nLab BrownAHT

This entry is about the article

• Kenneth S. Brown,

  Abstract Homotopy Theory and Generalized Sheaf Cohomology,

  Transactions of the American Mathematical Society,

  186 (1973) 419-458

  (jstor:1996573).

Brown was a student of Dan Quillen. He expands in this article work from his thesis on a notion of homotopical category – called a category of fibrant objects – with a bit less structure than that of a full model category, which he uses to study “generalized abelian sheaf cohomology”:

Following, or rather as a precursor of, the nPOV on cohomology he studies precisely the cohomology within ∞-stack (∞,1)-toposes and, in the second part of the article, that in (∞,1)-categories of (∞,1)-sheaves of spectra, see at sheaves of spectra.

As such, this article is a forerunner of the development of the full model category structures built for the purpose of modelling ∞-stack (∞,1)-toposes: the model structure on simplicial (pre)sheaves as first defined by Andre Joyal in his letter to Grothendieck and then later generalized from simplicial sheaves to simplicial presheaves and developed in great detail by Jardine (in its injective version) and Dan Dugger (in its projective version).

Dugger's theorem that every combinatorial model category is a left Bousfield localization of the projective global model structure on simplicial presheaves was finally understood by Jacob Lurie in the book Higher Topos Theory to be the model category-theoretic version of the intrinsic higher category theory statement that every presentable (∞,1)-category is the left localization of an (∞,1)-category of an (∞,1)-category of (∞,1)-presheaves, and that the left exact such localizations are precisely the (∞,1)-categories of (∞,1)-sheaves – the ∞-stack (∞,1)-toposes.

But the homotopy category of these (∞,1)-toposes – that category which remembers just the ∞-stack cohomology classes, and not the cocycles – is precisely what Kenneth Brown constructed, studied and identified correctly in its nature back then in 1973.

See also at Ken Brown's lemma.

category: reference