Topos Theory Olivia Caramello Geometric morphisms Locales and pointless topology Geometric morphisms as flat functors Topos Theory Geometric morphisms to Sh(C; J) Lectures 15-18 Characterizing Grothendieck toposes Giraud’s theorem Olivia Caramello Morphisms between sites The Comparison Lemma Points of toposes Separating sets of points of a topos The subterminal topology Topos-theoretic Galois theory For further reading Topos Theory Geometric morphisms Olivia Caramello Geometric The natural, topologically motivated, notion of morphism of morphisms Grothendieck toposes is that of geometric morphism. The natural Locales and pointless notion of morphism of geometric morphisms if that of geometric topology transformation. Geometric morphisms as flat functors Definition Geometric morphisms to (i) Let E and F be toposes. A geometric morphism f : E!F Sh(C; J) consists of a pair of functors f : E!F (the direct image of Characterizing ∗ Grothendieck f ) and f ∗ : F!E (the inverse image of f ) together with an toposes ∗ ∗ Giraud’s theorem adjunction f a f∗, such that f preserves finite limits. Morphisms between sites (ii) Let f and g : E!F be geometric morphisms. A geometric The Comparison transformation α : f ! g is defined to be a natural Lemma transformation a : f ∗ ! g∗. Points of toposes Separating sets of points of a topos • Grothendieck toposes and geometric morphisms between The subterminal them form a category, denoted by BTop. topology Topos-theoretic • Given two toposes E and F, geometric morphisms from E to Galois theory F and geometric transformations between them form a For further reading category, denoted by Geom(E; F). 2 / 45 Topos Theory Examples of geometric morphisms Olivia Caramello Geometric morphisms • A continuous function f : X ! Y between topological spaces Locales and pointless gives rise to a geometric morphism Sh(f ): Sh(X) ! Sh(Y ). topology The direct image Sh(f )∗ sends a sheaf F 2 Ob(Sh(X)) to the Geometric −1 morphisms as flat sheaf Sh(f )∗(F) defined by Sh(f )∗(F)(V ) = F(f (V )) for functors any open subset V of Y . The inverse image Sh(f )∗ acts on Geometric morphisms to étale bundles over Y by sending an étale bundle p : E ! Y (C; ) Sh J to the étale bundle over X obtained by pulling back p along Characterizing Grothendieck f : X ! Y . toposes Giraud’s theorem • Every Grothendieck topos E has a unique geometric Morphisms between sites morphism E! Set. The direct image is the global sections The Comparison functor Γ: E! Set, sending an object e 2 E to the set Lemma Hom (1 ; e), while the inverse image functor ∆ : Set !E Points of toposes E E Separating sets sends a set S to the coproduct t1E . of points of a s2S topos The subterminal • For any site (C; J), the pair of functors formed by the topology inclusion Sh(C; J) ,! [Cop; Set] and the associated sheaf Topos-theoretic op Galois theory functor a :[C ; Set] ! Sh(C; J) yields a geometric morphism op For further i : Sh(C; J) ! [C ; Set]. reading 3 / 45 Topos Theory Slice toposes Olivia Caramello Geometric morphisms Locales and The notion of Grothendieck topos is stable with respect to the pointless topology slice construction: Geometric morphisms as flat Proposition functors Geometric E E morphisms to (i) For any Grothendieck topos and any object P of , the Sh(C; J) slice category E=P is also a Grothendieck topos; more Characterizing R Grothendieck precisely, if E = Sh(C; J) then E=P ' Sh( P; JP ), where JP toposes is the Grothendieck topology on R P whose covering sieves Giraud’s theorem Morphisms are precisely the sieves whose image under the canonical between sites R projection functor πP : P !C is J-covering. The Comparison Lemma Points of toposes (ii) For any Grothendieck topos E and any morphism f : P ! Q ∗ Separating sets in E, the pullback functor f : E=Q !E=P has both a left of points of a topos adjoint (namely, the functor Σf given by composition with f ) The subterminal and a right adjoint πf . It is therefore the inverse image of a topology geometric morphism E=P !E=Q. Topos-theoretic Galois theory For further reading 4 / 45 Topos Theory The notion of locale Olivia Caramello To better understand the relationship between topological spaces and Geometric the associated toposes, it is convenient to introduce the notion of a morphisms locale. Locales and pointless topology Definition Geometric • A frame is a complete lattice A satisfying the infinite distributive morphisms as flat functors law Geometric a ^ _bi = _(a ^ bi ) morphisms to i2I i2I Sh(C; J) • A frame homomorphism h : A ! B is a mapping preserving finite Characterizing Grothendieck meets and arbitrary joins. toposes • We write Frm for the category of frames and frame Giraud’s theorem homomorphisms. Morphisms between sites The Comparison Fact Lemma A poset is a frame if and only if it is a complete Heyting algebra. Points of toposes Note that we have a functor Top ! Frmop which sends a topological Separating sets of points of a space X to its lattice O(X) of open sets and a continuous function topos f : X ! Y to the function O(f ): O(Y ) !O(X) sending an open The subterminal subset V of Y to the open subset f −(V ) of X. This motivates the topology following Topos-theoretic Galois theory Definition For further The category Loc of locales is the dual Frmop of the category of reading frames (a locale is an object of the category Loc). 5 / 45 Topos Theory Pointless topology Olivia Caramello Geometric morphisms Pointless topology is an attempt to do Topology without making Locales and pointless reference to the points of topological spaces but rather entirely in topology Geometric terms of their open subsets and of the inclusion relation between morphisms as flat functors them. For example, notions such as connectedness or Geometric compactness of a topological space can be entirely reformulated morphisms to Sh(C; J) as properties of its lattice of open subsets: Characterizing • Grothendieck A space X is connected if and only if for any a; b 2 O(X) toposes such that a ^ b = 0, a _ b = 1 implies either a = 1 or b = 1; Giraud’s theorem Morphisms • between sites A space X is compact if and only if whenever 1 = _ai in i2I The Comparison 0 Lemma O(X), there exist a finite subset I ⊆ I such that 1 = _ai . i2I0 Points of toposes Separating sets Pointless topology thus provides tools for working with locales as of points of a topos they were lattices of open subsets of a topological space (even The subterminal though not all of them are of this form). On the other hand, a topology locale, being a complete Heyting algebra, can also be studied by Topos-theoretic Galois theory using an algebraic or logical intuition. For further reading 6 / 45 Topos Theory The dual nature of the concept of locale Olivia Caramello Geometric morphisms Locales and pointless topology Geometric This interplay of topological and logical aspects in the theory of morphisms as flat functors locales is very interesting and fruitful; in fact, important Geometric morphisms to ‘topological’ properties of locales translate into natural logical Sh(C; J) properties, via the identification of locales with complete Heyting Characterizing Grothendieck algebras: toposes Giraud’s theorem Example Morphisms between sites Locales Complete Heyting algebras The Comparison Lemma Extremally disconnected locales Complete De Morgan algebras Points of toposes Separating sets Almost discrete locales Complete Boolean algebras of points of a topos The subterminal topology Topos-theoretic Galois theory For further reading 7 / 45 Topos Theory The adjunction between locales and spaces Olivia Caramello Geometric • For any topological space X, the lattice O(X) of its open sets morphisms is a locale. Locales and pointless • Conversely, with any locale F one can associate a topological topology space XF , whose points are the frame homomorphism Geometric F ! f0; 1g and whose open sets are the subsets of frame morphisms as flat functors homomorphisms F ! f0; 1g which send a given element Geometric f 2 F to 1. morphisms to Sh(C; J) • In fact, the assignments Characterizing Grothendieck X !O(X) toposes Giraud’s theorem and Morphisms between sites F ! XF The Comparison lift to an adjunction O a X between the category Top of Lemma − topological spaces and continuous maps and the category Points of toposes Loc = Frmop of locales. Separating sets of points of a • The topological spaces X such that the unit ηX : X ! XO(X) is topos a homeomorphism are precisely the sober spaces, while the The subterminal topology locales F such that the counit O(XF ) ! F is an isomorphism Topos-theoretic are the spatial locales. Galois theory • The adjunction thus restricts to an equivalence between the For further full subcategories on the sober spaces and on the spatial reading locales. 8 / 45 Topos Theory Sheaves on a locale Olivia Caramello Geometric Definition morphisms Given a locale L, the topos Sh(L) of sheaves on L is defined as Sh(L; JL), Locales and pointless where JL is the Grothendieck topology on L (regarded as a poset topology category) given by: Geometric morphisms as flat functors fai | i 2 Ig 2 JL(a) if and only if _ ai = a : i2I Geometric morphisms to Sh(C; J) Characterizing Grothendieck Theorem toposes Giraud’s theorem • For any locale L, there is a Heyting algebra isomorphism ∼ Morphisms L = SubSh(L)(1Sh(L)). between sites • The assignment L ! Sh(L) is the object-map of a full and faithful The Comparison Lemma (pseudo-)functor from the category Loc of locales to the category Points of toposes BTop of Grothendieck toposes.