Abstract homotopy theory and generalized sheaf cohomology

Brown, Kenneth S.

doi:10.1090/S0002-9947-1973-0341469-9

Abstract homotopy theory and generalized sheaf cohomology
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by Kenneth S. Brown PDF

Trans. Amer. Math. Soc. 186 (1973), 419-458 Request permission

Abstract:

Cohomology groups ${H^q}(X,E)$ are defined, where X is a topological space and E is a sheaf on X with values in Kan’s category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defined by means of the “homotopical algebra” of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen’s, and this is developed in Part I of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized cohomology theory) and the “local to global” spectral sequence (in sheaf cohomology theory).

References

Grothendieck topologies

Séminaire de géométrie algébrique

Cohomologie étale des schémas

M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, No. 100, Springer-Verlag, Berlin-New York, 1969. MR 0245577
A. K. Bousfield and D. M. Kan, Homotopy with respect to a ring, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 59–64. MR 0326734
Glen E. Bredon, Sheaf theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR 0221500
Dan Burghelea and Aristide Deleanu, The homotopy category of spectra. I, Illinois J. Math. 11 (1967), 454–473. MR 219063
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Albrecht Dold, Halbexakte Homotopiefunktoren, Lecture Notes in Mathematics, vol. 12, Springer-Verlag, Berlin-New York, 1966 (German). MR 0198464
Albrecht Dold and Dieter Puppe, Homologie nicht-additiver Funktoren. Anwendungen, Ann. Inst. Fourier (Grenoble) 11 (1961), 201–312 (German, with French summary). MR 150183

La suite spectrale de Adams

Structure multiplicative

11

P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125
Roger Godement, Topologie algébrique et théorie des faisceaux, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1252, Hermann, Paris, 1958 (French). Publ. Math. Univ. Strasbourg. No. 13. MR 0102797
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093
Daniel M. Kan, Semisimplicial spectra, Illinois J. Math. 7 (1963), 463–478. MR 153017
Daniel M. Kan, On the $k$-cochains of a spectrum, Illinois J. Math. 7 (1963), 479–491. MR 155319
Daniel M. Kan and George W. Whitehead, The reduced join of two spectra, Topology 3 (1965), no. suppl, suppl. 2, 239–261. MR 178463, DOI 10.1016/0040-9383(65)90077-7
Daniel M. Kan and George W. Whitehead, Orientability and Poincaré duality in general homology theories, Topology 3 (1965), 231–270. MR 190925, DOI 10.1016/0040-9383(65)90056-X
Klaus Lamotke, Semisimpliziale algebraische Topologie, Die Grundlehren der mathematischen Wissenschaften, Band 147, Springer-Verlag, Berlin-New York, 1968 (German). MR 0245005
J. Peter May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0222892
J. Milnor, On axiomatic homology theory, Pacific J. Math. 12 (1962), 337–341. MR 159327
Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432
George W. Whitehead, Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962), 227–283. MR 137117, DOI 10.1090/S0002-9947-1962-0137117-6