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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Coalgebras, sheaves, and cohomology

Author: D. H. Van Osdol
Journal: Proc. Amer. Math. Soc. 33 (1972), 257-263
MSC: Primary 18C15
MathSciNet review: 0294447
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Abstract: The category of sheaves on the topological space X with values in the algebraic category $ \mathcal{A}$ is shown to be cotripleable under the stalk category $ {\mathcal{A}^{\vert X\vert}}$. If K is a field then the category of K-coalgebras is shown to be cotripleable under the category of K-vector spaces. This makes possible the interpretation of the first group of the associated triple cohomology complex. In particular, for coalgebras our $ {H^1}$ is isomorphic to Jonah's $ {H^2}$.

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  • [1] Jonathan Mock Beck, Triples, algebras and cohomology, Repr. Theory Appl. Categ. 2 (2003), 1–59. MR 1987896
  • [2] Armand Borel, Linear algebraic groups, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0251042
  • [3] Seminar on triples and categorical homology theory (ETH 1966/1967)., Edited by B. Eckmann. Lecture Notes in Mathematics, No. 80, Springer-Verlag, Berlin-New York, 1969. MR 0240157
  • [4] Roger Godement, Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). MR 0102797
  • [5] David W. Jonah, Cohomology of coalgebras, Memoirs of the American Mathematical Society, No. 82, American Mathematical Society, Providence, R.I., 1968. MR 0229693
  • [6] E. G. Manes, A triple miscellany: Some aspects of the theory of algebras over a triple, Dissertation, Wesleyan University, Middletown, Conn., 1967.
  • [7] Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR 0202787
  • [8] Robert Paré, Absolute coequalizers, Category Theory, Homology Theory and their Applications, I (Battelle Institute Conference, Seattle, Wash., 1968, Vol. One), Springer, Berlin, 1969, pp. 132–145. MR 0244343
  • [9] Bodo Pareigis, Categories and functors, Translated from the German. Pure and Applied Mathematics, Vol. 39, Academic Press, New York-London, 1970. MR 0265428
  • [10] Moss E. Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR 0252485
  • [11] D. H. Van Osdol, Sheaves in regular categories, Exact Categories and Categories of Sheaves, Lecture Notes in Math., no. 236, Springer-Verlag, Berlin and New York, 1971, pp. 223-239.

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Keywords: Sheaves of algebras, cotripleable, coalgebras, principal objects, cogroup object, extension, Jonah cohomology
Article copyright: © Copyright 1972 American Mathematical Society

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