We define natural A ∞ -transformations and construct A ∞ -category of A ∞ -functors. The notion of non-strict units in an A ∞ -category is introduced. The 2-category of (unital) A ∞ -categories, (unital) functors and transformations is described. The study of higher homotopy associativity conditions for topological spaces began with Stasheff’s article [Sta63, I]. In a sequel to this paper [Sta6...

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 (generally non reversible) and fundamental n-categories (replacing the fundamental ngroupoids of the classical case). Finding a simple model of the latter is a non-trivial problem, whose solution gives r...

We develop Morita theory for finitary additive 2-representations of finitary 2-categories. As an application we describe Morita equivalence classes for 2-categories of projective functors associated to finite dimensional algebras and for 2-categories of Soergel bimodules.

In this note, we shall prove that two smooth projective varieties of dim 2n connected up by a Mukai flop have equivalent bounded derived categories of coherent sheaves. More precisely, let X and X be smooth projective varieties of dimension 2n such that there is a birational map φ : X −− → X obtained as the Mukai flop along a subvariety Y ⊂ X which is isomorphic to P. By definition, φ is decomp...

In this paper we re-develop the foundations of the category theory of quasicategories (also called ∞-categories) using 2-category theory. We show that Joyal’s strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected the...