Titles and some abstracts - Michael Barr: Cohomology of algebras Consider an adjunction F -| U, U: *A* --> *B* and F: *B* --> *A*. Suppose that *A* is a regular category and U a regular functor. Let T and G be the resultant triple and cotriple. Assume that B projective in *B* implies that TB is projective. Suppose the inclusion Ab(*A*) --> *A* has a left adjoint Diff. Suppose there is a chain complex functor C.: *A* --> ChComp(Ab(*A*)) that produces, for each A of *A*, for which UA is projective, a projective resolution of Diff(A). Suppose also that for each n >= 0, there is a ~C_n: *B* --> Ab(*A*) such that ~C_n o U = C_n (= here means natural equivalence). Then for any A of *A* such that UA is projective and any M of Ab(*A*), we have H^._G(A,M)=Ext^.(Diff(A),M). Applications include the well-known equivalence of the Hochschild cohomology of associative algebras with the cotriple cohomology for algebras that are projective over the ground ring, as well as that of the Eilenberg-Mac Lane cohomology of groups with the cotriple cohomology, but also gives the analogous result for Lie algebras, which has not, to my knowledge, been previously published. - David B. Benson: Paths in Higher-Dimensional Graphs Paths in higher-dimensional graphs have arrows between arrows. These paths abstract many aspects of programming languages and related semantical matters. Higher-dimensional graphs, a concept closely akin to computad, generate free n-categories -- and hence the free cellular omega-categories. The arrows of such categories are represented by paths as the equivalence classes of paths modulo interchange. Normal forms for the presentations of paths provide fast equality tests between the represented arrows, important for the goal of programming these ideas. Two normal forms are considered. Natural normal form arises directly from the definitions. Irreducible normal form is more efficient in the use of computer storage and appears to be related to Street's exponential wedge of r-equivalences. - Richard Blute: Proof Nets and Monoidal Categories - William Boshuck: Strong completeness of predicate modal logic or A duality theorem for predicate modal logic - Robin Cockett: TBA - Peter J. Freyd: Path Integrals - Robert Gordon: Tricategories - Michael Hebert: Syntactic characterizations of closure under pullbacks and of locally polypresentable categories Locally (finitely) polypresentable, resp. multipresentable, resp. presentable categories (LFPP, LFMP, LFP) are (finitely) accessible categories with wide pullbacks, resp. connected limits, resp. all (small) limits. LFPP are also fin. acc. cat. having all its slices LFP (Example: the algebraically closed fields). LFMP have been syntactically described by Johnstone in 1979, LFP by Coste in 1976. My first result is a syntactic description of LFPP in this spirit. If we now start with a given (finitary) language and want to stay within, and require the limits to be the "usual" ones, they correspond to elementary categories of models closed respectively under pullbacks (LFPP), equalizers and pullbacks (LFMP) and finite limits (LFP). Syntactic characterizations become somewhat more complicated; LFP case was solved by H.Volger in 1979 and LFMP case by myself in 1991. I will present the solution for the pullback case. Those two results solve problems respectively presented by F. Lamarche (in the context of Domain Theory) and by H.Volger (in the context of classical Model Theory, with motivations also from Database Theory). - Hongde Hu: A duality theorem on accessible exact categories - James R. Otto: Complexity classes from categories - Richard Squire: TBA - Walter Tholen: Factorization Systems as Eilenberg-Moore Algebras - Charles F. Wells: Graph Based Logic and Sketches - Mike Wendt: On Measurably Indexed Families of Hilbert Spaces. - Richard Wood: A somewhat circular characterization of the category of sets Years ago in a paper on total categories (those locally small B for which Y=yoneda:B--->PB has a left adjoint) I observed that for B=set_ there is U-|V-|W-|X-|Y and conjectured that this characterizes set_. Bob Rosebrugh and I have returned to this recently and hope to formally announce the result very soon. It builds on our work on completely distributive lattices. It probably makes good sense to say that a category B is "totally distributive" if it admits W-|X-|Y so the result can be seen as setting bounds on how "exact" a category can be. - Marek Zawadowski: Lax Descent Theorems for Left Exact Categories ==============================================================================