model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
This entry is about the article
Abstract Homotopy Theory and Generalized Sheaf Cohomology,
Transactions of the American Mathematical Society,
186 (1973) 419-458
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.
Last revised on June 16, 2022 at 07:52:07. See the history of this page for a list of all contributions to it.