**All titles are links to pdf files.**

This a partial, rough manuscript for a monograph, which is tentatively to be published by Springer-Verlag. The book is meant to be a basic account of the homotopy theories of simplicial sheaves and presheaves, and the stable homotopy theory of presheaves of spectra. Selected applications are included.

Comments and suggestions are welcome.

**Abstract:**
This paper describes features of the injective model
structure on cosimplicial spaces.

Standard results from non-abelian cohomology theory specialize to theories of torsors and stacks for cosimplicial groupoids. It is shown that the space of global sections of the stack completion of a cosimplicial groupoid G is weakly equivalent to the Bousfield-Kan total complex of BG, even though BG is not necessarily Bousfield-Kan fibrant. We show that the k-invariants for the Postnikov tower of a cosimplicial space X are naturally elements of stack cohomology for the stack associated to the fundamental groupoid of X.

Cocycle theoretic ideas and techniques are used throughout the paper.

**Abstract:**
Suppose that X is a simplicial presheaf on
the etale site for a field k. This note gives a pro equivalence
criterion which would imply that X satisfies Galois descent in global
sections, in the presence of a uniform bound on the Galois
cohomological dimension of k with respect to the sheaves of homotopy
groups of X.

**Abstract:**
The paper gives E_2 model structures in the
style of Dwyer-Kan-Stover and Goerss-Hopkins for
categories of simplicial objects in
pointed simplicial presheaves, presheaves of spectra and
presheaves of symmetric spectra on a small Grothendieck site.
Analogs of these results for unstable and stable motivic homotopy theory
are also displayed and proved. The key
technical device is a bounded approximation technique for objects in the
respective categories, which ultimately
depends on cardinality count methods previously seen in localization
theory.

**Note:** This paper will not be published.

**Abstract:**
This paper reviews the construction of the Hasse-Witt and
Stiefel-Whitney classes for an orthogonal representation of
a Galois group, and then gives a simplicial presheaf theoretic
demonstration of the Frohlich-Kahn-Snaith formula for the Hasse-Witt
invariant of the associated twisted form. A Steenrod squares argument
is used to show that this formula has an analogue in degree 3.
The mod 2 étale cohomology of the classifying simplicial scheme of the
automorphism group of an arbitrary non-degenerate symmetric bilinear
form is calculated, and the relation of this cohomology ring with the
Hasse-Witt classes of the Frohlich twisted form is discussed.

This is a corrected version of a paper that has been published (Expositiones Math. 10 (1992), 97-134). There is a bad printer error which makes the introduction of the published version completely unintelligible.

**Abstract:**
This paper proves a rigidity theorem for mod l Karoubi
L-theory, and then uses it to calculate the mod l Karoubi L-groups of
algebraically closed fields. All proofs and calculations given here
depend on the homotopy theory of simplicial sheaves.

This paper was written in 1983, and is unpublished. Karoubi proved and published similar results from a different point of view - see this preprint for further details.

Thesis, University of British Columbia (1981)