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: A dynamical system is a space X with a pairing from X x S to X for some parameter space S, and a map of such dynamical systems is an S-equivariant map. There is an injective and a projective model structure for the resulting category of spaces with S-action, and both are easily derived. These model structures are special cases of model structures for presheaf-valued diagrams X defined on a fixed presheaf of categories E which is enriched in simplicial sets.
Simultaneously varying the parameter category object E (or parameter space S) along with the diagrams X up to weak equivalence is more interesting, and requires new model structures for E-diagrams having weak equivalences defined by homotopy colimits, as well as a generalization of Thomason's model structure for small categories to a model structure for presheaves of simplicial categories. These new model structures exist for arbitrary presheaves of simplicial categories E and their categories of diagrams. We do not require our presheaves of simplicial categories E to have simplicially discrete objects.
Abstract: This paper shows that the category of bisimplicial presheaves carries a model structure for which the weak equivalences are defined by the diagonal functor and the cofibrations are monomorphisms. This model structure is the member having the most cofibrations of a large family of model structures with weak equivalences defined by the diagonal. The diagonal structure for bisimplicial presheaves specializes to a diagonal model structure for bisimplicial sets, for which the fibrations are the Kan fibrations.
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)
[UWO Math. Dept. home page]