Let J be a shape in some category Shp for which there is a functor : Shp Cat. A categorical transition system (or system) is a pair (J; (J) C) consisting of a shape labelled by a functor in a category in C. Systems generalize conventional labelled transition systems. By choosing a suitable universe of shapes, systems can model concurrent and asynchronous computation. By labelling in a category,...

Received In this paper we provide a categorical interpretation of the rst-order Hoare logic of a small programming language, by giving a weakest precondition semantics for the language. To this end, we extend the well-known notion of a ((rst-order) hyperdoctrine to include partial maps. The most important new aspect of the resulting partial ((rst order) hyperdoctrine is a diierent notion of mor...

Introduction Our objectives are topological versions of the Nielsen-Schreier Theorem on subgroups of free groups, and the Kurosh Theorem on subgroups of free products of groups. It is known that subgroups of free topological groups need not be free topological [2, 6, and 9]. However we might expect a subgroup theorem when a continuous Schreier transversal exists, and we give such a result in th...

We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κ-ary exact categories” are a reflective sub-2-category of “κ-ary sites”, for any regular cardinal κ. A κ-ary exact category is an exact category with disjoint and universal κ-small coproducts, and a κ-ary site is a site whose covering sieves are generated by κ-small families and ...

A theory of topological gravity is a homotopy-theoretic representation of the Segal-Tillmann topologification of a two-category with cobordisms as morphisms. This note describes some relatively accessible examples of such a thing, suggested by the wall-crossing formulas of Donaldson theory. 1 Gravity categories A cobordism category has manifolds as objects, and cobordisms as morphisms. Such cat...

A flow is homotopy continuous if it is indefinitely divisible up to S-homotopy. The full subcategory of cofibrant homotopy continuous flows has nice features. Not only it is big enough to contain all dihomotopy types, but also a morphism between them is a weak dihomotopy equivalence if and only if it is invertible up to dihomotopy. Thus, the category of cofibrant homotopy continuous flows provi...

We prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs isomorphism classes fibrations onto curves genus g?2. Moreover, in dimension at most four, we same normal surfaces with positive holomorphic Euler characteristic and admitting finite morphism to an abelian variety. Finally, study invariance class minimal base under condition all Ho...