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The cohomology groups of an epimorphism

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1348)

Abstract

If p:S→T is a U-split epimorphism in a monadic category ℂ (such as groups or algebras) and ℂ denotes the induced cotriple on ℂ then the cohomology groups H n (p,A) of the epimorphism with coefficients in a abelian group object A in ℂ have been defined by van Osdol ([18]) and interpreted in dimension 1 by Aznar and Cegarra ([1]), as isomorphism classes of 2-torsors which have p as their augmentation. In this paper it is shown that these groups are themselves “cotriple cohomology groups” for the cotriple on the category of simplicial objects of ℂ induced by ℂ and applied to the complex COSK o(p) with coefficients in the abelian group object K(A,1). Van Osdol's long exact sequence in the first variable associated with p is shown to be isomorphic to the standard second variable cotriple cohomology sequence, provided by the short exact sequence 0→K(A,0)→L(A,0)→K(A,1)→0 in this category. The interpretation of H 1(p,A) in terms of 2-torsors is shown to be a consequence of the standard interpretation of H 1 as isomorphism classes of 1-torsors, combined with the properties of the functor .

Keywords

  • Exact Sequence
  • Isomorphism Class
  • Cohomology Group
  • Short Exact Sequence
  • Homotopy Class

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper has been partially supported by a grant from CAICYT Proyecto de Investigación 3556-83C2-00

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© 1988 Springer-Verlag

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Lirola, A., Aznar, E.R., Bullejos, M. (1988). The cohomology groups of an epimorphism. In: Borceux, F. (eds) Categorical Algebra and its Applications. Lecture Notes in Mathematics, vol 1348. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0081362

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  • DOI: https://doi.org/10.1007/BFb0081362

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  • Print ISBN: 978-3-540-50362-0

  • Online ISBN: 978-3-540-45985-9

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