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
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This paper has been partially supported by a grant from CAICYT Proyecto de Investigación 3556-83C2-00
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References
Aznar,E.R. Cohomologia no abeliana en categorias de interes. Alxebra 33 (1981)
Barr,M.-Beck,J. Homology and standard constructions. Lec. Not. in Math. 80. Springer (1969)
Beck, J. Triples, Algebras and Cohomology. Dissert. Columbia (1967)
Bullejos, M. Cohomologia no abeliana (la sucesion exacta larga). Cuadernos de Algebra. Granada (1987)
Carrasco, P. Cohomologia de haces. Cuadernos de Algebra. Granada (1987)
Cegarra,A.M. Cohomologia Varietal. Alxebra (1980)
Cegarra, A.M.-Aznar, E.R. An exact sequence in the first variable for Torsor Cohomology: the 2-dimensional theory of obstructions. J.P. and Appl. Algebra 39, 197–250 (1986)
Duskin,J. Simplicial methods and the interpretation of triple cohomology. Mem. A.M.S. (2), 163 (1975)
Eilenberg,S.-Moore,J.C. Adjoint functors and triples. Ill. J. Math. 9 (1965)
Glenn, P. Realization of cohomology classes in arbitrary exact categories. J. P. Appl. Algebra 25 (1), 33–107 (1982)
Herrlich,H.-Strecker,G. Category Theory. Allyn&Bacon (1973)
Higgins,P.J. Categories and Groupoids. Van Nostrand Reinhred. Math. Studies 32 (1971)
Lirola,A. Cohomologia de torsores relativos. Mem. Lic. U. Granada (1982)
Loday, J.L. Cohomologie et groupe de Steinberg relatifs. J. Algebra 54, 178–202 (1978)
May,J.P. Simplicial objects in Algebraic Topology. Van Nostrand (1967)
R-Grandjean,A. Homologia en categorias exactas. Alxebra 4 (1970)
Rinehart, G.S. Satellites and Cohomology. J. Algebra 12, 295–329 (1969)
Van Osdol, D.H. Long exact sequences in the first variable for algebraic cohomology theories. J. P. Appl. Algebra 23 (3), 271–309 (1982)
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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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