An Introduction to Grothendieck Toposes

Home , Alexander Grothendieck, Grothendieck topology, Sheaf (mathematics), Subfunctor

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 ρU,V : F (U) → F (V ) subject to the conditions Sheaves on a site Geometric • ρU,U is the identity map F (U) → F (U) and morphisms • if W ⊆ V ⊆ U are three open subsets, then ρU,W = ρV ,W ◦ ρU,V . Toposes as mathematical The maps ρ are called restriction maps, and we sometimes write s| universes U,V V instead of (s), if s ∈ (U). Classifying ρU,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 {Vi | i ∈ I} is an open covering of U, and if topological space s,t ∈ (U) are elements such that s| = t| for all i, then s = t; Sheaves on a site F Vi Vi Geometric (ii) if U is an open set, if {Vi | i ∈ I} is an open covering of U, and if we morphisms have elements s ∈ (V ) for each i, with the property that for each Toposes as i F i mathematical i,j ∈ I, si |V ∩V = sj |V ∩V , then there is an element s ∈ F (U) universes i j i j (necessarily unique by (i)) such that s| = 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. Fraïssé theorem

Stone-type dualities These remarks led Grothendieck to introduce a signiﬁcant categorical General remarks generalization of the notion of sheaf, and hence the notion of Future directions Grothendieck topos. 8 / 66 An introduction to Grothendieck Limits and colimits in Sh(X) toposes

Olivia Caramello

Introduction

Toposes as generalized spaces Theorem Sheaves on a topological space op Sheaves on a site (i) The category Sh(X) is closed in [O(X) ,Set] under Geometric morphisms arbitrary (small) limits.

Toposes as op mathematical (ii) The associated sheaf functor a :[O(X) ,Set] → Sh(X) universes (having a right adjoint) preserves all (small) colimits. Classifying toposes

Toposes as • Part (i) follows from the fact that limits commute with limits, in bridges light of the characterization of sheaves in terms of limits. Examples of ‘bridges’ • From part (ii) it follows that Sh(X) has all small colimits, Topological Galois theory which are computed by applying the associated sheaf functor

Theories of to the colimit of the diagram considered with values in presheaf type [O(X)op,Set]. Topos-theoretic Fraïssé theorem

Stone-type dualities

General remarks

Future directions 9 / 66 An introduction to Grothendieck Adjunctions induced by points toposes

Olivia Caramello

Introduction Let x be a point of a topological space X. Toposes as generalized spaces Deﬁnition Sheaves on a topological space Let A be a set. Then the skyscraper sheaf Skyx (A) of A at x is the Sheaves on a site sheaf on X deﬁned as Geometric morphisms • Sky (A)(U) = A if x ∈ U Toposes as x mathematical universes • Skyx (A)(U) = 1 = {∗} if x ∈/ U Classifying toposes and in the obvious way on arrows. Toposes as The assignment A → Sky (A) is clearly functorial. bridges x

Examples of ‘bridges’ Theorem

Topological The stalk functor Stalkx : Sh(X) → Set at x is left adjoint to the Galois theory skyscraper functor Skyx : Set → Sh(X). Theories of presheaf type Topos-theoretic In fact, as we shall see later in the course, points in topos theory Fraïssé theorem are deﬁned as suitable kinds of functors (more precisely, colimit Stone-type dualities and ﬁnite-limit preserving ones). General remarks

Future directions 10 / 66 An introduction to Grothendieck Open sets as subterminal objects toposes

Olivia Caramello

Introduction Since limits in a category Sh(X) are computed as in the category Toposes as of presheaves [ (X)op,Set], a subobject of a sheaf F in Sh(X) is generalized O spaces just a subsheaf, that is a subfunctor which is a sheaf. Sheaves on a topological space Notice that a subfunctor S ⊆ F is a sheaf if and only if for every Sheaves on a site Geometric open covering {U ⊆ U | i ∈ I} and every element x ∈ F(U), morphisms i ∈ ( ) | ∈ ( ) Toposes as x S U if and only if x Ui S Ui . mathematical universes Deﬁnition Classifying toposes In a category with a terminal object, a subterminal object is an Toposes as object whose unique arrow to the terminal object is a bridges monomorphism. Examples of ‘bridges’ Topological Theorem Galois theory Let X be a topological space. Then we have a frame isomorphism Theories of presheaf type ∼ Topos-theoretic SubSh(X )(1) = O(X) . Fraïssé theorem

Stone-type dualities between the subterminal objects of Sh(X) and the open sets of X. General remarks

Future directions 11 / 66 An introduction to Grothendieck Sieves toposes

Olivia Caramello In order to ‘categorify’ the notion of sheaf of a topological space, the ﬁrst step is to introduce an abstract notion of covering (of an Introduction

Toposes as object by a family of arrows to it) in a category. generalized spaces Deﬁnition Sheaves on a topological space Sheaves on a site • Given a category C and an object c ∈ Ob(C ), a presieve P in Geometric morphisms C on c is a collection of arrows in C with codomain c. Toposes as mathematical • Given a category C and an object c ∈ Ob(C ), a sieve S in C universes on c is a collection of arrows in C with codomain c such that Classifying toposes

Toposes as f ∈ S ⇒ f ◦ g ∈ S bridges

Examples of ‘bridges’ whenever this composition makes sense. Topological • We say that a sieve S is generated by a presieve P on an Galois theory

Theories of object c if it is the smallest sieve containing it, that is if it is the presheaf type collection of arrows to c which factor through an arrow in P. Topos-theoretic Fraïssé theorem

Stone-type If S is a sieve on c and h : d → c is any arrow to c, then dualities ∗ General remarks h (S) := {g | cod(g) = d, h ◦ g ∈ S} Future directions is a sieve on d. 12 / 66 An introduction to Grothendieck Grothendieck topologies toposes

Olivia Caramello

Introduction

Toposes as generalized Deﬁnition spaces Sheaves on a • topological space A Grothendieck topology on a category C is a function J Sheaves on a site which assigns to each object c of C a collection J(c) of Geometric morphisms sieves on c in such a way that Toposes as mathematical (i)(maximality axiom) the maximal sieve Mc = {f | cod(f ) = c} is universes in J(c); Classifying (ii)(stability axiom) if S ∈ J(c), then f ∗(S) ∈ J(d) for any arrow toposes f : d → c; Toposes as bridges (iii)(transitivity axiom) if S ∈ J(c) and R is any sieve on c such that Examples of f ∗(R) ∈ J(d) for all f : d → c in S, then R ∈ J(c). ‘bridges’ Topological The sieves S which belong to J(c) for some object c of C are Galois theory said to be J-covering. Theories of presheaf type • A site is a pair (C ,J) where C is a small category and J is a Topos-theoretic Fraïssé theorem Grothendieck topology on C .

Stone-type dualities

General remarks

Future directions 13 / 66 An introduction to Grothendieck Examples of Grothendieck topologies toposes Olivia Caramello • For any (small) category C , the trivial topology on C is the Introduction Grothendieck topology in which the only sieve covering an Toposes as object c is the maximal sieve M . generalized c spaces • The dense topology D on a category C is deﬁned by: for a Sheaves on a topological space sieve S, Sheaves on a site Geometric morphisms

Toposes as S ∈ D(c) if and only if for any f : d → c there exists mathematical universes g : e → d such that f ◦ g ∈ S .

Classifying toposes If C satisﬁes the right Ore condition i.e. the property that any Toposes as two arrows f : d → c and g : e → c with a common codomain bridges c can be completed to a commutative square Examples of ‘bridges’

Topological • / d Galois theory

Theories of f presheaf type g Topos-theoretic e / c Fraïssé theorem Stone-type then the dense topology on C specializes to the atomic dualities topology on C i.e. the topology Jat deﬁned by: for a sieve S, General remarks Future directions S ∈ Jat (c) if and only if S 6= /0 . 14 / 66 An introduction to Grothendieck Examples of Grothendieck topologies toposes Olivia Caramello • If X is a topological space, the usual notion of covering in Topology Introduction gives rise to the following Grothendieck topology JO(X ) on the poset Toposes as category O(X): for a sieve S = {Ui ,→ U | i ∈ I} on U ∈ Ob(O(X)), generalized spaces Sheaves on a S ∈ JO(X )(U) if and only if ∪ Ui = U . topological space i∈I Sheaves on a site Geometric morphisms • More generally, given a frame (or complete Heyting algebra) H, we Toposes as can deﬁne a Grothendieck topology JH , called the canonical mathematical universes topology on H, by:

Classifying toposes {ai | i ∈ I} ∈ JH (a) if and only if ∨ ai = a . i∈I Toposes as bridges • Examples of Given a small category of topological spaces which is closed under ‘bridges’ finite limits and under taking open subspaces, one may define the Topological open cover topology on it by specifying as basis the collection of Galois theory open embeddings {Yi ,→ X | i ∈ I} such that ∪Yi = X. Theories of i∈I presheaf type • The Zariski topology on the opposite of the category Rngf.g. of Topos-theoretic Fraïssé theorem finitely generated commutative rings with unit is defined by: for any

Stone-type cosieve S in Rngf.g. on an object A, S ∈ Z(A) if and only if S dualities { : → | ≤ ≤ } contains a ﬁnite family ξi A Afi 1 i n of canonical maps General remarks : → { ,..., } ξi A Afi in Rngf.g. where f1 fn is a set of elements of A Future directions which is not contained in any proper ideal of A. 15 / 66 An introduction to Grothendieck Sheaves on a site toposes Olivia Caramello Deﬁnition op Introduction • A presheaf on a (small) category C is a functor P : C → Set. op Toposes as • Let P : C → Set be a presheaf on C and S be a sieve on an generalized spaces object c of C . Sheaves on a topological space A matching family for S of elements of P is a function which Sheaves on a site assigns to each arrow f : d → c in S an element xf ∈ P(d) in Geometric morphisms such a way that Toposes as mathematical P(g)(x ) = x for all g : e → d . universes f f ◦g

Classifying toposes An amalgamation for such a family is a single element x ∈ P(c) such that Toposes as bridges

Examples of P(f )(x) = xf for all f in S . ‘bridges’

Topological Galois theory Remark Theories of For any covering family F = {U ⊆ U | i ∈ I} in a topological space presheaf type i X and any presheaf F on X, giving a family of elements Topos-theoretic Fraïssé theorem si ∈ F (Ui ) such that for any i,j ∈ I si |Ui ∩Uj = sj |Ui ∩Uj is equivalent Stone-type to giving a family of elements {sW ∈ F (W ) | W ∈ SF } such that for dualities 0 any open set W ⊆ W , sW |W 0 = sW 0 , where SF is the sieve General remarks generated by F. In other words, it is the same as giving a Future directions matching family for SF of elements of F . 16 / 66 An introduction to Grothendieck Sheaves on a site toposes

Olivia Caramello

Introduction

Toposes as generalized spaces Sheaves on a topological space • Given a site ( ,J), a presheaf on is a J-sheaf if every Sheaves on a site C C Geometric matching family for any J-covering sieve on any object of morphisms C

Toposes as has a unique amalgamation. mathematical universes • The category Sh( ,J) of sheaves on the site ( ,J) is the full Classifying C C toposes subcategory of [C op,Set] on the presheaves which are Toposes as bridges J-sheaves.

Examples of ‘bridges’ • A Grothendieck topos is any category equivalent to the Topological Galois theory category of sheaves on a site.

Theories of presheaf type

Topos-theoretic Fraïssé theorem

Stone-type dualities

General remarks

Future directions 17 / 66 An introduction to Grothendieck Examples of toposes toposes

Olivia Caramello

Introduction The following examples show that toposes can be naturally Toposes as generalized attached to mathematical notions as different as (small) spaces Sheaves on a categories, topological spaces, or groups. topological space Sheaves on a site Geometric Examples morphisms Toposes as • op mathematical For any (small) category C , [C ,Set] is the category of universes sheaves Sh(C ,T ) where T is the trivial topology on C . Classifying toposes • For any topological space X, Sh(O(X),JO(X )) is equivalent to Toposes as bridges the usual category Sh(X) of sheaves on X. Examples of • For any (topological) group G, the category BG = Cont(G) of ‘bridges’ continuous actions of G on discrete sets is a Grothendieck Topological Galois theory topos (equivalent, as we shall see, to the category Theories of presheaf type Sh(Contt(G),Jat) of sheaves on the full subcategory Topos-theoretic Contt(G) on the non-empty transitive actions with respect to Fraïssé theorem the atomic topology). Stone-type dualities

General remarks

Future directions 18 / 66 An introduction to Grothendieck Basic properties of Grothendieck toposes toposes Olivia Caramello Grothendieck toposes satisfy all the categorical properties that

Introduction one might hope for:

Toposes as generalized Theorem spaces Sheaves on a Let (C ,J) be a site. Then topological space op Sheaves on a site • the inclusion Sh(C ,J) ,→ [C ,Set] has a left adjoint Geometric op morphisms a :[C ,Set] → Sh(C ,J) (called the associated sheaf functor), Toposes as mathematical which preserves ﬁnite limits. universes • The category Sh(C ,J) has all (small) limits, which are Classifying op toposes preserved by the inclusion functor Sh(C ,J) ,→ [C ,Set]; in

Toposes as particular, limits are computed pointwise and the terminal bridges op object 1Sh(C ,J) of Sh(C ,J) is the functor T : C → Set Examples of ‘bridges’ sending each object c ∈ Ob(C ) to the singleton {∗}. op Topological • The associated sheaf functor a :[C ,Set] → Sh(C ,J) Galois theory ( , ) Theories of preserves colimits; in particular, Sh C J has all (small) presheaf type colimits. Topos-theoretic Fraïssé theorem • The category Sh(C ,J) has exponentials, which are op Stone-type constructed as in the topos [C ,Set]. dualities • The category Sh(C ,J) has a subobject classiﬁer. General remarks • The category Sh(C ,J) has a separating set of objects. Future directions 19 / 66 An introduction to Grothendieck Geometric morphisms toposes Olivia Caramello The natural, topologically motivated, notion of morphism of

Introduction Grothendieck toposes is that of geometric morphism. The natural Toposes as notion of morphism of geometric morphisms if that of geometric generalized spaces transformation. Sheaves on a topological space Sheaves on a site Deﬁnition Geometric morphisms (i) Let E and F be toposes. A geometric morphism f : E → F Toposes as mathematical consists of a pair of functors f∗ : E → F (the direct image of universes f ) and f ∗ : F → E (the inverse image of f ) together with an Classifying ∗ ∗ toposes adjunction f a f∗, such that f preserves ﬁnite limits.

Toposes as bridges (ii) Let f and g : E → F be geometric morphisms. A geometric

Examples of transformation α : f → g is deﬁned to be a natural ‘bridges’ transformation a : f ∗ → g∗. Topological Galois theory (iii)A point of a topos E is a geometric morphism Set → E . Theories of presheaf type

Topos-theoretic • Grothendieck toposes and geometric morphisms between Fraïssé theorem them form a 2-category. Stone-type dualities • Given two toposes E and F , geometric morphisms from E to General remarks F and geometric transformations between them form a Future directions category, denoted by Geom(E ,F ). 20 / 66 An introduction to Grothendieck Examples of geometric morphisms toposes

Olivia Caramello • A continuous function f : X → Y between topological spaces Introduction gives rise to a geometric morphism Sh(f ): Sh(X) → Sh(Y ). Toposes as generalized The direct image Sh(f ) sends a sheaf F ∈ Ob(Sh(X)) to the spaces ∗ −1 Sheaves on a sheaf Sh(f )∗(F) deﬁned by Sh(f )∗(F)(V ) = F(f (V )) for topological space ∗ Sheaves on a site any open subset V of Y . The inverse image Sh(f ) acts on Geometric morphisms étale bundles over Y by sending an étale bundle p : E → Y to Toposes as mathematical the étale bundle over X obtained by pulling back p along universes f : X → Y . Classifying toposes • Every Grothendieck topos E has a unique geometric Toposes as → bridges morphism E Set. The direct image is the global sections

Examples of functor Γ: E → Set, sending an object e ∈ E to the set ‘bridges’ HomE (1E ,e), while the inverse image functor ∆ : Set → E Topological Galois theory F sends a set S to the coproduct 1E . Theories of presheaf type s∈S Topos-theoretic • For any site (C ,J), the pair of functors formed by the Fraïssé theorem op Stone-type inclusion Sh(C ,J) ,→ [C ,Set] and the associated sheaf dualities functor a :[C op,Set] → Sh(C ,J) yields a geometric General remarks morphism i : Sh(C ,J) → [C op,Set]. Future directions 21 / 66 An introduction to Grothendieck Geometric morphisms as ﬂat functors I toposes

Olivia Caramello

Introduction Toposes as Theorem generalized spaces Let C be a small category and E be a locally small cocomplete Sheaves on a topological space category. Then, for any functor A : C → E the functor Sheaves on a site op Geometric RA : E → [C ,Set] deﬁned for each e ∈ Ob(E ) and c ∈ Ob(C ) by: morphisms

Toposes as mathematical RA(e)(c) = HomE (A(c),e) universes Classifying op toposes has a left adjoint − ⊗C A :[C ,Set] → E . Toposes as bridges Definition Examples of ‘bridges’ • A functor A : C → E from a small category C to a Topological Galois theory Grothendieck topos E is said to be flat if the functor op Theories of − ⊗C A :[C ,Set] → E preserves finite limits. presheaf type

Topos-theoretic • The full subcategory of [C ,E ] on the ﬂat functors will be Fraïssé theorem denoted by Flat(C ,E ). Stone-type dualities

General remarks

Future directions 22 / 66 An introduction to Grothendieck Geometric morphisms as ﬂat functors II toposes

Olivia Caramello

Introduction

Toposes as generalized spaces Theorem Sheaves on a topological space Let C be a small category and E be a Grothendieck topos. Then Sheaves on a site we have an equivalence of categories Geometric morphisms op Toposes as Geom( ,[ ,Set]) ' Flat( , ) mathematical E C C E universes Classifying (natural in E ), which sends toposes Toposes as • a ﬂat functor A : C → E to the geometric morphism bridges op Examples of E → [C ,Set] determined by the functors RA and − ⊗C A, ‘bridges’ and Topological op Galois theory • a geometric morphism f : E → [C ,Set] to the ﬂat functor Theories of ∗ ∗ op presheaf type given by the composite f ◦ y of f :[C ,Set] → E with the op Topos-theoretic Yoneda embedding y : C → [C ,Set]. Fraïssé theorem

Stone-type dualities

General remarks

Future directions 23 / 66 An introduction to Grothendieck Geometric morphisms to Sh( ,J) I toposes C

Olivia Caramello

Introduction

Toposes as generalized spaces Sheaves on a Deﬁnition topological space Sheaves on a site Let E be a Grothendieck topos. Geometric morphisms • A family {fi : ai → a | i ∈ I} of arrows in E with common Toposes as mathematical codomain is said to be epimorphic if for any pair of arrows universes g,h : a → b with domain a, g = h if and only if g ◦ fi = h ◦ fi for Classifying toposes all i ∈ I. Toposes as bridges • If (C ,J) is a site, a functor F : C → E is said to be Examples of J-continuous if it sends J-covering sieves to epimorphic ‘bridges’ families. Topological Galois theory

Theories of The full subcategory of Flat(C ,E ) on the J-continuous ﬂat presheaf type functors will be denoted by FlatJ (C ,E ). Topos-theoretic Fraïssé theorem

Stone-type dualities

General remarks

Future directions 24 / 66 An introduction to Grothendieck Geometric morphisms to Sh( ,J) II toposes C Olivia Caramello Theorem Introduction For any site (C ,J) and Grothendieck topos E , the above-mentioned Toposes as generalized equivalence between geometric morphisms and ﬂat functors restricts spaces to an equivalence of categories Sheaves on a topological space Sheaves on a site Geometric Geom(E ,Sh(C ,J)) ' FlatJ (C ,E ) morphisms

Toposes as mathematical natural in E . universes Classifying Sketch of proof. toposes

Toposes as Appeal to the previous theorem bridges

Examples of • identifying the geometric morphisms E → Sh(C ,J) with the ‘bridges’ geometric morphisms E → [C op,Set] which factor through the Topological op Galois theory canonical geometric inclusion Sh(C ,J) ,→ [C ,Set], and

Theories of • using the characterization of such morphisms as the geometric presheaf type morphisms f : E → [C op,Set] such that the composite f ∗ ◦ y of Topos-theoretic ∗ Fraïssé theorem the inverse image functor f of f with the Yoneda embedding op Stone-type y : C → [C ,Set] sends J-covering sieves to colimits in E dualities (equivalently, to epimorphic families in E ). General remarks

Future directions 25 / 66 An introduction to Grothendieck Flat = filtering toposes Olivia Caramello Definition A functor F : C → E from a small category C to a Grothendieck Introduction topos E is said to be filtering if it satisfies the following conditions: Toposes as (a) For any object E of E there exist an epimorphic family generalized spaces {ei : Ei → E | i ∈ I} in E and for each i ∈ I an object bi of C and Sheaves on a a generalized element Ei → F(bi ) in E . topological space (b) For any two objects c and d in C and any generalized element Sheaves on a site hx,yi : E → F(c) × F(d) in E there is an epimorphic family Geometric morphisms {ei : Ei → E | i ∈ I} in E and for each i ∈ I an object bi of C with arrows ui : bi → c and vi : bi → d in C and a generalized Toposes as mathematical element zi : Ei → F(bi ) in E such that hF(ui ),F(vi )i ◦ zi = universes hx,yi ◦ ei for all i ∈ I. (c) For any two parallel arrows u,v : d → c in C and any Classifying toposes generalized element x : E → F(d) in E for which F(u) ◦ x = F(v) ◦ x, there is an epimorphic family Toposes as bridges {ei : Ei → E | i ∈ I} in E and for each i ∈ I an arrow wi : bi → d and a generalized element yi : Ei → F(bi ) such that Examples of u ◦ w = v ◦ w and F(w ) ◦ y = x ◦ e for all i ∈ I. ‘bridges’ i i i i i

Topological Galois theory Theorem A functor F : C → E from a small category C to a Grothendieck Theories of topos E is ﬂat if and only if it is ﬁltering. presheaf type

Topos-theoretic Remarks Fraïssé theorem • For any small category C , a functor P : C → Set is filtering if Stone-type and only if its category of elements R P is a filtered category dualities (equivalently, if it is a filtered colimit of representables). General remarks • For any small cartesian category C , a functor C → E is flat if Future directions and only if it preserves finite limits. 26 / 66 An introduction to Grothendieck Morphisms and comorphisms of sites toposes Olivia Caramello Geometric morphisms can be naturally induced by functors

Introduction between sites satisfying appropriate properties:

Toposes as generalized Definition spaces Sheaves on a • A morphism of sites ( ,J) → ( ,K ) is a functor F : → topological space C D C D 0 0 Sheaves on a site such that the composite l ◦ F, where l is the canonical Geometric morphisms functor D → Sh(D,K ), is flat and J-continuous. If C and D Toposes as mathematical have finite limits then F is a morphism of sites if and only if it universes preserves finite limits and is cover-preserving. Classifying toposes • A comorphism of sites (D,K ) → (C ,J) is a functor π : D → C Toposes as bridges which is cover-reflecting (in the sense that for any d ∈ D and

Examples of any J-covering sieve S on π(d) there is a K -covering sieve R ‘bridges’ on d such that π(R) ⊆ S). Topological Galois theory

Theories of presheaf type Theorem Topos-theoretic • ( , ) → ( , ) Fraïssé theorem Every morphism of sites C J D K induces a geometric

Stone-type morphism Sh(D,K ) → Sh(C ,J). dualities General remarks • Every comorphism of sites (D,K ) → (C ,J) induces a Future directions geometric morphism Sh(D,K ) → Sh(C ,J). 27 / 66 An introduction to Grothendieck Toposes as mathematical universes toposes

Olivia Caramello

Introduction W. Lawvere and M. Tierney discovered that a topos could not only Toposes as generalized be seen as a generalized space, but also as a mathematical spaces Sheaves on a universe in which one can do mathematics similarly to how one topological space Sheaves on a site does it in the classical context of sets, with the only important Geometric morphisms exception that one must argue constructively. In fact, the internal Toposes as logic of a topos, captured to a great extent by its subobject mathematical universes classiﬁer, is in general intuitionistic. Classifying toposes Amongst other things, this view of toposes as mathematical Toposes as bridges universes paved the way for: Examples of ‘bridges’ • Exploiting the inherent ‘ﬂexibility’ of the notion of topos to Topological construct ‘new mathematical worlds’ having particular Galois theory

Theories of properties. presheaf type

Topos-theoretic Fraïssé theorem • Considering models of any kind of (ﬁrst-order) mathematical

Stone-type theory not just in the classical set-theoretic setting, but inside dualities every topos, and hence ‘relativise’ Mathematics. General remarks

Future directions 28 / 66 An introduction to Grothendieck Subobjects in a Grothendieck topos toposes

Olivia Caramello

Introduction Since limits in a topos Sh(C ,J) are computed as in the presheaf Toposes as op generalized topos [C ,Set], a subobject of a sheaf F in Sh(C ,J) is just a spaces Sheaves on a subsheaf, that is a subfunctor which is a sheaf. topological space Sheaves on a site 0 Geometric Notice that a subfunctor F ⊆ F is a sheaf if and only if for every morphisms J-covering sieve S and every element x ∈ F(c), x ∈ F 0(c) if and Toposes as 0 mathematical only if F(f )(x) ∈ F (dom(f )) for every f ∈ S. universes

Classifying toposes Theorem Toposes as • bridges For any Grothendieck topos E and any object a of E , the Examples of poset SubE (a) of all subobjects of a in E is a complete ‘bridges’ Heyting algebra. Topological Galois theory • For any arrow f : a → b in a Grothendieck topos E , the Theories of ∗ presheaf type pullback functor f : SubE (b) → SubE (a) has both a left Topos-theoretic adjoint ∃f : SubE (a) → SubE (b) and a right adjoint Fraïssé theorem ∀f : SubE (a) → SubE (b). Stone-type dualities

General remarks

Future directions 29 / 66 An introduction to Grothendieck The Heyting operations on subobjects toposes Olivia Caramello Proposition Introduction The collection SubSh(C ,J)(E) of subobjects of an object E in Toposes as generalized Sh(C ,J) has the structure of a complete Heyting algebra with spaces ≤ ∈ Sheaves on a respect to the natural ordering A B if and only if for every c C , topological space A(c) ⊆ B(c). We have that Sheaves on a site Geometric morphisms • (A ∧ B)(c) = A(c) ∩ B(c) for any c ∈ C ; Toposes as • (A ∨ B)(c) = {x ∈ E(c) | {f : d → c | E(f )(x) ∈ A(d) ∪ B(d)} mathematical universes ∈ J(c)} for any c ∈ C ; Classifying toposes (the inﬁnitary analogue of this holds)

Toposes as • (A⇒B)(c) = {x ∈ E(c) | for every f : d → c,E(f )(x) ∈ A(d) bridges implies E(f )(x) ∈ B(d)} for any c ∈ C . Examples of • ‘bridges’ the bottom subobject 0 E is given by the embedding into E

Topological of the initial object 0 of Sh(C ,J) (given by: 0(c) = /0 if /0 ∈/ J(c) Galois theory and 0(c) = {∗} if /0 ∈ J(c)); Theories of • presheaf type the top subobject is the identity one.

Topos-theoretic Fraïssé theorem Remark Stone-type dualities From the Yoneda Lemma it follows that the subobject classiﬁer Ω in General remarks Sh(C ,J) has the structure of an internal Heyting algebra in Future directions Sh(C ,J). 30 / 66 An introduction to Grothendieck The interpretation of quantiﬁers toposes Olivia Caramello Let φ : E → F be a morphism in Sh(C ,J). Introduction • The pullback functor Toposes as generalized ∗ spaces φ : SubSh(C ,J)(F) → SubSh(C ,J)(E) Sheaves on a topological space ∗ −1 Sheaves on a site is given by: φ (B)(c) = φ(c) (B(c)) for any subobject Geometric morphisms B F and any c ∈ C . Toposes as mathematical • The left adjoint universes

Classifying toposes ∃φ : SubSh(C ,J)(E) → SubSh(C ,J)(F) Toposes as bridges is given by: ∃φ (A)(c) = {y ∈ E(c) | {f : d → c | (∃a ∈ Examples of ‘bridges’ A(d))(φ(d)(a) = E(f )(y))} ∈ J(c)} Topological for any subobject A E and any c ∈ C . Galois theory • The right adjoint Theories of presheaf type Topos-theoretic ∀φ : SubSh(C ,J)(E) → SubSh(C ,J)(F) Fraïssé theorem

Stone-type dualities is given by ∀φ (A)(c) = {y ∈ E(c) | for all f : d → −1 General remarks c,φ(d) (E(f )(y)) ⊆ A(d)} Future directions for any subobject A E and any c ∈ C . 31 / 66 An introduction to Grothendieck Model theory in toposes toposes Olivia Caramello We can consider models of arbitrary ﬁrst-order theories in any

Introduction Grothendieck topos, thanks to the rich categorical structure

Toposes as present on it. generalized spaces Sheaves on a The notion of model of a ﬁrst-order theory in a topos is a natural topological space Sheaves on a site generalization of the usual Taskian deﬁnition of a (set-based) Geometric morphisms model of the theory.

Toposes as mathematical Let Σ be a (possibly multi-sorted) first-order signature. A structure universes M over Σ in a category E with finite products is specified by the Classifying toposes following data: Toposes as bridges • any sort A of Σ is interpreted by an object MA of E Examples of • any function symbol f : A1,...,An → B of Σ is interpreted as an ‘bridges’ arrow Mf : MA1 × ··· × MAn → MB in E Topological Galois theory • any relation symbol R A1,...,An of Σ is interpreted as a Theories of subobject MR MA × ··· × MA in presheaf type 1 n E Topos-theoretic Any formula {~x . φ} in a given context ~x over Σ is interpreted as a Fraïssé theorem subobject [[~x . φ]] MA × ··· × MAn defined recursively on the Stone-type M 1 dualities structure of the formula. General remarks A model of a theory T over a first-order signature Σ is a structure Future directions over Σ in which all the axioms of T are satisfied. 32 / 66 An introduction to Grothendieck Geometric theories toposes

Olivia Caramello

Introduction

Toposes as Definition generalized spaces A geometric theory T is a theory over a first-order signature Σ Sheaves on a ( ` ) topological space whose axioms can be presented in the form φ ~x ψ , where φ Sheaves on a site and ψ are geometric formulae, that is formulae in the context ~x Geometric morphisms built up from atomic formulae over Σ by only using finitary Toposes as mathematical conjunctions, infinitary disjunctions and existential quantifications. universes Classifying Remark toposes

Toposes as Inverse image functors of geometric morphisms of toposes bridges always preserve models of a geometric theory (but in general not Examples of ‘bridges’ those of an arbitrary ﬁrst-order theory).

Topological Galois theory Most of the first-order theories naturally arising in Mathematics Theories of presheaf type are geometric; anyway, if a finitary first-order theory is not

Topos-theoretic geometric, one can always canonically associate with it a Fraïssé theorem geometric theory, called its Morleyization, having the same Stone-type dualities set-based models. General remarks

Future directions 33 / 66 An introduction to Grothendieck Classifying toposes toposes

Olivia Caramello

Introduction It was realized in the seventies (thanks to the work of several Toposes as people, notably including W. Lawvere, A. Joyal, G. Reyes and M. generalized spaces Makkai) that: Sheaves on a topological space Sheaves on a site Geometric • Every geometric theory T has a classifying topos ET which is morphisms characterized by the following representability property: for Toposes as mathematical any Grothendieck topos we have an equivalence of universes E

Classifying categories toposes Geom(E ,ET) ' T-mod(E ) Toposes as bridges natural in E , where Examples of ‘bridges’ - Geom(E ,ET) is the category of geometric morphisms E → ET Topological and Galois theory - T-mod(E ) is the category of T-models in E . Theories of presheaf type Topos-theoretic • The classifying topos of a geometric theory can be Fraïssé theorem T canonically built as the category Sh( ,J ) of sheaves on Stone-type CT T dualities the syntactic site (CT,JT) of T. General remarks

Future directions 34 / 66 An introduction to Grothendieck The syntactic category of a geometric theory toposes Olivia Caramello Deﬁnition (Makkai and Reyes 1977)

Introduction • Let T be a geometric theory over a signature Σ. The syntactic Toposes as category C of has as objects the ‘renaming’-equivalence generalized T T spaces classes of geometric formulae-in-context {~x . φ} over Σ and as Sheaves on a arrows {~x . φ} → {~y . ψ} (where the contexts ~x and ~y are topological space Sheaves on a site supposed to be disjoint without loss of generality) the Geometric -provable-equivalence classes [θ] of geometric formulae morphisms T θ(~x,~y) which are T-provably functional i.e. such that the Toposes as mathematical sequents universes (φ `~x (∃y)θ), Classifying (θ `~x,~y φ ∧ ψ), and toposes ((θ ∧ θ[~z/~y]) `~x,~y,~z (~y =~z)) Toposes as bridges are provable in T. Examples of • The composite of two arrows ‘bridges’

Topological [θ] [γ] Galois theory {~x . φ} / {~y . ψ} / {~z . χ} Theories of presheaf type is defined as the T-provable-equivalence class of the formula Topos-theoretic (∃~y)θ ∧ γ. Fraïssé theorem • The identity arrow on an object {~x . φ} is the arrow Stone-type dualities [φ∧x~0=~x] General remarks {~x . φ} / {~x0 . φ[~x0/~x]} Future directions 35 / 66 An introduction to Grothendieck The syntactic site toposes Olivia Caramello On the syntactic category of a geometric theory it is natural to put Introduction the Grothendieck topology defined as follows: Toposes as generalized Definition spaces Sheaves on a The syntactic topology J on the syntactic category C of a topological space T T Sheaves on a site geometric theory T is the geometric topology on it; in particular, Geometric morphisms ~ ~ Toposes as a small family {[θi ]: {xi . φi } → {y . ψ}} in CT is JT-covering mathematical universes if and only if Classifying toposes the sequent (ψ `~ ∨(∃~xi )θi ) is provable in . Toposes as y T bridges i∈I

Examples of ‘bridges’

Topological This notion is instrumental for identifying the models of the theory Galois theory T in any geometric category C (and in particular in any Theories of presheaf type Grothendieck topos) as suitable functors deﬁned on the syntactic Topos-theoretic category C with values in C ; indeed, these are precisely the Fraïssé theorem T J -continuous cartesian functors C → C . So if C is a Stone-type T T dualities Grothendieck topos they correspond precisely to the geometric General remarks morphisms from C to Sh(CT,JT). This topos therefore classiﬁes Future directions T. 36 / 66 An introduction to Grothendieck Morita equivalence toposes

Olivia Caramello

Introduction

Toposes as generalized spaces Sheaves on a topological space • Two mathematical theories are said to be Morita-equivalent if Sheaves on a site Geometric have the same classifying topos (up to equivalence): this morphisms means that they have equivalent categories of models in Toposes as mathematical every Grothendieck topos E , naturally in E . universes

Classifying toposes • Every Grothendieck topos is the classifying topos of some Toposes as geometric theory (and in fact, of inﬁnitely many theories). bridges

Examples of ‘bridges’ • So a Grothendieck topos can be seen as a canonical

Topological representative of equivalence classes of theories modulo Galois theory Morita-equivalence. Theories of presheaf type

Topos-theoretic Fraïssé theorem

Stone-type dualities

General remarks

Future directions 37 / 66 An introduction to Grothendieck Toposes as bridges toposes

Olivia Caramello • The notion of Morita-equivalence is ubiquitous in Introduction Mathematics; indeed, it formalizes in many situations the Toposes as generalized feeling of ‘looking at the same thing in different ways’, or spaces Sheaves on a ‘constructing a mathematical object through different topological space Sheaves on a site methods’. Geometric morphisms • In fact, many important dualities and equivalences in Toposes as mathematical Mathematics can be naturally interpreted in terms of universes Morita-equivalences. Classifying toposes • On the other hand, Topos Theory itself is a primary source of Toposes as bridges Morita-equivalences. Indeed, different representations of the Examples of same topos can be interpreted as Morita-equivalences ‘bridges’ between different mathematical theories. Topological Galois theory • Any two theories which are bi-interpretable in each other are Theories of presheaf type Morita-equivalent but, very importantly, the converse does Topos-theoretic not hold. Fraïssé theorem

Stone-type • We can expect most of the categorical equivalences between dualities categories of set-based models of geometric theories to lift to General remarks Morita equivalences. Future directions 38 / 66 An introduction to Grothendieck Toposes as bridges toposes Olivia Caramello • In the topos-theoretic study of theories, the latter are

Introduction represented by sites (of deﬁnition of their classifying topos or of some other topos naturally attached to them). Toposes as generalized • spaces The existence of theories which are Morita-equivalent to each Sheaves on a other translates into the existence of different sites of deﬁnition topological space (or, more generally, presentations) for the same Grothendieck Sheaves on a site Geometric topos. morphisms • Toposes as Grothendieck toposes can be effectively used as ‘bridges’ for mathematical transferring notions, properties and results across different universes Morita-equivalent theories: Classifying toposes 1 E ' E 0 Toposes as T T bridges

Examples of 0 ‘bridges’ T T Topological Galois theory • The transfer of information takes place by expressing Theories of topos-theoretic invariants in terms of the different sites of presheaf type deﬁnition (or, more generally, presentations) for the given Topos-theoretic Fraïssé theorem topos. Stone-type • As such, different properties (resp. constructions) arising in dualities the context of theories classiﬁed by the same topos are seen General remarks to be different manifestations of a unique property (resp. Future directions construction) lying at the topos-theoretic level. 39 / 66 An introduction to Grothendieck Toposes as bridges toposes

Olivia Caramello • This methodology is technically effective because the Introduction relationship between a topos and its representations is often Toposes as generalized very natural, enabling us to easily transfer invariants across spaces different representations (and hence, between different Sheaves on a topological space theories). Sheaves on a site Geometric morphisms • On the other hand, the ‘bridge’ technique is highly non-trivial, in Toposes as mathematical the sense that it often yields deep and surprising results. This universes is due to the fact that a given invariant can manifest itself in Classifying toposes signiﬁcanly different ways in the context of different

Toposes as presentations. bridges

Examples of • The level of generality represented by topos-theoretic ‘bridges’ invariants is ideal to capture several important features of Topological Galois theory mathematical theories and constructions. Indeed, many Theories of important invariants of mathematical structures are actually presheaf type invariants of toposes (think for instance of cohomology or Topos-theoretic Fraïssé theorem homotopy groups) and topos-theoretic invariants considered on Stone-type the classifying topos of a geometric theory often translate dualities ET T General remarks into interesting logical (i.e. syntactic or semantic) properties of Future directions T. 40 / 66 An introduction to Grothendieck Toposes as bridges toposes Olivia Caramello • The fact that topos-theoretic invariants specialize to important Introduction properties or constructions of natural mathematical interest is Toposes as a clear indication of the centrality of these concepts in generalized spaces Mathematics. In fact, whatever happens at the level of toposes Sheaves on a topological space has ‘uniform’ ramiﬁcations in Mathematics as a whole: for Sheaves on a site instance Geometric morphisms

Toposes as mathematical universes

Classifying toposes

Toposes as bridges

Examples of ‘bridges’

Topological Galois theory

Theories of presheaf type

Topos-theoretic Fraïssé theorem This picture represents the lattice structure on the collection of Stone-type dualities the subtoposes of a topos E inducing lattice structures on the General remarks collection of ‘quotients’ of geometric theories T, S, R classiﬁed Future directions by it. 41 / 66 An introduction to Grothendieck The ‘bridge-building’ technique toposes

Olivia Caramello • Decks of ‘bridges’: Morita-equivalences (or more generally Introduction morphisms or other kinds of relations between toposes) Toposes as generalized • Arches of ‘bridges’: Site characterizations for topos-theoretic spaces Sheaves on a invariants (or more generally ‘unravelings’ of topos-theoretic topological space Sheaves on a site invariants in terms of concrete representations of the relevant Geometric morphisms topos) Toposes as mathematical universes

Classifying toposes

Toposes as bridges

Examples of ‘bridges’

Topological Galois theory

Theories of presheaf type

Topos-theoretic Fraïssé theorem Stone-type The ‘bridge’ yields a logical equivalence (or an implication) dualities between the ‘concrete’ properties P(C ,J) and Q(D,K ), interpreted in General remarks this context as manifestations of a unique property I lying at the Future directions level of the topos. 42 / 66 An introduction to Grothendieck Topos-theoretic invariants toposes Olivia Caramello • By a topos-theoretic invariant we mean any notion which is Introduction invariant under categorical equivalence of toposes. Toposes as generalized spaces • The notion of a geometric morphism of toposes has notably Sheaves on a topological space allowed to build general comology theories starting from the Sheaves on a site categories of internal abelian groups or modules in toposes. In Geometric morphisms particular, the topos-theoretic viewpoint has allowed Toposes as mathematical Grothendieck to reﬁne and enrich the study of cohomology, up to universes the so-called ‘six-operation formalism’. Classifying toposes The cohomological invariants have had a tremendous impact on

Toposes as the development of modern Algebraic Geometry and beyond. bridges Examples of • On the other hand, also homotopy-theoretic invariants such as ‘bridges’ the fundamental group and the higher homotopy groups can be Topological Galois theory deﬁned as invarants of toposes. Theories of presheaf type • Still, these are by no means the only invariants that one can Topos-theoretic Fraïssé theorem consider on toposes: indeed, there are inﬁnitely many invariants

Stone-type of toposes (of algebraic, logical, geometric or whatever nature), dualities the notion of identity for toposes being simply categorical General remarks equivalence. Future directions 43 / 66 An introduction to Grothendieck A few selected applications toposes Olivia Caramello Since the theory of topos-theoretic ‘bridges’ was introduced in

Introduction 2010, several applications of it have been obtained in different Toposes as fields of Mathematics, such as: generalized spaces • Sheaves on a Model theory (topos-theoretic Fraïssé theorem) topological space Sheaves on a site • Proof theory (various results for first-order theories) Geometric morphisms • Algebra (topos-theoretic generalization of topological Galois Toposes as mathematical theory) universes • Topology (topos-theoretic interpretation/generation of Classifying toposes Stone-type and Priestley-type dualities) Toposes as bridges • Functional analysis (various results on Gelfand spectra and Examples of Wallman compactifications) ‘bridges’

Topological • Many-valued logics and lattice-ordered groups (two joint Galois theory papers with A. C. Russo) Theories of presheaf type • Cyclic homology, as reinterpreted by A. Connes (work on Topos-theoretic Fraïssé theorem “cyclic theories”, jointly with N. Wentzlaff) Stone-type • dualities Algebraic geometry (logical analysis of (co)homological

General remarks motives, cf. the paper “Syntactic categories for Nori motives”

Future directions joint with L. Barbieri-Viale and L. Lafforgue) 44 / 66 An introduction to Grothendieck Some examples of ‘bridges’ toposes

Olivia Caramello

Introduction

Toposes as generalized We shall now discuss a few ‘bridges’ established in the context of spaces Sheaves on a the above-mentioned applications: topological space Sheaves on a site Geometric morphisms • Topological Galois theory Toposes as mathematical universes • Theories of presheaf type Classifying toposes

Toposes as • Topos-theoretic Fraïssé theorem bridges

Examples of ‘bridges’ • Stone-type dualities Topological Galois theory

Theories of presheaf type The results are completely different... but the methodology is Topos-theoretic always the same! Fraïssé theorem

Stone-type dualities

General remarks

Future directions 45 / 66 An introduction to Grothendieck Topological Galois theory toposes

Olivia Caramello

Introduction Recall that classical topological Galois theory provides, given a Toposes as Galois extension F ⊆ L, a bijective correspondence between the generalized spaces intermediate field extensions (resp. finite field extensions) Sheaves on a topological space F ⊆ K ⊆ L and the closed (resp. open) subgroups of the Galois Sheaves on a site Geometric group AutF (L). morphisms Toposes as This admits the following categorical reformulation: the functor mathematical universes K → Hom(K ,L) defines an equivalence of categories Classifying toposes ( L)op ' Cont (Aut (L)), Toposes as LF t F bridges L Examples of where L is the category of finite intermediate field extensions ‘bridges’ F and Cont (Aut (L)) is the category of continuous non-empty Topological t F Galois theory transitive actions of AutF (L) on discrete sets. Theories of presheaf type A natural question thus arises: can we characterize the Topos-theoretic Fraïssé theorem categories C whose dual is equivalent to (or fully embeddable Stone-type into) the category of (non-empty transitive) actions of a dualities topological automorphism group? General remarks

Future directions 46 / 66 An introduction to Grothendieck The topos-theoretic interpretation toposes

Olivia Caramello

Introduction Key observation: the above equivalence extends to an Toposes as generalized equivalence of toposes spaces Sheaves on a op topological space Sh( L ,J ) ' Cont(Aut (L)), Sheaves on a site LF at F Geometric morphisms Lop Toposes as where Jat is the atomic topology on LF and Cont(AutF (L)) is mathematical universes the topos of continuous actions of AutF (L) on discrete sets.

Classifying toposes It is therefore natural to investigate our problem by using the Toposes as methods of topos theory: more speciﬁcally, we shall look for bridges conditions on a small category and on an object u of its Examples of C ‘bridges’ ind-completion for the existence of an equivalence of toposes of Topological Galois theory the form op Theories of Sh(C ,Jat ) ' Cont(Aut(u)) . presheaf type

Topos-theoretic We will then be able to obtain, starting from such an equivalence, Fraïssé theorem an answer to our question, and hence build Galois-type theories Stone-type dualities in a great variety of different mathematical contexts. General remarks

Future directions 47 / 66 An introduction to Grothendieck The key notions I toposes

Olivia Caramello • A category C is said to satisfy the amalgamation property Introduction (AP) if for every objects a,b,c ∈ C and morphisms f : a → b, Toposes as generalized g : a → c in C there exists an object d ∈ C and morphisms spaces 0 0 0 0 Sheaves on a f : b → d, g : c → d in C such that f ◦ f = g ◦ g: topological space Sheaves on a site Geometric f morphisms a / b Toposes as mathematical g f 0 universes

Classifying toposes c / d g0 Toposes as bridges Examples of • A category C is said to satisfy the joint embedding property ‘bridges’ (JEP) if for every pair of objects a,b ∈ C there exists an Topological Galois theory object c ∈ C and morphisms f : a → c, g : b → c in C : Theories of presheaf type a Topos-theoretic Fraïssé theorem f Stone-type dualities b c General remarks g / Future directions 48 / 66 An introduction to Grothendieck The key notions II toposes

Olivia Caramello

Introduction

Toposes as generalized • An object u ∈ Ind- is said to be -universal if for every spaces C C Sheaves on a a ∈ C there exists an arrow χ : a → u in Ind-C : topological space Sheaves on a site Geometric χ morphisms a / u Toposes as mathematical universes • An object u ∈ Ind-C is said to be C -ultrahomogeneous if for Classifying toposes any object a ∈ C and arrows χ1 : a → u, χ2 : a → u in Ind-C Toposes as there exists an automorphism j : u → u such that j ◦ χ1 = χ2: bridges

Examples of χ1 ‘bridges’ a / u Topological Galois theory j Theories of χ2 presheaf type u Topos-theoretic Fraïssé theorem

Stone-type dualities

General remarks

Future directions 49 / 66 An introduction to Grothendieck Topological Galois theory as a ‘bridge’ toposes Olivia Caramello Theorem Introduction Let C be a small category satisfying the amalgamation and joint Toposes as generalized embedding properties, et let u be a C -universal et spaces C -ultrahomogeneous object of the ind-completion Ind-C of C . Then Sheaves on a topological space there is an equivalence of toposes Sheaves on a site Geometric morphisms op Sh(C ,Jat ) ' Cont(Aut(u)), Toposes as mathematical universes where Aut(u) is endowed with the topology in which a basis of open Classifying neighbourhoods of the identity is given by the subgroups of the form toposes I = {α ∈ Aut(u) | α ◦ χ = χ} for χ : c → u an arrow in Ind- from an Toposes as χ C bridges object c of C . Examples of This equivalence is induced by the functor ‘bridges’ Topological op Galois theory F : C → Cont(Aut(u)) Theories of presheaf type which sends any object c of C on the set HomInd-C (c,u) (endowed Topos-theoretic Fraïssé theorem with the obvious action of Aut(u)) and any arrow f : c → d in C to the

Stone-type Aut(u)-equivariant map dualities

General remarks − ◦ f : HomInd-C (d,u) → HomInd-C (c,u) . Future directions 50 / 66 An introduction to Grothendieck Topological Galois theory as a ‘bridge’ toposes Olivia Caramello The following result arises from two ‘bridges’, respectively obtained by considering the invariant notions of atom and of arrow between atoms. Introduction Toposes as Theorem generalized spaces Under the hypotheses of the last theorem, the functor F is full and Sheaves on a topological space faithful if and only if every arrow of C is a strict monomorphism, and it is Sheaves on a site ( ( )) ( ( )) Geometric an equivalence on the full subcategory Contt Aut u of Cont Aut u morphisms on the non-empty transitive actions if C is moreover atomically complete. Toposes as mathematical universes

Classifying op toposes Sh(C ,Jat ) ' Cont(Aut(u)) Toposes as bridges

Examples of op ‘bridges’ C Contt (Aut(u)) Topological Galois theory This theorem generalizes Grothendieck’s theory of Galois categories and Theories of presheaf type can be applied for generating Galois-type theories in different fields of Topos-theoretic Mathematics, for example that of finite groups and that of finite graphs. Fraïssé theorem Stone-type Moreover, if a category C satisfies the first but not the second condition dualities of the theorem, our topos-theoretic approach gives us a fully explicit way General remarks to complete it, by means of the addition of ‘imaginaries’, so that also the Future directions second condition gets satisfied. 51 / 66 An introduction to Grothendieck Theories of presheaf type toposes Olivia Caramello Definition Introduction A geometric theory is said to be of presheaf type if it is classified by Toposes as generalized a presheaf topos. spaces Sheaves on a Theories of presheaf type are very important in that they constitute topological space Sheaves on a site the basic ‘building blocks’ from which every geometric theory can be Geometric morphisms built. Indeed, as every Grothendieck topos is a subtopos of a

Toposes as presheaf topos, so every geometric theory is a ‘quotient’ of a theory mathematical universes of presheaf type.

Classifying toposes These theories are the logical counterpart of small categories, in the

Toposes as sense that: bridges • Examples of For any theory of presheaf type T, its category T-mod(Set) of ‘bridges’ (set-based) models is equivalent to the ind-completion of the full Topological Galois theory subcategory f.p.T-mod(Set) on the finitely presentable models. Theories of • Any small category C is, up to idempotent splitting completion, presheaf type equivalent to the category f.p. -mod(Set) for some theory of Topos-theoretic T Fraïssé theorem presheaf type T. Stone-type dualities Moreover, any geometric theory T can be expanded to a theory General remarks classified by the topos [f.p.T-mod(Set),Set]. Future directions 52 / 66 An introduction to Grothendieck Theories of presheaf type toposes Olivia Caramello Every finitary algebraic (or, more generally, cartesian) theory is of

Introduction presheaf type, but this class contains many other interesting Toposes as mathematical theories including generalized spaces • Sheaves on a the theory of linear orders (classified by the simplicial topos) topological space • Sheaves on a site the theory of algebraic extensions of a given field Geometric • morphisms the theory of flat modules over a ring

Toposes as • the theory of lattice-ordered abelian groups with strong unit mathematical universes • the ‘cyclic theories’ (classiﬁed by the cyclic topos, the Classifying epicyclic topos and the arithmetic topos) toposes • the theory of perfect MV-algebras (or more generally of local Toposes as bridges MV-algebras in a proper variety of MV-algebras) Examples of • ‘bridges’ the geometric theory of ﬁnite sets

Topological Galois theory Any theory of presheaf type T gives rise to two different Theories of representations of its classifying topos, which can be used to presheaf type build ‘bridges’ connecting its syntax and semantics: Topos-theoretic Fraïssé theorem

Stone-type [f.p.T-mod(Set),Set] ' Sh(CT,JT) dualities

General remarks

Future directions op f.p.T-mod(Set) (CT,JT) 53 / 66 An introduction to Grothendieck Irreducible formulae and ﬁnitely presentable models toposes Olivia Caramello Deﬁnition Introduction Let T be a geometric theory over a signature Σ. Then a geometric Toposes as ~ generalized formula φ(x) over Σ is said to be T-irreducible if, regarded as an spaces object of the syntactic category of , it does not admit any Sheaves on a CT T topological space non-trivial J -covering sieves. Sheaves on a site T Geometric morphisms Theorem Toposes as mathematical Let T be a theory of presheaf type over a signature Σ. Then universes

Classifying (i) Any ﬁnitely presentable T-model in Set is presented by a toposes T-irreducible geometric formula φ(~x) over Σ; Toposes as bridges (ii) Conversely, any T-irreducible geometric formula φ(~x) over Σ Examples of ‘bridges’ presents a T-model. op Topological In fact, the category f.p. -mod(Set) is equivalent to the full Galois theory T subcategory C irr of C on the -irreducible formulae. Theories of T T T presheaf type Irreducible object Topos-theoretic [f.p.T-mod(Set),Set] ' Sh(C ,J ) Fraïssé theorem T T

Stone-type dualities CT,JT) General remarks op f.p.T-mod(Set) T-irreducible Every object formula Future directions 54 / 66 An introduction to Grothendieck A deﬁnability theorem toposes

Olivia Caramello

Introduction

Toposes as generalized Theorem spaces Sheaves on a Let T be a theory of preshef type and suppose that we are given, topological space Sheaves on a site for every ﬁnitely presentable Set-model M of T, a subset RM of Geometric n morphisms M in such a way that every T-model homomorphism Toposes as h : M → N maps RM into RN . Then there exists a geometric mathematical universes formula-in-context φ(x1,...,xn) such that RM = [[~x . φ]]M for each Classifying ﬁnitely presentable -model . toposes T M

Toposes as bridges

Examples of Subobject of UA1×···×UAn ‘bridges’ [f.p.T-mod(Set),Set] ' Sh(CT,JT) Topological Galois theory Theories of op C ,J ) presheaf type f.p.T-mod(Set) T T Functorial assignment Geometric formula A1 An Topos-theoretic M→RM ⊆MA1×···×MAn φ(x1 ,...,xn ) Fraïssé theorem

Stone-type dualities

General remarks

Future directions 55 / 66 An introduction to Grothendieck Topos-theoretic Fraïssé theorem toposes

Olivia Caramello

Introduction The following result, which generalizes Fraïssé’s theorem in Toposes as classical model theory, arises from a triple ‘bridge’. generalized spaces Sheaves on a Deﬁnition topological space Sheaves on a site A set-based model M of a geometric theory T is said to be Geometric morphisms homogeneous if for any arrow y : c → M in T-mod(Set) and any Toposes as arrow f in f.p. -mod(Set) there exists an arrow u in -mod(Set) mathematical T T universes such that u ◦ f = y: Classifying y toposes c / M Toposes as ? bridges f u Examples of ‘bridges’ d Topological Galois theory

Theories of presheaf type Theorem Topos-theoretic Fraïssé theorem Let T be a theory of presheaf type such that the category Stone-type f.p.T-mod(Set) is non-empty and has AP and JEP. Then the dualities 0 theory T of homogeneous T-models is complete and atomic. General remarks

Future directions 56 / 66 An introduction to Grothendieck Topos-theoretic Fraïssé theorem toposes

Olivia Caramello

Introduction

Toposes as Atomic topos generalized op Sh(f.p. -mod(Set) ,J ) ' Sh( 0 ,J 0 ) spaces T at CT T Sheaves on a topological space op Sheaves on a site (f.p.T-mod(Set) ,Jat ) 0 , 0 ) Geometric Atomic site i.e. CT JT 0 morphisms AP on f.p.T-mod(Set) Atomicity of T Toposes as mathematical universes Two-valued topos Classifying op ( ( ) , ) ' ( 0 , 0 ) toposes Sh f.p.T-mod Set Jat Sh CT JT Toposes as bridges op 0 ,J 0 ) Examples of (f.p.T-mod(Set) ,Jat ) CT T JEP on f.p. -mod(Set) Completeness of 0 ‘bridges’ T T

Topological Galois theory

Theories of Point of op presheaf type ( ( ) , ) ' ( 0 , 0 ) Sh f.p.T-mod Set Jat Sh CT JT Topos-theoretic Fraïssé theorem op 0 , 0 ) Stone-type (f.p.T-mod(Set) ,Jat ) CT JT 0 dualities homogeneous T-model in Set T -model in Set General remarks

Future directions 57 / 66 An introduction to Grothendieck Stone-type dualities through ‘bridges’ toposes Olivia Caramello The ‘bridge-building’ technique allows one to unify all the classical

Introduction Stone-type dualities between special kinds of preorders and partial

Toposes as orders, locales or topological spaces as instances of just one generalized spaces topos-theoretic phenomenon, and to generate many new such Sheaves on a topological space dualities. Sheaves on a site Geometric morphisms More precisely, this machinery generates Stone-type

Toposes as dualities/equivalences by functorializing ‘bridges’ of the form mathematical universes Classifying Sh(C ,J ) ' Sh(D,K ) toposes C D

Toposes as bridges

Examples of ‘bridges’

Topological C D Galois theory Theories of where presheaf type

Topos-theoretic • C is a preorder (regarded as a category), Fraïssé theorem • JC is a (subcanonical) Grothendieck topology on C , Stone-type dualities • C is a KD -dense full subcategory of D, and General remarks • JC is the induced Grothendieck topology (KD )|C on C . Future directions 58 / 66 An introduction to Grothendieck Stone-type dualities through ‘bridges’ toposes

Olivia Caramello

Introduction Our machinery relies on the following key points: Toposes as • The possibility of deﬁning Grothendieck topologies on posets generalized spaces in an intrinsic way which exploits the lattice-theoretic Sheaves on a topological space structure present on them. Sheaves on a site Geometric morphisms • The possibility of functorializing the assignments Toposes as C → Sh(C ,JC ) and D → Sh(D,KD ) by means of the mathematical universes (covariant or contravariant) theory of morphisms of sites. Classifying toposes • The possibility of recovering (under suitable hypotheses

Toposes as which are satisﬁed in a great number of cases) a given bridges preordered structure from the associated topos by means of Examples of ‘bridges’ a topos-theoretic invariant. Topological More precisely, if the topologies K (resp. J ) can be Galois theory D C

Theories of ‘uniformly described through an invariant C of families of presheaf type subterminals in a topos’ then the elements of D (resp. of C ) Topos-theoretic Fraïssé theorem can be recovered as the subterminal objects of the topos Stone-type Sh(D,KD ) (resp. Sh(C ,JC )) which satisfy a condition of dualities C-compactness. General remarks

Future directions 59 / 66 An introduction to Grothendieck A mathematical morphogenesis toposes

Olivia Caramello

Introduction • The essential ambiguity given by the fact that any topos is Toposes as associated in general with an inﬁnite number of theories or generalized spaces different sites allows to study the relations between different Sheaves on a topological space theories, and hence the theories themselves, by using Sheaves on a site Geometric toposes as ’bridges’ between these different presentations. morphisms

Toposes as mathematical • Every topos-theoretic invariant generates a veritable universes mathematical morphogenesis resuting from its expression in Classifying toposes terms of different representations of toposes, which gives rise Toposes as bridges in general to connections between properties or notions that

Examples of are completely different and apparently unrelated from each ‘bridges’ other Topological Galois theory Theories of • The mathematical exploration is therefore in a sense presheaf type ‘reversed’ since it is guided by the Morita-equivalences and Topos-theoretic Fraïssé theorem by topos-theoretic invariants, from which one proceeds to Stone-type dualities extract concrete information on the theoires that one wishes

General remarks to study.

Future directions 60 / 66 An introduction to Grothendieck The duality between ‘real’ and ‘imaginary’ toposes Olivia Caramello • The passage from a site (or a theory) to the associated topos can be regarded as a sort of ‘completion’ by the addition of Introduction ‘imaginaries’ (in the model-theoretic sense), which materializes Toposes as generalized the potential contained in the site (or theory). spaces Sheaves on a • The duality between the (relatively) unstructured world of topological space presentations of theories and the maximally strucured world of Sheaves on a site Geometric toposes is of great relevance as, on the one hand, the morphisms ‘simplicity’ and concreteness of theories or sites makes it easy Toposes as to manipulate them, while, on the other hand, computations are mathematical universes much easier in the ‘imaginary’ world of toposes thanks to their very rich internal structure and the fact that invariants live at Classifying toposes this level.

Toposes as bridges

Examples of IMAGINARY topos ‘bridges’ Morita equivalence Topological Galois theory choice of invariants lifting for computation Theories of presheaf type

Topos-theoretic Fraïssé theorem starting point generation = concrete fact Stone-type REAL of other (often quite dualities concrete results elementary) no General remarks direct Future directions deduction 61 / 66 An introduction to Grothendieck Some key features of toposes toposes

Olivia Caramello Here are some essential features of toposes, which account for their relevance in Mathematics: Introduction

Toposes as • Generality: Unlike most of the invariants used in Mathematics, generalized spaces the level of generality of topos-theoretic invariants is such as to Sheaves on a make them suitable for effectively comparing with each other topological space Sheaves on a site theories or objects coming from different ﬁelds of Mathematics. Geometric morphisms • Expressivity: On the other hand, many important invariants Toposes as mathematical arising in Mathematics can be expressed as topos-theoretic universes invariants (think for instance of the cohomological and Classifying homotopy-theoretic invariants). toposes Toposes as • Centrality: The fact that topos-theoretic invariants often bridges manifest as important properties or constructions of natural Examples of ‘bridges’ mathematical or logical interest is a clear indication of the

Topological centrality of these concepts in Mathematics. In fact, whatever Galois theory happens at the level of toposes has ‘uniform’ ramiﬁcations in Theories of presheaf type Mathematics as a whole. Topos-theoretic • Fraïssé theorem Technical ﬂexibility: Toposes are mathematical universes which

Stone-type are very rich in terms of internal structure; moreover, they have dualities a very-well behaved representation theory, which makes them General remarks extremely effective computational tools, in particular when they Future directions are considered as ‘bridges’. 62 / 66 An introduction to Grothendieck Toposes as ‘bridges’ and the Erlangen Program toposes

Olivia Caramello

Introduction The methodology ‘toposes as bridges’ is a vast extension of Toposes as Felix Klein’s Erlangen Program (A. Joyal) generalized spaces Sheaves on a topological space More speciﬁcally: Sheaves on a site Geometric morphisms • Every group gives rise to a topos (namely, the category of

Toposes as actions of it), but the notion of topos is much more general. mathematical universes • As Klein classiﬁed geometries by means of their Classifying toposes automorphism groups, so we can study ﬁrst-order geometric

Toposes as theories by studying the associated classifying toposes. bridges • Examples of As Klein established surprising connections between very ‘bridges’ different-looking geometries through the study of the Topological Galois theory algebraic properties of the associated automorphism groups, Theories of so the methodology ‘toposes as bridges’ allows to discover presheaf type non-trivial connections between properties, concepts and Topos-theoretic Fraïssé theorem results pertaining to different mathematical theories through Stone-type the study of the categorical invariants of their classifying dualities

General remarks toposes.

Future directions 63 / 66 An introduction to Grothendieck Future directions toposes

Olivia Caramello

Introduction The evidence provided by the results obtained so far shows that Toposes as toposes can effectively act as unifying spaces for transferring generalized spaces information between distinct mathematical theories and for Sheaves on a topological space generating new equivalences, dualities and symmetries across Sheaves on a site Geometric different ﬁelds of Mathematics. morphisms

Toposes as mathematical In fact, toposes have an authentic creative power in Mathematics, universes in the sense that their study naturally leads to the discovery of a Classifying toposes great number of notions and ‘concrete’ results in different

Toposes as mathematical ﬁelds, which are pertinent but often unsuspected. bridges Examples of In the next years, we intend to continue pursuing the development ‘bridges’ of these general unifying methodologies both at the theoretical Topological Galois theory level and at the applied level, in order to continue developing the Theories of presheaf type potential of toposes as fundamental tools in the study of

Topos-theoretic mathematical theories and their relations, and as key concepts Fraïssé theorem deﬁning a new way of doing Mathematics liable to bring distinctly Stone-type dualities new insights in a great number of different subjects.

General remarks

Future directions 64 / 66 An introduction to Grothendieck Future directions toposes Olivia Caramello Central themes in this programme will be: Introduction • investigation of important dualities or correspondences in Toposes as generalized Mathematics from a topos-theoretic perspective (in particular, spaces the theory of motives, class ﬁeld theory and the Langlands Sheaves on a topological space programme) Sheaves on a site Geometric • systematic study of invariants of toposes in terms of their morphisms presentations, and introduction of new invariants which Toposes as mathematical capture important aspects of concrete mathematical universes problems Classifying toposes • interpretation and generalization of important parts of

Toposes as classical and modern model theory in terms of toposes and bridges development of a functorial model theory Examples of • introduction of new methodologies for generating ‘bridges’ Morita-equivalences Topological Galois theory • development of general techniques for building spectra by Theories of using classifying toposes presheaf type

Topos-theoretic • generalization of the ‘bridge’ technique to the setting of Fraïssé theorem higher categories and toposes through the introduction of Stone-type higher geometric logic dualities General remarks • development of a relative theory of classifying toposes Future directions 65 / 66 An introduction to Grothendieck For further reading toposes

Olivia Caramello

Introduction Toposes as O. Caramello generalized spaces Grothendieck toposes as unifying ‘bridges’ in Mathematics, Sheaves on a topological space Mémoire d’habilitation à diriger des recherches, Sheaves on a site Geometric Université de Paris 7 (2016), morphisms available from my website www.oliviacaramello.com. Toposes as mathematical universes Classifying O. Caramello toposes Theories, Sites, Toposes: Relating and studying Toposes as bridges mathematical theories through topos-theoretic ‘bridges’, Examples of ‘bridges’ Oxford University Press (2017).

Topological Galois theory S. Mac Lane and I. Moerdijk. Theories of presheaf type Sheaves in geometry and logic: a ﬁrst introduction to topos Topos-theoretic Fraïssé theorem theory

Stone-type Springer-Verlag (1992). dualities

General remarks

Future directions 66 / 66