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Transactions of the American Mathematical Society

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Bicohomology theory


Author: Donovan H. Van Osdol
Journal: Trans. Amer. Math. Soc. 183 (1973), 449-476
MSC: Primary 18H15
DOI: https://doi.org/10.1090/S0002-9947-1973-0323873-8
MathSciNet review: 0323873
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Abstract: Given a triple T and a cotriple G on a category $ \mathcal{D}$ such that T preserves group objects in $ \mathcal{D}$, let P and M be in $ \mathcal{D}$ with M an abelian group object. Applying the ``hom functor'' $ \mathcal{D}( - , - )$ to the (co)simplicial resolutions $ {G^ \ast }P$ and $ {T^ \ast }M$ yields a double complex $ \mathcal{D}({G^ \ast }P,{T^ \ast }M)$. The nth homology group of this double complex is denoted $ {H^n}(P,M)$, and this paper studies $ {H^0}$ and $ {H^1}$. When $ \mathcal{D}$ is the category of bialgebras arising from a triple, cotriple, and mixed distributive law, a complete description of $ {H^0}$ and $ {H^1}$ is given. The applications include a solution of the singular extension problem for sheaves of algebras.


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  • [1] M. Barr, Cohomology in tensored categories, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 344-354. MR 35 #235. MR 0209337 (35:235)
  • [2] -, Cohomology and obstructions: Commutative algebras, Sem. Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, Berlin, 1969, p. 357. MR 42 #6075. MR 0271192 (42:6075)
  • [2a] M. Barr, Exact categories, Lecture Notes in Math., vol. 236, Springer-Verlag, Berlin and New York, 1971.
  • [3] -, Harrison cohomology, Hochschild cohomology, and triples, J. Algebra 8 (1968), 314-323. MR 36 #3851. MR 0220799 (36:3851)
  • [4] -, Shukla cohomology and triples, J. Algebra 5 (1967), 222-231. MR 36 #1510.
  • [5] M. Barr and J. Beck, Acyclic models and triples, Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965), Springer, New York, 1966, pp. 336-343. MR 39 #6955. MR 0245649 (39:6955)
  • [6] -, Homology and standard constructions, Sem. Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, Berlin, 1969, pp. 245-335. MR 41 #3562. MR 0258917 (41:3562)
  • [7] M. Barr and G. Rinehart, Cohomology as the derived functor of derivations, Trans. Amer. Math. Soc. 122 (1966), 416-426. MR 33 #159. MR 0191932 (33:159)
  • [8] J. Beck, Distributive laws, Sem. Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, Berlin, 1969, pp. 119-140. MR 39 #2842. MR 0241502 (39:2842)
  • [9] -, Triples, algebras, and cohomology, Dissertation, Columbia University, New York, 1967 (submitted).
  • [10] H. Cartan and S. Eilenberg, Homological algebra, Princeton Univ. Press, Princeton, N. J., 1956. MR 17, 1040. MR 0077480 (17:1040e)
  • [11] B. Eckmann and P. Hilton, Group-like structures in general categories. I. Multiplications and comultiplications, Math. Ann. 145 (1961/62), 227-255. MR 25 #108. MR 0136642 (25:108)
  • [12] S. Eilenberg and S. Mac Lane, Cohomology theory in abstract groups. I, Ann. of Math. (2) 48 (1947), 51-78. MR 8, 367.
  • [13] S. Eilenberg and J. Moore, Adjoint functors and triples, Illinois J. Math. 9 (1965), 381-398. MR 32 #2455. MR 0184984 (32:2455)
  • [14] R. Godement, Topologie algébrique et théorie des faisceaux, Actualités Sci. Indust., no. 1252, Hermann, Paris, 1958. MR 21 #1583. MR 0102797 (21:1583)
  • [15] J. Gray, Extensions of sheaves of algebras, Illinois J. Math. 5 (1961), 159-174. MR 23 #A186. MR 0122854 (23:A186)
  • [16] P. Grillet, Regular categories, Lecture Notes in Math., vol. 236, Springer-Verlag, Berlin and New York, 1971.
  • [17] A Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. (2) 9 (1957), 119-221. MR 21 #1328. MR 0102537 (21:1328)
  • [18] G. Hochschild, On the cohomology groups of an associative algebra, Ann. of Math. (2) 46 (1945), 58-67. MR 6, 114. MR 0011076 (6:114f)
  • [19] D. Kan, Adjoint functors, Trans. Amer. Math. Soc. 87 (1958), 294-329. MR 24 #A1301. MR 0131451 (24:A1301)
  • [20] S. Mac Lane, Homology, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 28 #122. MR 0349792 (50:2285)
  • [21] E. Manes, A triple theoretic construction of compact algebras, Sem. Triples and Categorical Homology Theory (ETH, Zürich, 1966/67), Springer, Berlin, 1969, pp. 91-118. MR 39 #5657. MR 0244342 (39:5657)
  • [22] B. Mitchell, Theory of categories, Pure and Appl. Math., vol. 17, Academic Press, New York, 1965. MR 34 #2647. MR 0202787 (34:2647)
  • [23] B. Pareigis, Kategorien und Funktoren, Mathematische Leitfäden, Teubner, Stuttgart, 1969; English transl., Pure and Appl. Math., vol. 39, Academic Press, New York, 1970. MR 42 #337a, b. MR 0265427 (42:337a)
  • [25] U. Shukla, Cohomologie des algèbres associatives, Ann. Sci. École Norm. Sup. (3) 78 (1961), 163-209. MR 24 #A2605. MR 0132769 (24:A2605)
  • [26] M. Sweedler, Hopf algebras, Math. Lecture Note Series, Benjamin, New York, 1969. MR 40 #5705. MR 0252485 (40:5705)
  • [27] D. Van Osdol, Remarks concerning $ {\operatorname{Ext}^ \ast }(M, - )$, Bull. Amer. Math. Soc. 76 (1970), 612-617. MR 42 #6081. MR 0271198 (42:6081)
  • [28] -, Sheaves in regular categories, Lecture Notes in Math., vol. 236, Springer-Verlag, Berlin and New York, 1971.
  • [29] -, Some applications of triples to sheaf theory, Dissertation, University of Illinois, Chicago, Ill., 1969 (submitted).
  • [30] H. Wolff, V-localizations and V-triples, Dissertation, University of Illinois, Chicago, Ill., 1970 (submitted).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0323873-8
Keywords: Triple, principal homogeneous object, cotriplable, distributive law, double complex, homology, algebras, sheaves
Article copyright: © Copyright 1973 American Mathematical Society

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