{"title":"单子的标准等级","authors":"Flavien Breuvart, Dylan McDermott, Tarmo Uustalu","doi":"10.4204/EPTCS.380.1","DOIUrl":null,"url":null,"abstract":"We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system), there is a canonical grading of T. Our application is to graded monads and models of computational effects. We demonstrate our results by characterizing the canonical gradings of a number of monads, for which C is endofunctors with composition. We also show that we can obtain canonical grades for algebraic operations.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"52 1","pages":"1-21"},"PeriodicalIF":0.0000,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Canonical Gradings of Monads\",\"authors\":\"Flavien Breuvart, Dylan McDermott, Tarmo Uustalu\",\"doi\":\"10.4204/EPTCS.380.1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system), there is a canonical grading of T. Our application is to graded monads and models of computational effects. We demonstrate our results by characterizing the canonical gradings of a number of monads, for which C is endofunctors with composition. We also show that we can obtain canonical grades for algebraic operations.\",\"PeriodicalId\":11810,\"journal\":{\"name\":\"essentia law Merchant Shipping Act 1995\",\"volume\":\"52 1\",\"pages\":\"1-21\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-07-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"essentia law Merchant Shipping Act 1995\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4204/EPTCS.380.1\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"essentia law Merchant Shipping Act 1995","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4204/EPTCS.380.1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We define a notion of grading of a monoid T in a monoidal category C, relative to a class of morphisms M (which provide a notion of M-subobject). We show that, under reasonable conditions (including that M forms a factorization system), there is a canonical grading of T. Our application is to graded monads and models of computational effects. We demonstrate our results by characterizing the canonical gradings of a number of monads, for which C is endofunctors with composition. We also show that we can obtain canonical grades for algebraic operations.
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