Holmstrom Simplicial sheaves

Model structures from nLab, see maybe also the entry on global model structure, and local model str, and also the links under Simplicial presheaves

arXiv:1109.5477 A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type from arXiv Front: math.AG by Ilan Barnea, Tomer M. Schlank In [Isa], Isaksen showed that a proper model category $\cC$, induces a model structure on the pro category $Pro(\cC)$. In this paper we generalize Isaksen’s theorem to the case when $\cC$ possess a weaker structure, which we call a “weak fibration category”. Namely, we show that if $\cC$ is a weak fibration category, that satisfies an extra condition, there is a naturally induced model structure on $Pro(\cC)$. We then apply our theorem to the case when $\cC$ is the weak fibration category of simplicial sheafs on a Grothendieck site, where both weak equivalences and fibrations are local as in [Jar]. This gives a new model structure on the category of pro simplicial sheaves. Using this new model structure we give a definition of the étale homotopy type of Artin and Mazur [AM], as the result of applying a derived functor. Our definition actually gives as object in $Pro(\cS)$ and not just in $Pro(Ho(\cS))$ as in [AM]. Our definition also extends naturally to a relative notion of the étale homotopy type, as considered for example in [HaSc].

Dwyer and Kan have a number of papers on diagrams of simplicial sets, see e.g. http://www.ams.org/mathscinet-getitem?mr=744846. In “Function complexes etc” http://www.ams.org/mathscinet-getitem?mr=705421, they show that there is a model structure on any functor cat with target ´sSet´. They also compute homotopy type of function complexes. Another potentially important paper is “Equivalences between homotopy theories of diagrams” where they give conditions on a functor for the induced map on simplicial presheaves to give an equivalence of homotopy theories.

Various things of Gillet - check all these for any motivational text on simplicial sheaves. Motivation for simplicial sheaves include various constructions for stacks and Q-varieties, for example Chow groups, K-theory of simplicial schemes, line bundles on stacks, and more. In “Filtrations on higher algebraic Kth” he uses the homotopy theory of simplicial objects in a topos.

http://mathoverflow.net/questions/5179/global-fibrations-of-simplicial-sheaves

arXiv:1009.2930 Classifying spaces and fibrations of simplicial sheaves from arXiv Front: math.AT by Matthias Wendt In this paper, we discuss the construction of classifying spaces of fibre sequences in model categories of simplicial sheaves. One construction proceeds via Brown representability and provides a classification in the pointed model category. The second construction is given by the classifying space of the monoid of homotopy self-equivalences of a simplicial sheaf and provides the unpointed classification.

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See file simplicial.tex, and the MR batches in same folder.

Goal: Write survey on simplicial sheaves and schemes.

Sources: Lurie Higher topoi, Thomason article, Morel-Voevodsky, and much more.

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Do a complete search on simplicial schemes/varieties/scheme/variety/sheaf/sheaves. Work out which CTs can be defined on simplicial schemes.

Key authors: Jardine, Deligne, Voevodsky, Quillen, maybe Grothendieck? Who else?

One aspect of simplicial schemes is their relation to RoS, as in Hodge III

It should be really interesting to make a systematic study of the functor category from Rings to simplicial sets. Algebraic K-theory belongs to this category, and possibly every cohomology theory “belongs to” or factors through this category. Every simplicial set and every simplicial scheme also belongs here. What is the relation to various cats of motives? Should look at Jouanolou’s trick, and also at the formalism of unramified cohomology. Could it be that $Fct(Rings, Sset)$ is a “better” cat than simplicial schemes, for some purposes? What does it mean for a functor like this to live “over” some base?

Thinking of schemes as functors, is it easy to define each type of CT? Example: If K-theory is a simplicial scheme, what $K_*(K)$? Is this at all related to cohomology operations, or is it just the notation playing tricks?

Cox: Homotopy theory of simplicial schemes

Notion of dimension for simplicial schemes? At least there should be a notion of cohomological dimension.

Concrete (very simple) examples of the simplicial smooth projective variety corresponding to a noncomplete/singular variety?

Notion of birational equivalence??

Check appendix (C2, C3???) to Jannsen’s LNM notes.

Exercise: Go through Baues: Homotopy types, and generalize as much as possible to simplicial sheaves. E.g. E-M spaces, Hurewicz.

Huber-Wildeshaus: Treats spaces wrt the Zariski topology. Lots of things, and references to check. Some notes from section B.3: Consider a CT defined by a pseudo-flasque complex of presheaves $F$. There are three points of view on how to define cohomology groups from $F$, and all three give the same result. If $F$ is part of a twisted duality theory, then we define Chern classes from K-cohomology of spaces to cohomology with coeffs in $F$.

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Start the complete search here. Will search for simplicial schemes/varieties/scheme/variety/sheaf/sheaves, and for descent in various forms. Sources: MathSciNet, Google Scholar, and arXiv fulltext. Phrase: “simplicial scheme” OR “simplicial schemes” OR “simplicial variety” OR “simplicial varieties” OR “simplicial sheaf” OR “simplicial sheaves” OR “cohomological descent”.

MathSciNet: 106 hits for this phrase. Stored in folder in Dropbox.

SGA4, Tome 2, Expose 5, App 2, for cohomological descent.

Hodge III.

David Cox: Homotopy theory of simplicial schemes

Friedlander: Etale homotopy of simplicial schemes

Carlson http://www.ams.org/mathscinet-getitem?mr=808740 Polyhedral resolutions of algebraic varieties (“explicit” simplicial resolutions“).

Jarding: Simplicial objects in a Grothendieck topos

Guillen and Puerta http://www.ams.org/mathscinet-getitem?mr=808740: On cubic schemes and cubic resolutions, and applications to Hodge theory. See also http://www.ams.org/mathscinet-getitem?mr=972983 for more of the same.

Jardine: Simplicial presheaves.

Jardine: Universal Hasse-Witt classes

Nisnevich, related to Kato-Saito (Global CFT of arithmetic schemes) and Thomason-Trobaugh

Have reached (but not read) Joshua: Generalized Verdier duality…

Am reading the reviews, have reached Gillet: Homological Descent… After finishing, check the list again quickly to make sure nothing is left out here.

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Note: Can extend DM to simplicial schemes (Deglise in Oberwolfach talk). What about the six operations? This should be of interest in a complete reworking of SH and DM for generalized schemes, including the compactification of Spec(Z).

nLab page on Simplicial sheaves?