We prove that every stable, presentable model category can be enriched in a natural way over symmetric spectra. As a consequence of the general theory, every object in such a model category has an associated homotopy endomorphism ring spectrum. Basic properties of these invariants are established.

Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A 'directed space', e.g. an ordered topological space, has directed homotopies (which are generally non-reversible) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). Finding a simple model of the latter is a non-trivial problem, whose solution gives relevant information on the given 'space'; a problem which is also of interest in general Category Theory, as it requires equivalence relations which are more general than categorical equivalence. Taking on a previous work on 'The shape of a category up to directed homotopy', we study now the fundamental 2-category of a directed space. All the notions of 2-category theory used here are explicitly reviewed.

This paper is an exposition of the ideas and methods of Cisinksi, in the context of A-presheaves on a small Grothendieck site, where A is an arbitrary test category in the sense of Grothendieck. The homotopy theory for the category of simplicial presheaves and each of its localizations can be modelled by A-presheaves in the sense that there is a corresponding model structure for A-presheaves with an equivalent homotopy category. The theory specializes, for example, to the homotopy theories of cubical sets and cubical presheaves, and gives a cubical model for motivic homotopy theory. The applications of Cisinski's ideas are explained in some detail for cubical sets.

Working in the context of categorical groups, we show that the semidirect product provides a biequivalence between actions and points. From this biequivalence, we deduce a two-dimensional classification of split extensions of categorical groups, as well as the universal property of the holomorph of a categorical group. We also discuss the link between the holomorph and inner autoequivalences.

$Let P^n(p)$ be an $n$- dimensional mod $p$ Moore space and $V^n$ be the mapping cone of an Adams map $A : P^{n-1}(p) \rightarrow P^{n-2p+1}(p)$. This paper gives an unstable splitting of $V^m \bigwedge V ^n$ for a prime $p \geq 5$. The proof is based on explicit calculations of $[V^{n+2p-1}, V^n]$. As an application, we define a Samelson product on $[V^*, \Omega X]$ and prove that it satisfies anticommutativity and the Jacobi identity.

The holomorphic (or semi-topological) K-theory of a smooth projective variety sits between the algebraic K-theory of the variety and the topological K-theory of the underlying topological space. We describe how to define a family of dbar operators on holomorphic K-theory in a manner analogous to Atiyah's construction of a family of elliptic operators in topological K-theory. In the process, we prove a result akin to Bott periodicity for holomorphic mapping spaces. These results first appeared in the author's Stanford University Ph.D. thesis under the direction of Ralph Cohen.

We establish a general method to produce cofibrant approximations in the model category U _S (C,D) of S-valued C-indexed diagrams with D-weak equivalences and D-fibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial model category, and D subset C are small categories. An application to the notion of homotopy colimit is presented.

We study Hopf algebra structures on polynomial algebras generated by coloured trees. In particular, we establish the existence of a 2n-parameter family of such structures for trees coloured by n colours.