An introduction to Grothendieck toposes Olivia Caramello Introduction Toposes as generalized spaces Sheaves on a topological space An introduction to Grothendieck toposes Sheaves on a site Geometric morphisms Toposes as mathematical universes Olivia Caramello Classifying toposes University of Insubria (Como) and IHÉS Toposes as bridges Examples of ‘bridges’ Topics in Category Theory Topological Edinburgh, 11-13 March 2020 Galois theory Theories of presheaf type Topos-theoretic Fraïssé theorem Stone-type dualities General remarks Future directions An introduction to Grothendieck The “unifying notion” of topos toposes Olivia Caramello Introduction “It is the topos theme which is this “bed” or “deep river” Toposes as where come to be married geometry and algebra, topology and arithmetic, generalized spaces mathematical logic and category theory, the world of the “continuous” and Sheaves on a that of “discontinuous” or discrete structures. It is what I have conceived of topological space most broad to perceive with finesse, by the same language rich of Sheaves on a site Geometric geometric resonances, an “essence” which is common to situations morphisms most distant from each other coming from one region or another Toposes as of the vast universe of mathematical things”. mathematical universes A. Grothendieck Classifying toposes Toposes as bridges Topos theory can be regarded as a unifying subject in Mathema- Examples of tics, with great relevance as a framework for systematically inves- ‘bridges’ Topological tigating the relationships between different mathematical theories Galois theory and studying them by means of a multiplicity of different points of Theories of presheaf type view. Its methods are transversal to the various fields and com- Topos-theoretic plementary to their own specialized techniques. In spite of their Fraïssé theorem generality, the topos-theoretic techniques are liable to generate in- Stone-type dualities sights which would be hardly attainable otherwise and to establish General remarks deep connections that allow effective transfers of knowledge bet- Future directions ween different contexts. 2 / 66 An introduction to Grothendieck The multifaceted nature of toposes toposes Olivia Caramello Introduction The role of toposes as unifying spaces is intimately tied to their Toposes as multifaceted nature. generalized spaces Sheaves on a topological space For instance, a topos can be seen as: Sheaves on a site Geometric morphisms • a generalized space Toposes as mathematical universes • a mathematical universe Classifying toposes Toposes as • a theory modulo ‘Morita-equivalence’ bridges Examples of ‘bridges’ Topological In this course we shall review each of these classical points of Galois theory view, and then present the more recent theory of topos-theoretic Theories of presheaf type ‘bridges’, which combines all of them to provide tools for making Topos-theoretic Fraïssé theorem toposes effective means for studying mathematical theories from Stone-type multiple points of view, relating and unifying theories with each dualities other and constructing ‘bridges’ across them. General remarks Future directions 3 / 66 An introduction to Grothendieck A bit of history toposes Olivia Caramello Introduction • Toposes were originally introduced by Alexander Grothendieck in the Toposes as early 1960s, in order to provide a mathematical underpinning for the generalized spaces ‘exotic’ cohomology theories needed in algebraic geometry. Every Sheaves on a topological space topological space gives rise to a topos and every topos in Sheaves on a site Geometric Grothendieck’s sense can be considered as a ‘generalized space’. morphisms Toposes as • At the end of the same decade, William Lawvere and Myles Tierney mathematical universes realized that the concept of Grothendieck topos also yielded an Classifying abstract notion of mathematical universe within which one could carry toposes out most familiar set-theoretic constructions, but which also, thanks to Toposes as bridges the inherent ‘flexibility’ of the notion of topos, could be profitably Examples of exploited to construct ‘new mathematical worlds’ having particular ‘bridges’ Topological properties. Galois theory • A few years later, the theory of classifying toposes added a further Theories of presheaf type fundamental viewpoint to the above-mentioned ones: a topos can be Topos-theoretic Fraïssé theorem seen not only as a generalized space or as a mathematical universe, Stone-type but also as a suitable kind of first-order theory (considered up to a dualities general notion of equivalence of theories). General remarks Future directions 4 / 66 An introduction to Grothendieck Toposes as unifying ‘bridges’ toposes Olivia Caramello Introduction Since the times of my Ph.D. studies I have been developing a Toposes as theory and a number of techniques which allow to effectively use generalized spaces toposes as unifying spaces. Sheaves on a topological space Sheaves on a site The key idea is that the possibility of representing a topos in a Geometric morphisms multitude of different ways can be effectively exploited for building Toposes as unifying ‘bridges’ between theories having an equivalent, or mathematical universes strictly related, mathematical content. Classifying toposes These ‘bridges’ allow effective and often deep transfers of Toposes as notions, ideas and results across the theories. bridges Examples of ‘bridges’ In spite of the number of applications in different fields obtained Topological throughout the last years, the potential of these methods has just Galois theory started to be explored. Theories of presheaf type In fact, ‘bridges’ have proved useful not only for connecting Topos-theoretic Fraïssé theorem different theories with each other, but also for working inside a Stone-type given mathematical theory and investigating it from a multiplicity dualities General remarks of different points of view. Future directions 5 / 66 An introduction to Grothendieck Presheaves on a topological space toposes Olivia Caramello Definition Introduction Let X be a topological space. A presheaf F on X consists of the data: Toposes as generalized (i) for every open subset U of X, a set F (U) and spaces (ii) for every inclusion V ⊆ U of open subsets of X, a function Sheaves on a topological space rU;V : F (U) ! F (V ) subject to the conditions Sheaves on a site Geometric • rU;U is the identity map F (U) ! F (U) and morphisms • if W ⊆ V ⊆ U are three open subsets, then rU;W = rV ;W ◦ rU;V . Toposes as mathematical The maps r are called restriction maps, and we sometimes write sj universes U;V V instead of (s), if s 2 (U). Classifying rU;V F toposes A morphism of presheaves F ! G on a topological space X is a Toposes as collection of maps (U) ! (U) which is compatible with respect to bridges F G restriction maps. Examples of ‘bridges’ Topological Remark Galois theory Categorically, a presheaf F on X is a functor F : O(X)op ! Set, where Theories of presheaf type O(X) is the poset category corresponding to the lattice of open sets of Topos-theoretic the topological space X (with respect to the inclusion relation). Fraïssé theorem A morphism of presheaves is then just a natural transformation Stone-type dualities between the corresponding functors. op General remarks So we have a category [O(X) ;Set] of presheaves on X. Future directions 6 / 66 An introduction to Grothendieck Sheaves on a topological space toposes Olivia Caramello Definition Introduction A sheaf F on a topological space X is a presheaf on X satisfying the Toposes as additional conditions generalized spaces Sheaves on a (i) if U is an open set, if fVi | i 2 Ig is an open covering of U, and if topological space s;t 2 (U) are elements such that sj = tj for all i, then s = t; Sheaves on a site F Vi Vi Geometric (ii) if U is an open set, if fVi | i 2 Ig is an open covering of U, and if we morphisms have elements s 2 (V ) for each i, with the property that for each Toposes as i F i mathematical i;j 2 I, si jV \V = sj jV \V , then there is an element s 2 F (U) universes i j i j (necessarily unique by (i)) such that sj = s for each i. Classifying Vi i toposes A morphism of sheaves is defined as a morphism of the underlying Toposes as bridges presheaves. Examples of ‘bridges’ Examples Topological Galois theory • the sheaf of continuous real-valued functions on any topological Theories of space presheaf type • the sheaf of regular functions on a variety Topos-theoretic Fraïssé theorem • the sheaf of differentiable functions on a differentiable manifold Stone-type • the sheaf of holomorphic functions on a complex manifold dualities General remarks In each of the above examples, the restriction maps of the sheaf are the Future directions usual set-theoretic restrictions of functions to a subset. 7 / 66 An introduction to Grothendieck Sheaves from a categorical point of view toposes Olivia Caramello Sheaves arising in Mathematics are often equipped with more structure Introduction than the mere set-theoretic one; for example, one may wish to consider Toposes as sheaves of modules (resp. rings, abelian groups, ...) on a topological generalized space X. spaces Sheaves on a The natural categorical way of looking at these notions is to consider topological space Sheaves on a site them as models of certain (geometric) theories in a category Sh(X) of Geometric morphisms sheaves of sets. Toposes as mathematical Remarks universes op Classifying • Categorically, a sheaf is a functor O(X) ! Set which satisfies toposes certain conditions expressible in categorical language entirely in Toposes as ( ) bridges terms of the poset category O X and of the usual notion of Examples of covering on it. The category Sh(X) of sheaves on a topological ‘bridges’ space X is thus a full subcategory of the category [O(X)op;Set] of Topological presheaves on X. Galois theory • Theories of Many important properties of topological spacesX can be naturally presheaf type formulated as (invariant) properties of the categories Sh(X) of Topos-theoretic sheaves of sets on the spaces.