Francis Borceux

Quantales and Their Sheaves

A quantale is a complete lattice provided with a binary multiplication which, in particular, distributes over ∨ in each variable. First we study the quantales, their morphisms, their points and the relations with the theory of locales: the case where the multiplication is just the meet. We describe the presheaves and sheaves on a quantale and exhibit a link between the corresponding toposes and quantum logic.

Algebra in a localic topos with applications to ring theory

Categories of algebraic sheaves.- The formal initial segments.- Localizations and algebraic sheaves.- Integral theories and characterization theorem.- Spectrum of a theory.- Applications to module theory.- Pure representation of rings.- Gelfand rings.

Euclid’s Elements

A review of all thirteen books of the Elements, which imposed themselves as “the geometric Bible” for some two millenniums.

An Essay On Noncommutative Topology

Abstract A quantum space is a set provided with a family of open subsets, stable under arbitrary unions and a noncommutative finite “intersection”. A basic example is given bythe spectrum of a C ∗-algebra, where the noncommutative “intersection” is induced by the product of ideals. We develop the basic theory of quantum spaces and some applications to C ∗-algebras.

Universal algebra in a closed category

Abstract The development of finitary universal algebra is carried out in a suitable closed category called a π-category. The π-categories are characterized by their completeness and cocompleteness and some product-colimit commutativities. We establish the existence of left adjoints to algebraic functors, completeness and cocompleteness of algebraic categories, a structure-semantics adjunction, a characterization theory for algebraic categories and the existence of the theory generated by a presentation. The conditions on the closed category are sufficiently weak to be satisfied by any (complete and cocomplete) cartesian closed category, semi-additive category, commutatively algebraic category and also the categories of semi-normed spaces, normed spaces and Banach spaces.

Short Introduction to Enriched Categories

This text aims to be a short introduction to some of the basic notions in ordinary and enriched category theory. With reasonable detail but always in a compact fashion, we have brought together in the first part of this paper the definitions and basic properties of such notions as limit and colimit constructions in a category, adjoint functors between categories, equivalences and monads. In the second part we pass on to enriched category theory: it is explained how one can “replace” the category of sets and mappings, which plays a crucial role in ordinary category theory, by a more general symmetric monoidal closed category, and how most results of ordinary category theory can be translated to this more general setting. For a lack of space we had to omit detailed proofs, but instead we have included lots of examples which we hope will be helpful. In any case, the interested reader will find his way to the references, given at the end of the paper.

A theory of quantal sets

Abstract A quantale is a complete lattice provided with a (generally non commutative) binary multiplication which, in particular, distributes over arbitrary suprema in each variable. That notion is intended to replace locales in the spectral constructions involving non commutative algebraic structures. We propose a corresponding notion of quantale-valued set, which generalizes the classical notions of boolean or heyting-valued sets. We study the category obtained in this way and show that it can be seen as a fibration over the original quantale, whose fibres are toposes.

Enriched accessible categories

We consider category theory enriched in a locally finitely presentable symmetric monoidal closed category V. We define the V-filtered colimits as those colimits weighted by a V-flat presheaf and consider the corresponding notion of V-accessibIe category. We prove that V-accessible categories coincide with the categories of V-flat presheaves and also with the categories of V-points of the categories of V-presheaves. Moreover, the V-locally finitely presentable categories are exactly the V-cocomplete finitely accessible ones. To prove this last result, we show that the Cauchy completion of a small V-category C is equivalent to the category of V-finitely presentable V-flat presheaves on C.

On the notion of bimodel for functorial semantics

We discuss the notion of bimodel in order to obtain a classification of the equivalences between categories of models in the sense of functorial semantics.

A characterization of quasi-toposes

A locally presentable category 6 is the category of models of some theory F which can be described using a-limits, for a regular cardinal a. Our aim is to find conditions which ensure good properties of the category d of models: when is 6 Cartesian closed? When is d a quasi-topos? When is d a topos? and so on. In this paper, we focus our attention on some semantic conditions. A locally a-presentable category 8 can thus be presented as the category of cc-left exact presheaves on a small category