-online":{"date-parts":[[2026,1,8]]},"reference":[{"key":"e_1_2_1_1_1","unstructured":"J. Ad\u00e1mek H. Herrlich and G. Strecker. 1990. Abstract and Concrete Categories\u2014The Joy of Cats. Wiley."},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129514000103"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.2168\/LMCS-11(1:3)2015"},{"key":"e_1_2_1_4_1","unstructured":"Nathanael Arkor and Dylan McDermott. [n. d.]. The formal theory of relative monads. arxiv:2302.14014 Updated version of Arkor-McDermot"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2024.107676"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2012.02.039"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2018.02.039"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1006\/aima.1999.1877"},{"key":"e_1_2_1_9_1","doi-asserted-by":"crossref","first-page":"985","DOI":"10.70930\/tac\/ekkdj1hd","article-title":"Skew-Monoidal Reflection And Lifting Theorems","volume":"30","author":"Lack Stephen","year":"2015","unstructured":"Stephen Lack and Ross Street. 2015. Skew-Monoidal Reflection And Lifting Theorems. Theory and Applications of Categories, 30, 28 (2015), 985\u20131000. http:\/\/www.tac.mta.ca\/tac\/volumes\/30\/28\/30-28abs.html","journal-title":"Theory and Applications of Categories"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(01)00105-0"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.48550\/arXiv.math\/0305049"},{"key":"e_1_2_1_12_1","doi-asserted-by":"crossref","unstructured":"P. B. Levy. 2004. Call-By-Push-Value. A Functional\/Imperative Synthesis. Springer.","DOI":"10.1007\/978-94-007-0954-6"},{"key":"e_1_2_1_13_1","unstructured":"Will Offord. 2024. Skew-Monoidal Categories: Logical and Graphical Calculi. https:\/\/golem.ph.utexas.edu\/category\/2024\/07\/skewmonoidal_categories_logica.html posted at the n-Category Caf\u00e9"},{"key":"e_1_2_1_14_1","doi-asserted-by":"crossref","first-page":"496","DOI":"10.70930\/tac\/0w1wkwjx","article-title":"Sequential Multicategories","volume":"29","author":"Pisani Claudio","year":"2014","unstructured":"Claudio Pisani. 2014. Sequential Multicategories. Theory and Applications of Categories, 29, 19 (2014), 496\u2013541. http:\/\/www.tac.mta.ca\/tac\/volumes\/29\/19\/29-19abs.html","journal-title":"Theory and Applications of Categories"},{"key":"e_1_2_1_15_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0956796809007321"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2012.06.027"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2018.11.017"},{"key":"e_1_2_1_18_1","volume-title":"The Free Encyclopedia. https:\/\/en.wikipedia.org\/w\/index.php?title=Extranatural_transformation&oldid=1147520438 accessed 10-July-2025","author":"Wikipedia","unstructured":"Wikipedia contributors. 2023. Extranatural transformation \u2014 Wikipedia, The Free Encyclopedia. https:\/\/en.wikipedia.org\/w\/index.php?title=Extranatural_transformation&oldid=1147520438 accessed 10-July-2025"}],"container-title":["Proceedings of the ACM on Programming Languages"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3776727","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T19:02:11Z","timestamp":1767898931000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3776727"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,8]]},"references-count":18,"journal-issue":{"issue":"POPL","published-print":{"date-parts":[[2026,1,8]]}},"alternative-id":["10.1145\/3776727"],"URL":"https:\/\/doi.org\/10.1145\/3776727","relation":{},"ISSN":["2475-1421"],"issn-type":[{"value":"2475-1421","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,8]]},"assertion":[{"value":"2025-07-10","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-11-06","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-01-08","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}