Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental ngroupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applicatio...

This is a continuation of the paper by the same author published in this journal, v. 19 (2007), no. 6. This paper is a continuation of the paper [4] and makes a whole with it. We retain the notation, the definitions, and the assumptions of [4], and we continue the numeration of sections, theorems and lemmas. The proofs of Theorems 2–4 in §6 of [4] were in fact missing; it was said only that one...

Most often, in a categorical semantics for a programming language, the substitution of terms is expressed by composition and finite products. However this does not deal with the order of evaluation of arguments, which may have major consequences when there are side-effects. In this paper Cartesian effect categories are introduced for solving this issue, and they are compared with strong monads,...

Let k be a field and Q a finite quiver without oriented cycles. Let kQ be the path algebra of Q and mod kQ the category of k-finite-dimensional right kQ-modules. The cluster category CQ was introduced in [1] (for general Q) and, independently, in [4] (for Q of type An). It is defined as the orbit category of the bounded derived category D(mod kQ) under the action of the automorphism Σ−1 ◦ S, wh...

Two talks by Bertrand Toën at the IMA 2004 Summer programme on n-Categories: foundations and applications. (Notes taken by Joachim Kock, revised by B.T.) This version 2 (June 2005) is identical to the original except that a correction to 1.3 is given in a footnote.