Holmstrom Descent

nLab entry, check this and also this

Conrad: Cohomological descent (hypercover.pdf)

http://ncatlab.org/nlab/show/descent+for+simplicial+presheaves

http://ncatlab.org/nlab/show/hypercompletion

http://ncatlab.org/nlab/show/Carlos+Simpson

Descent: See Borceaux vol2 chapter for a categorical viewpoint.

Deglise uses something for rigid cohomology

Guillen and Navarro Aznar uses something called a “descent category” in their “extension” paper.

Friedlander, Walker: Semitopological K-theory of real varieties. Proves Nisnevich descent! Review says that the corresponding fact for alg K-th was proved by Thomason.

Giraud: Technique de descente, 1964.

R. Street, Descent, Oberwolfach preprint, mentioned by Toby Bartels at nLab

rectified infinity-stack at nLab

A must-read for descent: Toen: Thm de RR pour les champs de DM. File Toen web publ rrchamp.pdf. Discusses K-theory and RR for DM stacks, but also various notions of descent, including basic stuff on simplicial presheaves, homological descent, and etale descent.

For the notion of descent data, see Toen course in cours folder under Toen, chapter 5. Also in Vistoli maybe.

Possibly interesting: Descent theory for derived categories. Alexei D Elagin 2009 Russ. Math. Surv. 64 748-749

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Email from Peter, July 09:

I stick to the slogan: descent means glueing is respected… what ever that means. So it is some sheaf condition. Somehow descent for a functor (aka this functor being a (higher) stack) means, after Toen and many others, that F(X) = holim(cech simpl. object of any (hyper)covering of X). See the notes, hopefully… For seeing how Galois descent is an instance of that I found the relevant chapter in Waterhouse illuminating - I don’t claim I understood it but it seems to put one on the right track.

Brown rep.: After Neeman’s article (“Brown rep. and Grothendieck duality via Bousfield’s techniques”) Brown rep. holds for compactly generated triangulated categories which SH(Z) should be; I think connected schemes are generators for sPre(Sm/k) - every presheaf is a colim of Homs, i.e. of schemes (and non-connected schemes are coproducts of connected sch.s), every simplicial set is a colim of its simplices, now one would have to combine these, maybe one needs additionally the constant sheaves with value $\Delta^n$ as generators (?). Then hence they would be generating for SH(Z) (localizing only makes them a “bigger” subset of all objects). Intuition says that these are compact in SH(Z) but I know no proof.

One can shift the difficulty thus: Neeman proves in his book that Brown rep. holds for “well-generated” categories (see Krause for a definition) and Rosicky says in the introduction to his article that he shows that combinatorial model cats give well-generated triangulated cats. Now here the Brown rep. theorem is probably more difficult but it is easy to see that SH(Z) falls under its scope. Roughly: Combinatorial = accessible i.e. everything is filtered colim of presentable objects, i.e. objects s.t. Hom out of them commutes with filtered colims. This is clearly the case for sPre(C), C any cat: It is a functor category, namely $Set^(C \times \Delta^op)$ and in such everythin is filtered colim of representables…

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A bibliography:

• J. Bénabou and J. Roubaud paper Monades et Descente jour Comptes Rendus Acad. Sc. Paris 270, série A \\\\\\\\pages 96–98 \\\\\\\\yr 1970 \\\\\\\\lang French
• R. Betti \\\\\\\\paper Adjointness in descent theory \\\\\\\\jour J. Pure Appl. Algebra 116, No.1-3 \\\\\\\\pages 41–47 \\\\\\\\yr 1997
• I. Bucur \\\\\\\\book Selected Topics in Algebra \\\\\\\\publaddr D. Reidel Publishing Company (Dordrecht, Boston, Lancaster) \\\\\\\\yr 1984
• J. Duskin\\\\\\\\paper An Outline of a Theory of Higher Dimensional Descent \\\\\\\\jour Bull. Soc. Math. de Belgique (Serie A) \\\\\\\\vol Tome XLI Fasc. 2 \\\\\\\\pages 249–277 \\\\\\\\yr 1989
• Duskin \\\\\\\\paper The Theory of Descent \\\\\\\\paperinfo (xx+230 pp.) unpublished appendix
• Duskin \\\\\\\\talk Triples and Descent preprint of talks given at Oberwolfach \\\\\\\\yr Sep. 17-23 1995.
• A. Grothendieck\\\\\\\\paper Technique de Déscente et Théoréms d\\\\\\\'Existence en Géométrie Algébrique. Généralités. Déscente par Morphisms Fidélment Plat \\\\\\\\inbook Séminaire Bourbaki, $12^e$ no.190 \\\\\\\\yr 59–60 \\\\\\\\lang French
• \\\\\\\\bysame \\\\\\\\paper Categories Fibrées et Déscente,Déscente Fidelment plat, and Déscente des Morphisms etales \\\\\\\\inbook S.G.A. 1 exposé VI,VIII and IX,respectively Lec. Notes in Math. \\\\\\\\#224 \\\\\\\\yr 1971\\\\\\\\lang French
• G. Janelidze and W. Tholen \\\\\\\\paper Facets of descent. I. \\\\\\\\jour [J] Appl. Categ. Struct. 2, No.3, \\\\\\\\pages 245–281 \\\\\\\\yr 1994.
• G. Janelidze and W. Tholen \\\\\\\\paper Facets of descent. II. \\\\\\\\jour [J] Appl. Categ. Struct. 5, No.3, \\\\\\\\pages 229–248 \\\\\\\\yr 1997
• A. Joyal and M. Tierney \\\\\\\\paper An Extension of the Galois Theory of Grothendieck \\\\\\\\jour Memoirs Amer. Math. Soc. 309 \\\\\\\\yr 1984
• M .A. Knus, and M. Ojanguren \\\\\\\\book Théorie de la Dé scente et Algébras de Azumaya \\\\\\\\bookinfo Lec. Notes in Math. \\\\\\\\#389 \\\\\\\\yr 1974 \\\\\\\\lang French
• W. Lawvere \\\\\\\\book Theory of Categories over a Base Topos \\\\\\\\publaddr Mimeographed notes, Perugia \\\\\\\\yr 1972–73 \\\\\\\\pages 30 and 70
• I. Moerdijk \\\\\\\\paper Descent Theory for Toposes \\\\\\\\jour Bull. Soc. Math. de Belgique (Series A)\\\\\\\\vol Tome XLI Fasc. 2 \\\\\\\\pages 373–391 \\\\\\\\yr 1989
• M. Sobraland W. Tholen \\\\\\\\paper Effective Descent Morphisms and Effective Equivalence Relations \\\\\\\\inbook Can. Math. Soc. Conference Proceedings \\\\\\\\vol 13 \\\\\\\\pages 421–433 \\\\\\\\yr 1992
• W.C. Waterhouse \\\\\\\\book Introduction to Affine Group Schemes \\\\\\\\bookinfo Graduate texts in math. \\\\\\\\#66 \\\\\\\\yr 1979 \\\\\\\\publaddr Springer-Verlag (New York)

nLab page on Descent