Olivia Caramello's website


Unifying theory

A brief historical digression

The concept of (Grothendieck) topos was introduced by Alexandre Grothendieck in the early 1960s, in order to provide a mathematical underpinning for the 'exotic' cohomology theories needed in algebraic geometry. Every topological space X gives rise to a topos (the category of sheaves of sets on X) and every topos in Grothendieck’s sense can be considered as a 'generalized space'.

At the end of the same decade, William Lawvere and Myles Tierney realized that the concept of Grothendieck topos also yielded an abstract notion of mathematical universe within which one could carry out most familiar set-theoretic constructions, but which also, thanks to the inherent 'flexibility' of the notion of topos, could be profitably exploited to construct ‘new mathematical worlds’ having particular properties.

A few years later, the theory of classifying toposes added a further fundamental viewpoint to the above-mentioned ones: a topos can be seen not only as a generalized space or as a mathematical universe, but also as a suitable kind of first-order theory (considered up to a general notion of equivalence of theories).

In fact, every first-order mathematical theory (of a general specified form - technically speaking, a geometric theory) admits a canonical topos-theoretic model lying in the classifying topos of the theory, satisfying the universal property that every topos-theoretic model of the theory can be obtained as a pullback of this canonical model along a unique morphism of toposes. Such canonical models of mathematical theories do not exist in the classical set-theoretic setting, but any classical (i.e., set-based) model of the theory can be seen as a point of its classifying topos.

Unfortunately, apart from a few isolated exceptions, the study of Grothendieck toposes in the context of Logic was essentially not pursued afterwards. In fact, a great part of the efforts of the categorical community in the past years had been focused on the study of elementary toposes, a more general (elementarily axiomatizable) class of categories in which one can consider models of arbitrary finitary first-order theories, but in which one cannot do geometry in a Grothendieckian sense due to the fact that elementary toposes do not in general admit sites of definition. Even though the study of these more general concepts has shed important light on the original concept of topos (in fact, it was seen that several concepts in Grothendieck's topos theory admitted an 'elementary' formulation), the concept of elementary topos has not been fruitful as a source of 'concrete' applications in different fields of Mathematics, essentially because of the lack of a non-trivial representation theory 'from below' analogous to that provided by sites (which in the context of Grothendieck toposes play the role of 'carriers of concrete mathematical information').