{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:22:38Z","timestamp":1760242958207,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2015,1,19]],"date-time":"2015-01-19T00:00:00Z","timestamp":1421625600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001663","name":"Volkswagen Foundation","doi-asserted-by":"publisher","award":["I\/85989"],"award-info":[{"award-number":["I\/85989"]}],"id":[{"id":"10.13039\/501100001663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004801","name":"Shota Rustaveli National Science Foundation","doi-asserted-by":"publisher","award":["DI\/12\/5-103\/11"],"award-info":[{"award-number":["DI\/12\/5-103\/11"]}],"id":[{"id":"10.13039\/501100004801","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The definition of Azumaya algebras over commutative rings \\(R\\) requires the tensor product of modules over \\(R\\) and the twist map for the tensor product of any two \\(R\\)-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category \\(\\mathbb{A}\\) by considering a monad \\((F,m,e)\\) on \\(\\mathbb{A}\\) endowed with a distributive law \\(\\lambda: FF\\to FF\\) satisfying the Yang\u2013Baxter equation (BD%please define -law). This allows to introduce an opposite monad \\((F^\\lambda,m\\cdot \\lambda,e)\\) and a monad structure on \\(FF^\\lambda\\). The quadruple \\((F,m,e,\\lambda)\\) is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category \\(\\mathbb{A}\\) and the category of \\(FF^\\lambda\\)-modules. Properties and characterizations of these monads are studied, in particular for the case when \\(F\\) allows for a right adjoint functor. Dual to Azumaya monads, we define Azumaya comonads and investigate the interplay between these notions. In braided categories (V\\(,\\otimes,I,\\tau)\\), for any V-algebra \\(A\\), the braiding induces a BD-law \\(\\tau_{A,A}:A\\otimes A\\to A\\otimes A\\), and \\(A\\) is called left (right) Azumaya, provided the monad \\(A\\otimes-\\) (resp. \\(-\\otimes A\\)) is Azumaya. If \\(\\tau\\) is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide.<\/jats:p>","DOI":"10.3390\/axioms4010032","type":"journal-article","created":{"date-parts":[[2015,1,19]],"date-time":"2015-01-19T10:53:54Z","timestamp":1421664834000},"page":"32-70","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Azumaya Monads and Comonads"],"prefix":"10.3390","volume":"4","author":[{"given":"Bachuki","family":"Mesablishvili","sequence":"first","affiliation":[{"name":"Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 6, Tamarashvili Str., Tbilisi Centre for Mathematical Sciences, Chavchavadze Ave. 75, 3\/35, Tbilisi 0177, Georgia"}]},{"given":"Robert","family":"Wisbauer","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Heinrich Heine University, D\u00fcsseldorf 40225, Germany"}]}],"member":"1968","published-online":{"date-parts":[[2015,1,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1090\/S0002-9939-1975-0393195-5","article-title":"The Brauer group of a closed category","volume":"50","year":"1975","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1007\/BFb0077339","article-title":"Non-additive ring and module theory IV. The Brauer group of a symmetric monoidal category","volume":"549","author":"Pareigis","year":"1976","journal-title":"Lect. 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