Preprints: J.F. Jardine

All titles are links to pdf files.

Cluster graphs

Most recent revision: July 4, 2017

Abstract Cluster graphs are defined as subgraphs of nerves of translation categories by removing edges associated to branch points, for arrays of sets F of all dimensions. The path components of the cluster graph for F are the clusters of the array F. Combinatorial scoring functions are defined for clusters in dimensions one and two.

Complexity reduction for path categories

Most recent revision: July 19, 2016

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.

Pro-equivalences of diagrams

Most recent revision: November 7, 2016 (very slight revision of Nov. 23, 2015 document)

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.

Path categories and quasi-categories

Most recent revision: December 14, 2015

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.

Galois descent criteria

Most recent revision: January 30, 2013

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.

The Barratt-Priddy-Quillen Theorem

Most recent revision: September 15, 2009

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.

E_2 model structures for presheaf categories

Most recent revision: September 24, 2001

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.

Cohomological invariants associated to symmetric bilinear forms

Most recent revision: November 29, 1998

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.

A rigidity theorem for L-theory

Most recent revision: November 29, 1998

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.

Algebraic homotopy theory, groups and K-theory

Most recent revision: September 9, 2005

Thesis, University of British Columbia (1981)

[UWO Math. Dept. home page]