Holmstrom C15 Simplicial techniques

Schlank and Barnea has a nice treatment of simplicial sheaves and presheaves and their model structures. See material throughout the paper including the last section on Verdier’s thm.

There are several papers by Voevodsy where he talks about simplicial schemes, but it seems like he always relies on working over a field with RoS whenever something is difficult.

Som general remarks about simplicial sheaves and the Nisnevich topology are in Voevodsky’s Seattle lectures.

Some motivation: develop tools for progress in arithmetic. Interuniversal homotopy? Resolutions using alterations? Mimick theorems over RoS fields in the arithmetic setting?

Q: For what cats $C$ is $\Delta^{op}C$ a model cat?

\section{Simplicial techniques in algebraic geometry}

Wildeshaus and Huber: Polylog article. Contains lots of good material on cohomology theories in general, simplicial schemes, and more.

• Hypercohomology spectra and Thomason’s descent theorem (1997)

Etale cohomology of simplicial schemes

See the comprehensive MR search.

Hypercovers and simplicial presheaves, by Dugger, Isaksen, Hollander.

For some facts on hypercoverings (such as hypercoverings being cofinal among all locally contractible simplicial sheaves), Schlank and Barnea refers to Jardine: Higher spinor classes.

There is a nice paper of Hovey Palmieri Strickland on axiomatic stable homotopy theory.

Cubical cohomology

nLab page on C15 Simplicial techniques