If you need an accessible version of this item please contact JSTOR User Support

Abstract Homotopy Theory and Generalized Sheaf Cohomology

Kenneth S. Brown
Transactions of the American Mathematical Society
Vol. 186 (Dec., 1973), pp. 419-458
DOI: 10.2307/1996573
Stable URL: http://www.jstor.org/stable/1996573
Page Count: 40
  • Download PDF
  • Cite this Item

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support
Abstract Homotopy Theory and Generalized Sheaf Cohomology
Preview not available

Abstract

Cohomology groups Hq(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).

Page Thumbnails

  • Thumbnail: Page 
419
    419
  • Thumbnail: Page 
420
    420
  • Thumbnail: Page 
421
    421
  • Thumbnail: Page 
422
    422
  • Thumbnail: Page 
423
    423
  • Thumbnail: Page 
424
    424
  • Thumbnail: Page 
425
    425
  • Thumbnail: Page 
426
    426
  • Thumbnail: Page 
427
    427
  • Thumbnail: Page 
428
    428
  • Thumbnail: Page 
429
    429
  • Thumbnail: Page 
430
    430
  • Thumbnail: Page 
431
    431
  • Thumbnail: Page 
432
    432
  • Thumbnail: Page 
433
    433
  • Thumbnail: Page 
434
    434
  • Thumbnail: Page 
435
    435
  • Thumbnail: Page 
436
    436
  • Thumbnail: Page 
437
    437
  • Thumbnail: Page 
438
    438
  • Thumbnail: Page 
439
    439
  • Thumbnail: Page 
440
    440
  • Thumbnail: Page 
441
    441
  • Thumbnail: Page 
442
    442
  • Thumbnail: Page 
443
    443
  • Thumbnail: Page 
444
    444
  • Thumbnail: Page 
445
    445
  • Thumbnail: Page 
446
    446
  • Thumbnail: Page 
447
    447
  • Thumbnail: Page 
448
    448
  • Thumbnail: Page 
449
    449
  • Thumbnail: Page 
450
    450
  • Thumbnail: Page 
451
    451
  • Thumbnail: Page 
452
    452
  • Thumbnail: Page 
453
    453
  • Thumbnail: Page 
454
    454
  • Thumbnail: Page 
455
    455
  • Thumbnail: Page 
456
    456
  • Thumbnail: Page 
457
    457
  • Thumbnail: Page 
458
    458