All titles are links to pdf files.
Abstract: This paper displays model structures for the category of pro objects in simplicial presheaves on an arbitrary small Grothendieck site. The first of these is an analogue of the Edwards-Hastings model structure for pro objects in simplicial sets, in which the cofibrations are monomorphisms of pro-objects and the weak equivalences are specified by comparisons of function complexes. Other model structures are built from the Edwards-Hastings structure by using Bousfield-Friedlander localization techniques, as in Biedermann's recent work on model structures for n-types. In particular, I display an n-type structure for pro objects, and also a model structure in which the map from a pro object to its Postnikov tower is formally inverted.
Abstract: This paper defines and gives the basic properties of the path category P(X), where X could be either a simplicial or a cubical set. The paper describes an approach to computing this invariant which involves a resolution in simplicial categories. If the input object K is either a finite oriented simplicial complex or a finite cubical complex, the resolution contains a finite 2-category which "resolves" the path category P(K), and is itself computable from a generators and relations structure.
Warning:. This paper replaces the paper "Representability theorems for simplicial presheaves". The proof Lemma 1 which appears in the former paper fails.
Abstract: Suppose that M is a simplicial model category and that F is a contravariant simplicial functor defined on M which takes values in pointed simplicial sets. This note displays conditions on the simplicial model category M and the functor F such that F is representable up to weak equivalence. The conditions on F are homotopy coherent versions of the classical conditions for Brown representability, while M should have the fundamental properties of the category of presheaves of spectra on a Grothendieck site. The theorem applies to functors defined on all categories of presheaves of spectra of symmetric spectra, and Bousfield localizations of these model structures.
Abstract: This paper gives a characterization of homotopy fibres of inverse image maps on groupoids of torsors which are induced by geometric morphisms, in terms of both pointed torsors and pointed cocycles, suitably defined. Cocycle techniques are used to give a complete description of such fibres, when the underlying geometric morphism is the canonical stalk on the classifying topos of a profinite group G. If the torsors in question are defined with respect to a constant group H, then the path components of the fibre can be identified with the set of continuous maps from the profinite group G to the group H. More generally, when H is not constant, this set of path components is the set of continuous maps from a pro object in sheaves of groupoids to H, which pro object can be viewed as a "Grothendieck fundamental groupoid".
Abstract: This short note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem which approximates morphisms [X,Y] in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where Y is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result, in that it is pointed (in a very broad sense) and there is no requirement for the source object X to be locally fibrant.
This paper is to appear in the Canadian Mathematical Bulletin.
Abstract: This paper gives a closed model structure for the category of cubical sets, suitably defined, and displays an equivalence of the associated homotopy category with the ordinary homotopy category of topological spaces, or simplicial sets.
Note: This paper will not be published, as the main results have much better proofs in Cisinski's thesis. See also Categorical homotopy theory, by J.F. Jardine.
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 is a short expository paper, which is meant to illuminate the assertion of Morel and Voevodsky that there is a closed model structure on the category of sheaves for the smooth Nisnevich site of a field such that the associated homotopy category is equivalent to the motivic homotopy category for simplicial sheaves. Actually, slightly more is true, in that the Morel-Voevodsky result is proved by first demonstrating a similar result relating presheaves and simplicial presheaves.
A shorter version of the material presented in this preprint appears as an appendix to the paper "Motivic symmetric spectra" (Doc. Math. 5 (2000), 445-552).
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)