Tannakian Categories. P. Deligne and J.S. Milne   Top
Expository Notes
A Primer of Commutative Algebra
Motives---Grothendieck's Dream
What is a Shimura Variety?
Introduction to Shimura Varieties
Shimura Varieties and Moduli
Tannakian Categories
The Work of Tate
pdf (first posted 13.05.11, 72 pages; latest version 15.08.12 minor corrections).


This is a corrected TeXed version of the article "Deligne, P., and Milne, J.S., Tannakian Categories, in Hodge Cycles, Motives, and Shimura Varieties, LNM 900, 1982, pp. 101-228". The numbering is unchanged from the original. Significant changes to the text have been noted in the footnotes. All footnotes have been added by the second author. First posted on the web May 13, 2011.


  1. Tensor categories.
  2. Neutral Tannakian categories.
  3. Fibre functors; the general notion of a Tannakian category.
  4. Polarizations.
  5. Graded Tannakian categories.
  6. Motives for absolute Hodge cycles
    Appendix: Terminology from nonabelian cohomology.