{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T09:58:52Z","timestamp":1747216732050,"version":"3.40.5"},"reference-count":24,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","license":[{"start":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T00:00:00Z","timestamp":1698624000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Compositionality"],"abstract":"<jats:p>A traced monad is a monad on a traced symmetric monoidal category that lifts the traced symmetric monoidal structure to its Eilenberg-Moore category. A long-standing question has been to provide a characterization of traced monads without explicitly mentioning the Eilenberg-Moore category. On the other hand, a symmetric Hopf monad is a symmetric bimonad whose fusion operators are invertible. For compact closed categories, symmetric Hopf monads are precisely the kind of monads that lift the compact closed structure to their Eilenberg-Moore categories. Since compact closed categories and traced symmetric monoidal categories are closely related, it is a natural question to ask what is the relationship between Hopf monads and traced monads. In this paper, we introduce trace-coherent Hopf monads on traced monoidal categories, which can be characterized without mentioning the Eilenberg-Moore category. The main theorem of this paper is that a symmetric Hopf monad is a traced monad if and only if it is a trace-coherent Hopf monad. We provide many examples of trace-coherent Hopf monads, such as those induced by cocommutative Hopf algebras or any symmetric Hopf monad on a compact closed category. We also explain how for traced Cartesian monoidal categories, trace-coherent Hopf monads can be expressed using the Conway operator, while for traced coCartesian monoidal categories, any trace-coherent Hopf monad is an idempotent monad. We also provide separating examples of traced monads that are not Hopf monads, as well as symmetric Hopf monads that are not trace-coherent.<\/jats:p>","DOI":"10.32408\/compositionality-5-10","type":"journal-article","created":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T14:56:54Z","timestamp":1698677814000},"page":"10","source":"Crossref","is-referenced-by-count":1,"title":["Traced Monads and Hopf Monads"],"prefix":"10.46298","volume":"5","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3460-8615","authenticated-orcid":false,"given":"Masahito","family":"Hasegawa","sequence":"first","affiliation":[{"name":"Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4124-3722","authenticated-orcid":false,"given":"Jean-Simon Pacaud","family":"Lemay","sequence":"additional","affiliation":[{"name":"School of Mathematical and Physical Sciences, Macquarie University, Sydney, New South Wales, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2023,10,30]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"S. 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