The monads of classical algebra are seldom weakly cartesian

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• 作者：Maria Manuel Clementino (1)
  Dirk Hofmann (2)
  George Janelidze (3)
  
• 关键词：Monad ; Cartesian ; Weakly cartesian ; Weak pullback ; Variety of algebras ; Semiring ; Semimodule ; Subtractive ; 18C15 ; 18C20 ; 08A62 ; 16Y60
• 刊名：Journal of Homotopy and Related Structures
• 出版年：2014
• 出版时间：April 2014
• 年：2014
• 卷：9
• 期：1
• 页码：175-197
• 全文大小：276 KB
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• 作者单位：Maria Manuel Clementino (1)
  Dirk Hofmann (2)
  George Janelidze (3)
  
  1. Department of Mathematics, CMUC, University of Coimbra, 3001-501聽, Coimbra, Portugal
  2. Departamento de Matem谩tica, CIDMA, Universidade de Aveiro, 3810-193聽, Aveiro, Portugal
  3. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, Cape Town, South Africa
  
• ISSN：1512-2891

文摘

This paper begins a systematic study of weakly cartesian properties of monads that determine familiar varieties of universal algebras. While these properties clearly fail to hold for groups, rings, and many other related classical algebraic structures, their analysis becomes non-trivial in the case of semimodules over semirings, to which our main results are devoted. In particular necessary and sufficient conditions on a semiring S, under which the free semimodule monad has: (a) its underlying functor weakly cartesian, (b) its unit a weakly cartesian natural transformation, (c) its multiplication a weakly cartesian natural transformation, are obtained.