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

**Abstract:**
This paper displays complexity reduction techniques for calculations of path categories (or fundamental categories) P(K) for finite simplicial and cubical complexes K. The central technique involves identifying inclusions of complexes for which the induced functor of path categories is fully faithful. Refinements of cubical complex structures are discussed. A first method for parallelizing the calculation of path categories for cubical complexes is introduced.

**Abstract:**
This paper presents a model structure for
natural transformations of diagrams of simplicial presheaves of a
fixed shape, in which the weak equivalences are defined by analogy
with pro-equivalences between pro-objects.

**Abstract:**
This paper was written in support of the
recent paper The Local Joyal
Model Structure, by Nicholas Meadows. Meadows' paper constructs a
presheaf-theoretic version of Joyal's quasi-category model structure,
for which a map of simplicial presheaves is a weak equivalence if it
is a stalkwise categorical weak equivalence.

The present paper is an exposition of the Joyal structure, which displays proofs of technical details that are used in the Meadows paper. It was also written with a view to setting up potential applications in concurrency theory. The main new technical result is a characterization of weak equivalences of quasi-categories as maps which induce equivalences of a certain infinite family of naturally defined groupoids.

**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:**
This preprint is a polished version of notes for a colloquium given at
the University of Bremen in November, 2007.
It gives a combinatorial
proof of the Barratt-Priddy-Quillen theorem which asserts that the
stable homotopy groups of spheres are isomophic to the homotopy groups
of the spaces obtained by applying the Quillen plus construction to
the classifying space of the infinite symmetric group.
This was my take on the subject at the time that the lecture was given, but the ideas are essentially the same as those found in the following paper:

Christian Schlichtkrull. *The homotopy infinite symmetric product represents stable homotopy*. Algebr. Geom. Topol. 7, 1963-1977, 2007.

**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)