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Home , Coequalizer, Kernel (category theory), Pointless topology

Theory

Olivia Caramello

Geometric

Locales and pointless

Geometric morphisms as flat Topos Theory Geometric morphisms to Sh(C, J) Lectures 15-18

Characterizing Grothendieck 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 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 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 , 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 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 ∈ 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 Γ: E → Set, sending an object e ∈ 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 t1E . of points of a s∈S 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 A satisfying the infinite distributive morphisms as flat functors law

Geometric a ∧ ∨bi = ∨(a ∧ bi ) morphisms to i∈I i∈I Sh(C, J) • A frame 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 . Morphisms between sites The Comparison Fact Lemma A poset is a frame if and only if it is a complete . Points of toposes Note that we have a functor Top → Frmop which sends a topological Separating sets of points of a 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 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 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 ∈ 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 i∈I The Comparison 0 Lemma O(X), there exist a finite subset I ⊆ I such that 1 = ∨ai . i∈I0 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 , 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 → {0, 1} and whose open sets are the subsets of frame morphisms as flat functors homomorphisms F → {0, 1} which send a given element

Geometric f ∈ 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 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

Topos-theoretic are the spatial locales. Galois theory • The adjunction thus restricts to an equivalence between the For further full 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 {ai | i ∈ I} ∈ JL(a) if and only if ∨ ai = a . i∈I 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.

Separating sets of points of a The assignment L → Sh(L) indeed brings (pointless) Topology into the topos world of Grothendieck toposes; in fact, important topological properties of The subterminal topology locales can be expressed as topos-theoretic invariants (i.e. properties of

Topos-theoretic toposes which are stable under categorical equivalence) of the Galois theory corresponding toposes of sheaves. These invariants can in turn be used For further reading to give definitions of topological properties for Grothendieck toposes.

9 / 45 Topos Theory A general hom-tensor adjunction I Olivia Caramello

Geometric morphisms Locales and Theorem pointless topology Let C be a small category, E be a locally small cocomplete Geometric morphisms as flat category and A : C → E a functor. Then we have an adjunction functors Geometric op o morphisms to LA :[C , Set] / E : RA Sh(C, J) Characterizing op Grothendieck where the right adjoint RA : E → [C , Set] is defined for each toposes Giraud’s theorem e ∈ Ob(E) and c ∈ Ob(C) by:

Morphisms between sites RA(e)(c) = HomE (A(c), e) The Comparison Lemma op Points of toposes and the left adjoint LA :[C , Set] → E is defined by

Separating sets of points of a topos LA(P) = colim(A ◦ πP ),

The subterminal topology R where πP is the canonical projection functor P → C from the Topos-theoretic R Galois theory category of elements P of P to C.

For further reading

10 / 45 Topos Theory A general hom-tensor adjunction II Olivia Caramello

Geometric Remarks morphisms • The functor LA can be considered as a generalized tensor Locales and product, since, by the construction of colimits in terms of pointless topology and , we have the following diagram: Geometric morphisms as flat θ functors ` 0 ` φ A(c ) / A(c) / LA(P), τ / Geometric c∈C,p∈P(c) c∈C, p∈P(c) morphisms to u:c0→c Sh(C, J) Characterizing where Grothendieck θ(c, p, u, x) = (c0, P(u)(p), x) toposes Giraud’s theorem and Morphisms τ(c, p, u, x) = (c, p, A(u)(x)) . between sites For this reason, we shall also denoteL by The Comparison A Lemma op − ⊗C A :[C , Set] → E . Points of toposes

Separating sets of points of a • We can rewrite the above coequalizer as follows: topos θ φ The subterminal ` 0 0 ` P(c) × HomC(c , c) × A(c ) / P(c) × A(c) / P ⊗C A . topology τ / c,c0∈C c∈C Topos-theoretic Galois theory From this we see that this definition is symmetric in P and A, For further that is reading ∼ P ⊗C A = A ⊗Cop P . 11 / 45 Topos Theory A couple of corollaries Olivia Caramello

Geometric Corollary morphisms

Locales and Every presheaf is a colimit of representables. More precisely, for pointless op topology any presheaf P : C → Set, we have

Geometric morphisms as flat ∼ functors P = colim(yC ◦ πP ),

Geometric morphisms to op where yC : C → [C , Set] is a Yoneda embedding and πP is the Sh(C, J) R Characterizing canonical projection P → C. Grothendieck toposes Giraud’s theorem Corollary op Morphisms For any small category C, the topos [C , Set] is the free between sites

The Comparison cocompletion of C (via the Yoneda embedding yC); that is, any Lemma functor A : C → E to a cocomplete category E extends, uniquely Points of toposes up to isomorphism, to a colimit-preserving functor [Cop, Set] → E Separating sets of points of a along yC: topos A The subterminal C / E topology ;

Topos-theoretic yC Galois theory For further op  reading [C , Set]

12 / 45 Topos Theory Separating sets of objects Olivia Caramello

Geometric Definition morphisms A separating set of objects for a Grothendieck topos E is a set C of Locales and pointless objects of E such that for any object A of E, the collection of arrows topology from objects in C to A is epimorphic. Geometric morphisms as flat functors Proposition Geometric morphisms to For any site (C, J), the collection of objects of the form lJ (c) (for Sh(C, J) c ∈ C), where Characterizing Grothendieck lJ : C → Sh(C, J) toposes op Giraud’s theorem is the composite of the Yoneda embedding yC : C → [C , Set] with op Morphisms the associated sheaf functor aJ :[C , Set] → Sh(C, J), is a between sites separating set of objects for the topos Sh(C, J). The Comparison Lemma The following theorem, which we shall prove below, provides a sort Points of toposes of converse to this proposition. Separating sets of points of a Theorem topos For any set of objects C of E which is separating, we have an The subterminal topology equivalence can Topos-theoretic E' Sh(C, J |C) Galois theory E can For further where J |C is the Grothendieck topology induced on C (regarded reading E as a full of E). 13 / 45 Topos Theory Geometric morphisms as flat functors I Olivia Caramello

Geometric morphisms Definition Locales and • A functor A : C → E from a small category C to a locally small pointless topology topos E with small colimits is said to be flat if the functor Geometric op morphisms as flat − ⊗C A :[C , Set] → E preserves finite limits. functors • The full subcategory of [C, E] on the flat functors will be Geometric morphisms to denoted by Flat(C, E). Sh(C, J)

Characterizing Grothendieck toposes Theorem Giraud’s theorem Let C be a small category and E be a Grothendieck topos. Then we Morphisms have an equivalence of categories between sites The Comparison op Lemma Geom(E, [C , Set]) ' Flat(C, E) Points of toposes Separating sets (natural in E), which sends of points of a topos • a flat functor A : C → E to the geometric morphism The subterminal op topology E → [C , Set] determined by the functors RA and − ⊗C A, and op Topos-theoretic • a geometric morphism f : E → [C , Set] to the flat functor Galois theory ∗ ∗ op given by the composite f ◦ yC of f :[C , Set] → E with the For further op reading Yoneda embedding yC : C → [C , Set].

14 / 45 Topos Theory Flat = filtering Olivia Caramello

Geometric Definition morphisms A functor F : C → E from a small category C to a Grothendieck topos E is said to be filtering if it satisfies the following conditions: Locales and pointless (a) For any object E of E there exist an epimorphic family topology {ei : Ei → E | i ∈ I} in E and for each i ∈ I an object bi of C and Geometric a generalized element Ei → F(bi ) in E. morphisms as flat (b) For any two objects c and d in C and any generalized element functors hx, yi : E → F(c) × F(d) in E there is an epimorphic family Geometric {ei : Ei → E | i ∈ I} in E and for each i ∈ I an object bi of C morphisms to with arrows u : b → c and v : b → d in C and a generalized Sh(C, J) i i i i element zi : Ei → F(bi ) in E such that hF(ui ), F(vi )i ◦ zi = Characterizing hx, yi ◦ ei for all i ∈ I. Grothendieck toposes (c) For any two parallel arrows u, v : d → c in C and any Giraud’s theorem generalized element x : E → F(d) in E for which F(u) ◦ x = F(v) ◦ x, there is an epimorphic family Morphisms between sites {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 The Comparison u ◦ w = v ◦ w and F(w ) ◦ y = x ◦ e for all i ∈ I. Lemma i i i i i Points of toposes Theorem Separating sets A functor F : C → E from a small category C to a Grothendieck of points of a topos E is flat if and only if it is filtering. topos Remarks The subterminal topology • For any small category C, a functor P : C → Set is filtering if R Topos-theoretic and only if its category of elements P is a filtered category Galois theory (equivalently, if it is a filtered colimit of representables). For further • For any small cartesian category C, a functor C → E is flat if reading and only if it preserves finite limits.

15 / 45 Topos Theory Geometric morphisms to Sh(C, J) I Olivia Caramello

Geometric We shall characterize the geometric morphisms to Sh(C, J) by morphisms identifying them with the geometric morphisms to [Cop, Set] which Locales and factor through the canonical geometric morphism pointless Sh(C, J) ,→ [Cop, Set]. topology

Geometric Definition morphisms as flat A subtopos of a topos E is a geometric morphism of the form functors

Geometric shj (E) ,→ E morphisms to (C, ) Sh J for a local operator j on E. Characterizing Grothendieck Recall that we proved that the subtoposes of a topos Sh(C, J) are toposes in bijective correspondence with the Grothendieck on C Giraud’s theorem which contain J. Morphisms between sites Proposition

The Comparison Let f : F → E be a geometric morphism and j a local operator on Lemma E. Then the following conditions are equivalent: Points of toposes (i) f factors through shj (E) ,→ E Separating sets (ii) f∗ sends all objects of F to j-sheaves in E. of points of a (iii) f ∗ maps c -dense in E to in F. topos j

The subterminal topology Corollary op Topos-theoretic Let f : E → [C , Set] be a geometric morphism. The the following Galois theory conditions are equivalent: For further (i) f factors through Sh(C, J) ,→ [Cop, Set]. reading ∗ (ii) f ◦ yC maps all J-covering sieves to epimorphic families in E. 16 / 45 Topos Theory Geometric morphisms to Sh(C, J) I Olivia Caramello

Geometric Definition morphisms If (C, J) is a site, a functor F : C → E to a Grothendieck topos is Locales and said to be J-continuous if it sends J-covering sieves to epimorphic pointless families. topology The full subcategory of Flat(C, E) on the J-continuous flat functors Geometric morphisms as flat will be denoted by FlatJ (C, E). functors Theorem Geometric morphisms to For any site (C, J) and Grothendieck topos E, the Sh(C, J) above-mentioned equivalence between geometric morphisms and Characterizing flat functors restricts to an equivalence of categories Grothendieck toposes Geom(E, Sh(C, J)) ' Flat (C, E) Giraud’s theorem J Morphisms natural in E. between sites The Comparison Sketch of proof. Lemma Appeal to the previous theorem Points of toposes • identifying the geometric morphisms E → Sh(C, J) with the Separating sets op of points of a geometric morphisms E → [C , Set] which factor through the topos canonical geometric inclusion Sh(C, J) ,→ [Cop, Set], and The subterminal • using the characterization of such morphisms as the topology geometric morphisms f : E → [Cop, Set] such that the Topos-theoretic composite f ∗ ◦ y of the inverse image functor f ∗ of f with the Galois theory Yoneda embedding y : C → [Cop, Set] sends J-covering sieves For further to epimorphic families in E. reading

17 / 45 Topos Theory Further properties of Grothendieck toposes I Olivia Caramello

Geometric Let E be a Grothendieck topos. Then morphisms (i) E is locally small. Locales and pointless topology (ii) E is well-powered. Geometric morphisms as flat (iii) Every in E is effective (in the sense that functors it is the pair of its own coequalizer), and every Geometric morphisms to is the coequalizer of its kernel pair. In Sh(C, J)

Characterizing particular, E is co-well-powered. Grothendieck toposes (iv) Finite limits commute with filtered colimits. Giraud’s theorem ∗ Morphisms (v) The ‘change of base’ functor f : E/A → E/B along any between sites arrow f : B → A in E preserves arbitrary colimits. The Comparison Lemma

Points of toposes (vi) Coproducts are disjoint.

Separating sets of points of a (vii) E has a separating set of objects (if E = Sh(C, J) then the topos object of the form lJ (c) for c ∈ C, where lJ is the composite of The subterminal op topology the Yoneda embedding C → [C , Set] with the associated op Topos-theoretic sheaf functor aJ :[C , Set], form a separating set). Galois theory For further (viii) E has a coseparating set of objects (if E = Sh(C, J) then the reading objects of the form ΩlJ (c) for c ∈ C form a coseparating set). 18 / 45 Topos Theory Further properties of Grothendieck toposes II Olivia Caramello Let us recall the following result: Geometric morphisms Theorem Locales and Let C be a locally small and complete (resp. cocomplete) category. pointless If C is well-powered (resp. co-well-powered) and admits a topology coseparating (resp. separating) set of objects then it has an initial Geometric (resp. a terminal) object. morphisms as flat functors This can be notably applied to categories of the form R F where F op Geometric is a contravariant functor C → Set defined on a locally small, morphisms to complete and well-powered category C with a coseparating set of Sh(C, J) objects which sends colimits in C to limits to deduce that F is Characterizing representable (and the dual result). Also, this latter result can be Grothendieck toposes used to show the Special Adjoint Functor Theorem: any functor Giraud’s theorem F : C → D defined on a locally small, complete (resp. cocomplete) Morphisms and well-powered category C with a coseparating (resp. between sites separating) set of objects with values in a locally small category D The Comparison preserves all small limits (resp. colimits) if and only if it has a left Lemma (resp. right) adjoint. Points of toposes From the above properties of Grothendieck toposes it thus follows Separating sets that: of points of a topos • A covariant (resp. contravariant) functor on a Grothendieck The subterminal topos with values in Set is representable if and only if it topology preserves arbitrary (small) limits (resp. sends colimits to Topos-theoretic limits). Galois theory • A functor between Grothendieck toposes admits a left adjoint For further (resp. a right adjoint) if and only if it preserves arbitrary (small) reading limits (resp. colimits). 19 / 45 Topos Theory The canonical topology on a Grothendieck topos Olivia Caramello can Geometric Recall that the canonical topology JC on a category C is the morphisms largest one for which all representables in C are sheaves. Its Locales and pointless covering sieves are precisely the universally effective-epimorphic topology ones. Geometric morphisms as flat functors Proposition

Geometric can morphisms to Let E be a Grothendieck topos. Then a sieve is JE -covering if Sh(C, J) and only if it contains a small epimorphic family. Characterizing Grothendieck toposes Sketch of proof. Giraud’s theorem Morphisms • Every small epimorphic family generates a sieve which is between sites universally effective-epimorphic, and every sieve which The Comparison Lemma contains a universally effective-epimorphic one is universally Points of toposes effective-epimorphic. Separating sets of points of a • On the other hand, E being well-powered, every epimorphic topos sieve contains a small epimorphic family (notice that if two The subterminal topology arrows f , g with common codomain A have the same image Topos-theoretic Galois theory then for any arrows ξ, χ : A → B, we have ξ ◦ f = χ ◦ f if and

For further only if ξ ◦ g = χ ◦ g). reading

20 / 45 Topos Theory Giraud’s theorem Olivia Caramello It is possible to axiomatically characterize Grothendieck toposes, Geometric morphisms thanks to a theorem due to J. Giraud: Locales and pointless Theorem topology The following conditions are equivalent: Geometric morphisms as flat functors (i) E is a Grothendieck topos.

Geometric (ii) E is a locally small category with finite limits in which colimits morphisms to Sh(C, J) exist and are preserved by the change of base functors,

Characterizing coproducts are disjoint, equivalence relations are effective and Grothendieck toposes are the coequalizers of their kernel pairs, and Giraud’s theorem there is a separating set of objects. Morphisms between sites The Comparison Sketch of proof. Lemma

Points of toposes The implication (i)⇒(ii) was established above. For the converse, Separating sets we shall prove that if C is a separating set of objects for E, regarded of points of a topos as a full subcategory of E, then we have an equivalence can The subterminal E' Sh(C, JE |C). This equivalence will be given by the geometric topology can morphism E → Sh(C, J |C) corresponding to the inclusion functor Topos-theoretic E Galois theory A of C into E (notice that this functor is flat since it is filtering, and it For further is filtering since C is separating for E). reading

21 / 45 Topos Theory Sketch of proof I Olivia Caramello

Geometric morphisms Locales and In order to prove that the functors pointless topology can RA = HomE (A(−), −): E → Sh(C, JE |C) and Geometric can morphisms as flat LA : Sh(C, JE |C) → E are quasi-inverses to each other, we verify functors that Geometric morphisms to Sh(C, J) • The counit  : L (R (e)) = lim c is an isomorphism for e A A −→c→e Characterizing Grothendieck every e ∈ E (this follows from the fact that C is separating in toposes light of the exactness properties needed to establish the fact Giraud’s theorem

Morphisms that the covering sieves for the canonical topology on a between sites Grothendieck topos are precisely those which contain small The Comparison Lemma epimorphic families).

Points of toposes

Separating sets • The unit ηP : P → RA(LA(P)) is an isomorphism for every of points of a can topos P ∈ Sh(C, JE |C). For this, since ηP is clearly an The subterminal isomorphism for R a representable, it clearly suffices to prove topology can that RA : E → Sh(C, J |C) preserves arbitrary colimits. Topos-theoretic E Galois theory

For further reading

22 / 45 Topos Theory Sketch of proof II Olivia Caramello

Geometric To prove that RA preserves colimits, one can resort to the morphisms can following criterion for a cocone {D(i) → C | i ∈ I} in Sh(C, JE |C) Locales and can pointless on a diagram D : I → Sh(C, J |C) to be colimiting in terms of the topology E canonical arrow ξ : colim (D) → C in [Cop, Set] from the colimit Geometric p morphisms as flat colim (D) of D in [Cop, Set]: this is the case if and only if the arrow functors p op a can (ξ) is an isomorphism, that is if and only if in [C , Set] Geometric JE |C morphisms to Sh(C, J) (1) Im(ξ) is c can -dense, and Characterizing JE |C Grothendieck toposes (2) the diagonal colim (D) → K to the domain Giraud’s theorem p ξ

Morphisms Kξ of the kernel pair of ξ is locally surjective. between sites

The Comparison In order to verify (2), one can in turn use the following Lemma characterization of colimits in E: if E = colimi∈I (Ei ) then, Points of toposes ` denoting by ξi the canonical coproduct arrows Ei → i∈I Ei and Separating sets ` of points of a by π : Ei → E the canonical arrow, for any arrows x : F → Ei topos i∈I and y : F → E , we have that π ◦ ξ ◦ x = π ◦ ξ ◦ y if and only if The subterminal j i j topology there exists an epimorphic family {ek : Fk → F | k ∈ K } in E such Topos-theoretic that (ξi ◦ x ◦ ek , ξj ◦ y ◦ ek ) belongs to the equivalence relation on Galois theory ` For further HomE (Fk , i∈I Ei ) generated by the pairs (ξi ◦ a, ξj ◦ Eu ◦ a) for an reading arrow u : i → j in I and a generalized element a : Fk → Ei . 23 / 45 Topos Theory Essential geometric morphisms I Olivia Caramello

Geometric Definition morphisms

Locales and A geometric morphism f : E → F is said to be essential if the pointless ∗ topology inverse image functor f : F → E has a left adjoint.

Geometric morphisms as flat Theorem functors

Geometric • Every functorf : C → D induces an essential geometric morphisms to Sh(C, J) morphism op op Characterizing E(f ):[C , Set] → [D , Set], Grothendieck toposes ∗ Giraud’s theorem whose inverse image functor f is given by composition with op Morphisms f . between sites The Comparison ∗ Lemma • In fact, the inverse image f is simultaneouslya ‘tensor

Points of toposes product’ functor LA1 and a ‘hom functor’ RA2 , where Separating sets of points of a op topos A1 = HomD(f (−), −): D → [C , Set]

The subterminal topology and Topos-theoretic op Galois theory A2 = HomD(−, f (−)) : C → [D , Set], For further reading so it has a left adjoint LA2 and a right adjoint RA1 .

24 / 45 Topos Theory Essential geometric morphisms II Olivia Caramello

Geometric Remarks morphisms • The direct image of the geometric morphism E(f ), namely the Locales and right adjoint to f ∗, is the right lim along f op, pointless ←−f op topology given by the following formula: Geometric morphisms as flat lim(F)(b) = lim F(a), functors ←− ←− f op φ:fa→b Geometric morphisms to Sh(C, J) where the is taken over the opposite of the comma

Characterizing category (f ↓b). Grothendieck • The left adjoint to f ∗ is the left Kan extension lim along f op, toposes −→f op Giraud’s theorem given by the following formula: Morphisms between sites ( )( ) = ( ), −→lim F b −→lim F a The Comparison f op φ:b→fa Lemma

Points of toposes where the colimit is taken over the opposite of the comma

Separating sets category (b ↓f ). of points of a topos

The subterminal Proposition topology If C and D are Cauchy-complete categories, a geometric morphism Topos-theoretic [Cop, Set] → [Dop, Set] is of the form E(f ) for some functor f : C → D Galois theory if and only if it is essential; in this case, f can be recovered from For further ( ) reading E f (up to isomorphism) as the restriction to the full subcategories of representables of the left adjoint to the inverse image of E(f ). 25 / 45 Topos Theory Morphisms of sites I Olivia Caramello By using the characterization of filtering functors with values in a Geometric Grothendieck topos as functors which send certain families to morphisms epimorphic families and the fact that the image under the Locales and associated sheaf functor of a family of natural transformations with pointless common codomain is epimorphic if and only if the family is locally topology jointly surjective, we obtain the following result: Geometric Corollary morphisms as flat 0 0 functors Let (C, J) and (C , J ) be essentially small sites, and l : C → Sh(C, J), l0 : C0 → Sh(C0, J0) be the canonical functors Geometric (given by the composite of the relevant Yoneda embedding with the morphisms to associated sheaf functor). Then, given a functor F : C → C0, the Sh(C, J) following conditions are equivalent: Characterizing (i) A induces a geometric morphism u : Sh(C0, J0) → Sh(C, J) Grothendieck toposes making the following square commutative: Giraud’s theorem F 0 Morphisms C / C between sites l l0 ∗ The Comparison  u 0 0 Lemma Sh(C, J) / Sh(C , J );

Points of toposes (ii) The functor F is a morphism of sites (C, J) → (C0, J0) in the Separating sets sense that it satisfies the following properties: of points of a (1) A sends every J-covering family in C into a J0-covering family in topos C0. 0 C0 0 The subterminal (2) Every object c of admits a J -covering family topology 0 0 ci −→ c , i ∈ I , Topos-theoretic 0 0 Galois theory by objects ci of C which have morphisms

0 For further ci −→ F(ci ) reading to the images under A of objects ci of C. 26 / 45 Topos Theory Morphisms of sites II Olivia Caramello 0 Geometric (3) For any objects c1, c2 of C and any pair of morphisms of C morphisms

Locales and 0 0 0 0 pointless f1 : c −→ F(c1) , f2 : c −→ F(c2) , topology 0 Geometric there exists a J -covering family morphisms as flat functors g0 : c0 −→ c0 , i ∈ I , Geometric i i morphisms to Sh(C, J) and a family of pairs of morphisms of C Characterizing Grothendieck i i toposes f1 : bi −→ c1 , f2 : bi → c2 , i ∈ I , Giraud’s theorem Morphisms and of morphisms of C0 between sites

The Comparison 0 0 Lemma hi : ci −→ F(bi ) , i ∈ I , Points of toposes

Separating sets making the following squares commutative: of points of a topos g0 g0 0 i 0 0 i 0 The subterminal c / c c / c topology i i

Topos-theoretic 0 0 0 0 hi f1 hi f2 Galois theory  F(f i )   F(f i )  For further 1 2 reading F(bi ) / F(c1) F(bi ) / F(c2)

27 / 45 Topos Theory Morphisms of sites III Olivia Caramello 0 Geometric (4) For any pair of arrows f1, f2 : c ⇒ d of C and any arrow of C morphisms f 0 : b0 −→ F(c) Locales and pointless topology satisfying 0 0 Geometric F(f1) ◦ f = F(f2) ◦ f , morphisms as flat 0 functors there exist a J -covering family

Geometric 0 0 0 morphisms to gi : bi −→ b , i ∈ I , Sh(C, J)

Characterizing and a family of morphisms of C Grothendieck toposes hi : bi −→ c , i ∈ I , Giraud’s theorem Morphisms satisfying between sites f1 ◦ hi = f2 ◦ hi , ∀ i ∈ I , The Comparison 0 Lemma and of morphisms of C

Points of toposes 0 0 hi : bi −→ F(bi ) , i ∈ I , Separating sets of points of a topos making commutative the following squares:

The subterminal g0 topology 0 i 0 bi / b Topos-theoretic Galois theory 0 0 hi f

For further  F(hi )  reading F(bi ) / F(c)

28 / 45 Topos Theory Morphisms of sites IV Olivia Caramello

Geometric If F is a morphism of sites (C, J) → (D, K ), we denote by morphisms Sh(F): Sh(D, K ) → Sh(C, J) the geometric morphism which it Locales and pointless induces. topology

Geometric Remarks morphisms as flat functors • If (C, J) and (D, K ) are cartesian sites (that is, C and D are Geometric morphisms to cartesian categories) then a functor C → D which is cartesian Sh(C, J) and sends J-covering sieves to K -covering sieves is a Characterizing Grothendieck morphism of sites (C, J) → (D, K ). toposes Giraud’s theorem • If J and K are subcanonical then a geometric morphism Morphisms between sites g : Sh(D, K ) → Sh(C, J) is of the form Sh(f ) for some f if and ∗ The Comparison only if the inverse image functor g sends representables to Lemma representables; if this is the case then f is isomorphic to the Points of toposes restriction of g∗ to the full subcategories of representables. Separating sets of points of a topos The subterminal Corollary topology

Topos-theoretic The assignment L → Sh(L) from locales to Grothendieck toposes Galois theory is a full and faithful 2-functor. For further reading

29 / 45 Topos Theory The Comparison Lemma I Olivia Caramello

Geometric morphisms Definition Locales and pointless Let D be a full subcategory of a small category C, and let J be a topology Grothendieck topology on C. Then D is said to be J-dense if for Geometric morphisms as flat every object c ∈ C there exists a sieve S ∈ J(c) generated by a functors family of arrows whose domains lie in D. Geometric morphisms to Sh(C, J) Theorem (The Comparison Lemma) Characterizing Grothendieck Let (C, J) be a site and D be a J-dense subcategory of C. Then toposes Giraud’s theorem the sieves in D of the form R ∩ arr(D) for a J-covering sieve R in Morphisms C form a Grothendieck topologyJ | on D, called the induced between sites D

The Comparison topology, and, denoted by i : D → C the canonical inclusion Lemma functor, the geometric morphism Points of toposes

Separating sets op op of points of a E(i):[D , Set] → [C , Set], topos

The subterminal topology restricts to an equivalence of categories

Topos-theoretic Galois theory E(i)| : Sh(D, J|D) ' Sh(C, J) . For further reading

30 / 45 Topos Theory The Comparison Lemma II Olivia Caramello

Geometric morphisms Locales and Corollary pointless topology

Geometric • Let B be a basis of a frame L, i.e. a subset B ⊂ L such that morphisms as flat functors every element in L can be written as a join of elements in B;

Geometric then we have an equivalence of categories morphisms to Sh(C, J) L Characterizing Sh(L) ' Sh(B, J |B), Grothendieck toposes L Giraud’s theorem where J is the canonical topology on L. Morphisms between sites • Let C be a preorder and J be a subcanonical topology on C. The Comparison Then we have an equivalence of categories Lemma

Points of toposes Sh(C, J) ' Sh(IdJ (C)), Separating sets of points of a topos where IdJ (C) is the frame of J-ideals on C (regarded as a The subterminal topology locale). In fact, this result holds also without the Topos-theoretic subcanonicity assumption. Galois theory

For further reading

31 / 45 Topos Theory Comorphisms of sites Olivia Caramello

Geometric morphisms

Locales and pointless Definition topology A comorphism of sites (D, K ) → (C, J) is a functor D → C which is Geometric morphisms as flat cover-reflecting (in the sense that for any d ∈ D and any functors J-covering sieve S on π(d) there is a K -covering sieve R on d Geometric morphisms to such that π(R) ⊆ S). Sh(C, J)

Characterizing Grothendieck Proposition toposes Giraud’s theorem Every comorphism of sites π : D → C induces a flat and Morphisms J-continuous functor Aπ : C → Sh(D, K ) given by between sites

The Comparison Lemma Aπ(c) = aK (HomC(π(−), c)) Points of toposes

Separating sets and hence a geometric morphism of points of a topos

The subterminal f : Sh(D, K ) → Sh(C, J) topology Topos-theoretic ∗ ∼ Galois theory with inverse image f (F) = aK (F ◦ π) for any J-sheaf F on C.

For further reading

32 / 45 Topos Theory Points of toposes Olivia Caramello

Geometric morphisms Definition Locales and pointless By a point of a topos E, we mean a geometric morphism Set → E. topology

Geometric morphisms as flat Examples functors Geometric • For any site (C, J), the points of the topos Sh(C, J) morphisms to Sh(C, J) correspond precisely to the J-continuous flat functors Characterizing C → Set; Grothendieck toposes • Giraud’s theorem For any locale L, the points of the topos Sh(L) correspond

Morphisms precisely to the frame homomorphisms L → {0, 1}; between sites • The Comparison For any small category C and any object c of C, we have a Lemma op op point ec : Set → [C , Set] of the topos [C , Set], whose Points of toposes inverse image is the evaluation functor at c. Separating sets of points of a topos

The subterminal Fact topology Any set of points P of a Grothendieck topos E indexed by a set X Topos-theoretic Galois theory via a function ξ : X → P can be identified with a geometric For further morphism ξ˜ :[X, Set] → E. reading

33 / 45 Topos Theory Separating sets of points Olivia Caramello

Geometric morphisms

Locales and pointless topology Definition Geometric • E E morphisms as flat Let be a topos and P be a collection of points of indexed functors by a set X via a function ξ : X → P. We say that P is Geometric morphisms to separating for E if the points in P are jointly surjective, i.e. if Sh(C, J) the inverse image functors of the geometric morphisms in P Characterizing Grothendieck jointly reflect isomorphisms (equivalently, if the geometric toposes ˜ Giraud’s theorem morphism ξ :[X, Set] → E is surjective, that is its inverse Morphisms image reflects isomorphisms). between sites

The Comparison • A topos is said to have enough points if the collection of all Lemma the points of E is separating for E. Points of toposes

Separating sets of points of a topos Fact The subterminal A Grothendieck topos has enough points if and only if there exists topology a set of points of E which is separating for E. Topos-theoretic Galois theory

For further reading

34 / 45 Topos Theory The subterminal topology Olivia Caramello

Geometric The following notion provides a way for endowing a given set of morphisms points of a topos with a natural topology. Locales and pointless topology Definition

Geometric Let ξ : X → P be an indexing of a set P of points of a Grothendieck morphisms as flat E functors topos E by a set X. We define the subterminal topology τξ as the Geometric image of the function φE : SubE (1) → P(X) given by morphisms to Sh(C, J) ∗ ∼ Characterizing φE (u) = {x ∈ X | ξ(x) (u) = 1Set} . Grothendieck toposes Giraud’s theorem E We denote the space X endowed with the topology τ by Xτ E . Morphisms ξ ξ between sites

The Comparison Lemma The interest of this notion lies in its level of generality, as well as in Points of toposes its formulation as a topos-theoretic invariant admitting a ‘natural Separating sets behaviour’ with respect to sites. of points of a topos

The subterminal Fact topology If P is a separating set of points for E then the frame O(X E ) of open τξ Topos-theoretic Galois theory sets of X E is isomorphic to SubE (1) (via φE ). τξ For further reading

35 / 45 Topos Theory Categories of toposes paired with points Olivia Caramello

Geometric The construction of the subterminal topology can be made functorial. morphisms

Locales and Definition pointless Top topology The category p toposes paired with points has as objects the

Geometric pairs (E, ξ), where E is a Grothendieck topos and ξ : X → P is an morphisms as flat 0 functors indexing of a set of points P of E, and whose arrows (E, ξ) → (F, ξ ), 0 Geometric where ξ : X → P and ξ : Y → Q, are the pairs (f , l) where f : E → F morphisms to Sh(C, J) is a geometric morphism and l : X → Y is a function such that the

Characterizing diagram Grothendieck E(l) toposes [X, Set] [Y , Set] Giraud’s theorem /

Morphisms ˜ ˜0 between sites ξ ξ The Comparison  f  Lemma E / F Points of toposes commutes (up to isomorphism). Separating sets of points of a topos Theorem The subterminal topology We have a functor Topp → Top (where Top is the category of Topos-theoretic topological spaces) sending an object (E, ξ) of Top to the space X E p τξ Galois theory and an arrow (f , l):(E, ξ) → (F, ξ0) in Top to the continuous For further p reading function l : Xτ E → Xτ F . ξ ξ0 36 / 45 Topos Theory Examples of subterminal topologies I Olivia Caramello

Geometric morphisms Definition Locales and Let (C, ≤) be a preorder category. A J-prime filter on C is a subset pointless topology F ⊆ ob(C) such that F is non-empty, a ∈ F implies b ∈ F Geometric whenever a ≤ b, for any a, b ∈ F there exists c ∈ F such that morphisms as flat functors c ≤ a and c ≤ b, and for any J-covering sieve {ai → a | i ∈ I} in C Geometric morphisms to if a ∈ F then there exists i ∈ I such that ai ∈ F. Sh(C, J) Characterizing Theorem Grothendieck toposes Let C be a preorder and J be a Grothendieck topology on it. Then Giraud’s theorem J the space X Sh(C,J) has as set of points the collection F of the Morphisms τ C between sites J-prime filters on C and as open sets the sets the form The Comparison Lemma F = { ∈ F J ∩ 6= ∅}, Points of toposes I F C | F I

Separating sets of points of a where I ranges among the J-ideals on C. In particular, a topos

The subterminal sub-basis for this topology is given by the sets topology Topos-theoretic F = { ∈ F J ∈ }, Galois theory c F C | c F

For further reading where c varies among the objects of C.

37 / 45 Topos Theory Examples of subterminal topologies II Olivia Caramello

Geometric morphisms • The Alexandrov topology( E = [P, Set], where P is a preorder Locales and pointless and ξ is the indexing of the set of points of E corresponding topology to the elements of P) Geometric morphisms as flat functors • The Stone topology for distributive lattices( E = Sh(D, Jcoh)

Geometric and ξ is an indexing of the set of all the points of E, where D morphisms to Sh(C, J) is a distributive lattice and Jcoh is the coherent topology on it) Characterizing op Grothendieck • A topology for meet-semilattices( E = [M , Set] and ξ is an toposes indexing of the set of all the points of E, where M is a Giraud’s theorem

Morphisms meet-semilattice) between sites • The space of points of a locale( E = Sh(L) for a locale L and The Comparison Lemma ξ is an indexing of the set of all the points of E) Points of toposes • A logical topology( E = Sh(C , J ) is the classiying topos of a Separating sets T T of points of a geometric theory and ξ is any indexing of the set of all the topos T

The subterminal points of E i.e. models of T) topology • The Zariski topology Topos-theoretic Galois theory

For further ... reading

38 / 45 Topos Theory The topos Cont(G) Olivia Caramello

Geometric Proposition morphisms Given a topological G, the category Cont(G) of continuous Locales and actions of G on discrete sets and G-equivariant maps between pointless them is a Grothendieck topos, equivalent to the topos topology Sh(Contt (G), Jat) of sheaves with respect to the atomic topology Geometric Jat on the full subcategory Contt (G) on the non-empty transitive morphisms as flat actions. functors

Geometric Remark morphisms to In fact, the same result holds by replacing Contt (G) with the full Sh(C, J) subcategory ContU (G) of Cont(G) on the actions of the form G/U Characterizing where U ∈ C for a cofinal system U of open subgroups of G (that is, Grothendieck a collection of subgroups such that every open subgroup contains a toposes member of U). Giraud’s theorem Recall that classical topological Galois theory provides, given a Morphisms between sites Galois extension F ⊆ L, a bijective correspondence between the intermediate field extensions (resp. finite field extensions) The Comparison Lemma F ⊆ K ⊆ L and the closed (resp. open) subgroups of the Galois group AutF (L). This admits the following categorical reformulation: Points of toposes the functor K → Hom(K , L) defines an equivalence of categories Separating sets L op of points of a (LF ) ' Contt (AutF (L)), topos

The subterminal which extends to an equivalence of toposes topology L op Sh(LF , Jat ) ' Cont(AutF (L)) . Topos-theoretic Galois theory A natural question thus arises: can we characterize the categories For further C whose dual is equivalent to (or fully embeddable into) the reading category of (non-empty transitive) actions of a topological automorphism group? 39 / 45 Topos Theory Topos-theoretic Galois theory Olivia Caramello

Geometric Theorem morphisms Let C be a small category satisfyingAP and JEP, and let u be a Locales and pointless C-universal et C-ultrahomogeneous object of the ind-completion topology

Geometric Ind-C of C. Then there is an equivalence of toposes morphisms as flat functors op Geometric Sh(C , Jat ) ' Cont(Aut(u)), morphisms to Sh(C, J) where Aut(u) is endowed with the topology in which a basis of Characterizing Grothendieck open neighbourhoods of the identity is given by the subgroups of toposes Giraud’s theorem the form Iχ = {α ∈ Aut(u) | α ◦ χ = χ} for χ : c → u an arrow in Morphisms between sites Ind-C from an object c of C.

The Comparison This equivalence is induced by the functor Lemma Points of toposes F : Cop → Cont(Aut(u)) Separating sets of points of a topos which sends any object c of C to the set HomInd-C(c, u) (endowed The subterminal topology with the obvious action of Aut(u)) and any arrow f : c → d in C to

Topos-theoretic the Aut(u)-equivariant map Galois theory

For further reading − ◦ f : HomInd-C(d, u) → HomInd-C(c, u) .

40 / 45 Topos Theory Restricting to sites Olivia Caramello

Geometric morphisms

Locales and pointless The following result arises from two ‘bridges’, respectively topology obtained by considering the invariant notions of atom and of arrow Geometric morphisms as flat between atoms. functors

Geometric op morphisms to Sh(C , Jat ) ' Cont(Aut(u)) Sh(C, J)

Characterizing Grothendieck toposes op Giraud’s theorem C Contt (Aut(u))

Morphisms between sites

The Comparison Lemma Theorem

Points of toposes Under the hypotheses of the last theorem, the functor F is full and

Separating sets faithful if and only if every arrow of C is a strict monomorphism, of points of a topos and it is an equivalence on the full subcategory Contt (Aut(u)) of The subterminal Cont(Aut(u)) on the non-empty transitive actions if C is moreover topology atomically complete. Topos-theoretic Galois theory

For further reading

41 / 45 Topos Theory Applications Olivia Caramello

Geometric morphisms

Locales and pointless topology • A natural source of ultrahomogenenous and universal objects Geometric morphisms as flat is provided by Fraïssé’s construction in Model Theory and its functors categorical generalizations. Geometric morphisms to For instance, if the category C is countable and all its arrows Sh(C, J) are then there always exists a C-universal Characterizing Grothendieck and C-ultrahomogeneous object in Ind-C. toposes Giraud’s theorem

Morphisms • This theorem generalizes Grothendieck’s theory of Galois between sites categories (which corresponds to the particular case when The Comparison Lemma the fundamental group is profinite). Points of toposes Separating sets • It can be applied for generating Galois-type theories in of points of a topos different fields of Mathematics, which do not fit in the The subterminal formalism of Galois categories. topology

Topos-theoretic Galois theory

For further reading

42 / 45 Topos Theory Examples Olivia Caramello

Geometric morphisms

Locales and pointless topology Natural categories with monic arrows: C equal to the category of Geometric morphisms as flat functors • Finite sets and injections Geometric • morphisms to Finite graphs and embeddings Sh(C, J) • Finite groups and injective homomorphisms Characterizing Grothendieck toposes Giraud’s theorem Natural categories with epic arrows: Cop equal to the category of Morphisms between sites • Finite sets and surjections The Comparison Lemma • Finite groups and surjective homomorphisms Points of toposes

Separating sets • Finite graphs and homomorphisms which are surjective both of points of a topos at the level of vertices and at the level of edges.

The subterminal topology

Topos-theoretic Galois theory

For further reading

43 / 45 Topos Theory Categories of ‘imaginaries’ Olivia Caramello

Geometric morphisms

Locales and pointless topology • If a category C satisfies the first but not the second condition Geometric of our last theorem, our topos-theoretic approach gives us a morphisms as flat functors fully explicit way to complete it, by means of the addition of Geometric ‘imaginaries’, so that also the second condition gets satisfied. morphisms to Sh(C, J) Characterizing • This is the case for instance for the categories considered Grothendieck toposes above; so we get notions of ‘imaginary finite set’, ‘imaginary Giraud’s theorem finite group’ etc. Morphisms between sites

The Comparison • The objects of the atomic completion admit an explicit Lemma description in terms of equivalence relations in the topos Points of toposes op op Sh(C , Jat ) on objects coming from the site C . Separating sets of points of a topos • In a joint work with L. Lafforgue we give an alternative The subterminal topology concrete description of the atomic completion.

Topos-theoretic Galois theory

For further reading

44 / 45 Topos Theory For further reading Olivia Caramello

Geometric morphisms

Locales and pointless topology O. Caramello. Geometric morphisms as flat Theories, Sites, Toposes: Relating and studying functors

Geometric mathematical theories through topos-theoretic ‘bridges’ morphisms to Sh(C, J) Oxford University Press, 2017.

Characterizing Grothendieck P. T. Johnstone. toposes Giraud’s theorem Sketches of an Elephant: a topos theory compendium, vols. 1

Morphisms and 2 between sites Oxford University Press, 2002. The Comparison Lemma S. Mac Lane and I. Moerdijk. Points of toposes Sheaves in geometry and logic: a first introduction to topos Separating sets of points of a theory topos

The subterminal Springer-Verlag, 1992. topology

Topos-theoretic Galois theory

For further reading

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