{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T07:06:26Z","timestamp":1761980786421,"version":"build-2065373602"},"reference-count":25,"publisher":"Cambridge University Press (CUP)","issue":"6","license":[{"start":{"date-parts":[[2014,3,11]],"date-time":"2014-03-11T00:00:00Z","timestamp":1394496000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2014,12]]},"abstract":"<jats:p>In this paper we characterise the categories of Lawvere theories and equational theories that correspond to the categories of analytic and polynomial monads on<jats:italic>Set<\/jats:italic>, and hence also to the categories of the symmetric and rigid operads in<jats:italic>Set<\/jats:italic>. We show that the category of analytic monads is equivalent to the category of regular-linear theories. The category of polynomial monads is equivalent to the category of rigid theories, that is, regular-linear theories satisfying an additional global condition. This solves a problem posed by A. Carboni and P. T. Johnstone. The Lawvere theories corresponding to these monads are identified<jats:italic>via<\/jats:italic>some factorisation systems. We also show that the categories of analytic monads and finitary endofunctors on<jats:italic>Set<\/jats:italic>are monadic over the category of analytic functors. The corresponding monad for analytic monads distributes over the monad for finitary endofunctors and hence the category of (finitary) monads on<jats:italic>Set<\/jats:italic>is monadic over the category of analytic functors. This extends a result of M. Barr.<\/jats:p>","DOI":"10.1017\/s0960129513000868","type":"journal-article","created":{"date-parts":[[2014,3,11]],"date-time":"2014-03-11T14:12:18Z","timestamp":1394547138000},"source":"Crossref","is-referenced-by-count":13,"title":["Theories of analytic monads"],"prefix":"10.1017","volume":"24","author":[{"given":"STANIS\u0141AW","family":"SZAWIEL","sequence":"first","affiliation":[]},{"given":"MAREK","family":"ZAWADOWSKI","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,11]]},"reference":[{"key":"S0960129513000868_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(72)90001-1"},{"key":"S0960129513000868_ref23","doi-asserted-by":"publisher","DOI":"10.1023\/A:1016340806503"},{"key":"S0960129513000868_ref19","unstructured":"Szawiel S. and Zawadowski M. (2010) Representing multicategories as monads. Talk at CT 2010, Genova. 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(2013) Weights for Objects of Monoids. arXiv:1306.3215 [math.CT]."},{"key":"S0960129513000868_ref4","first-page":"215","article-title":"T-categories (cat\u00e9gories dans un triple).","volume":"12","author":"Burroni","year":"1971","journal-title":"Cahiers de Topologie et G\u00e9om\u00e9trie Diff\u00e9rentielle Cat\u00e9goriques"},{"key":"S0960129513000868_ref13","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(01)00014-7"},{"key":"S0960129513000868_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0083084"},{"key":"S0960129513000868_ref18","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525896"},{"key":"S0960129513000868_ref11","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-4049(99)00179-6"},{"key":"S0960129513000868_ref17","doi-asserted-by":"crossref","unstructured":"Lawvere F. W. (1963) Functorial Semantics of Algebraic Theories and Some Algebraic Problems in the context of Functorial Semantics of Algebraic Theories, Ph.D. thesis, Columbia University. (Reprinted in Theory and Applications of Categories (2004) 5 1\u2013121.)","DOI":"10.1073\/pnas.50.5.869"},{"key":"S0960129513000868_ref10","unstructured":"Gould M. R. (2010) Coherence for Categorified Operadic Theories. arXiv:1002.0879v1 [math.CT]."},{"key":"S0960129513000868_ref2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01111838"},{"key":"S0960129513000868_ref5","doi-asserted-by":"crossref","unstructured":"Boja\u0144czyk M. , Szawiel S. and Zawadowski M. (2014) Rigidity is undecidable. 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