Topos Theory

Home , Grothendieck topology, Sieve (category theory), Subfunctor, Subobject

Olivia Caramello

Sheaves on a site Grothendieck topologies Grothendieck toposes

Basic properties of Grothendieck toposes Subobject lattices Topos Theory Balancedness The epi-mono factorization Lectures 7-14: Sheaves on a site The closure operation on subobjects Monomorphisms and epimorphisms Exponentials Olivia Caramello The subobject classiﬁer

Local operators

For further reading Topos Theory Sieves Olivia Caramello In order to ‘categorify’ the notion of sheaf of a topological space, Sheaves on a site Grothendieck the first step is to introduce an abstract notion of covering (of an topologies Grothendieck object by a family of arrows to it) in a category. toposes Basic properties Definition of Grothendieck toposes Subobject lattices • Given a category C and an object c ∈ Ob(C), a presieve P in Balancedness C on c is a collection of arrows in C with codomain c. The epi-mono factorization The closure • Given a category C and an object c ∈ Ob(C), a sieve S in C operation on subobjects on c is a collection of arrows in C with codomain c such that Monomorphisms and epimorphisms Exponentials The subobject f ∈ S ⇒ f ◦ g ∈ S classifier Local operators whenever this composition makes sense. For further reading • We say that a sieve S is generated by a presieve P on an object c if it is the smallest sieve containing it, that is if it is the collection of arrows to c which factor through an arrow in P.

If S is a sieve on c and h : d → c is any arrow to c, then h∗(S) := {g | cod(g) = d, h ◦ g ∈ S} is a sieve on d. 2 / 43 Topos Theory Grothendieck topologies I Olivia Caramello

Sheaves on a site Deﬁnition Grothendieck topologies Grothendieck • A Grothendieck topology on a category C is a function J toposes which assigns to each object c of C a collection J(c) of sieves Basic properties of Grothendieck on c in such a way that toposes Subobject lattices (i)(maximality axiom) the maximal sieve Mc = {f | cod(f ) = c} is Balancedness The epi-mono in J(c); factorization ∈ ( ) ∗( ) ∈ ( ) The closure (ii)(stability axiom) if S J c , then f S J d for any arrow operation on subobjects f : d → c; Monomorphisms and epimorphisms (iii)(transitivity axiom) if S ∈ J(c) and R is any sieve on c such that ∗ Exponentials f (R) ∈ J(d) for all f : d → c in S, then R ∈ J(c). The subobject classiﬁer The sieves S which belong to J(c) for some object c of C are Local operators said to be J-covering. For further reading • A site is a pair (C, J) where C is a small category and J is a Grothendieck topology on C.

Notice the following basic properties: • If R, S ∈ J(c) then R ∩ S ∈ J(c); • If R and R’ are sieves on an object c such that R0 ⊇ R then R ∈ J(c) implies R0 ∈ J(c). 3 / 43 Topos Theory Grothendieck topologies II Olivia Caramello

Sheaves on a site The notion of a Grothendieck topology can be put in the following Grothendieck topologies alternative (but equivalent) form: Grothendieck toposes Definition Basic properties of Grothendieck A Grothendieck topology on a category C is an assignment J toposes Subobject lattices sending any object c of C to a collection J(c) of sieves on c in Balancedness The epi-mono such a way that factorization The closure (a) the maximal sieve Mc belongs to J(c); operation on subobjects Monomorphisms and (b) for each pair of sieves S and T on c such that T ∈ J(c) and epimorphisms Exponentials S ⊇ T , S ∈ J(c); The subobject classifier (c) if R ∈ J(c) then for any arrow g : d → c there exists a sieve Local operators S ∈ J(d) such that for each arrow f in S, g ◦ f ∈ R; For further reading (d) if the sieve S generated by a presieve {fi : ci → c | i ∈ I} belongs to J(c) and for each i ∈ I we have a presieve {gij : dij → ci | j ∈ Ii } such that the sieve Ti generated by it belongs to J(ci ), then the sieve R generated by the family of composites { fi ◦ gij : dij → c | i ∈ I, j ∈ Ii } belongs to J(c). The sieve R defined in (d) will be called the composite of the sieve S with the sieves Ti for i ∈ I and denoted by S ∗ {Ti | i ∈ I}. 4 / 43 Topos Theory Bases for a Grothendieck topology Olivia Caramello

Sheaves on a site Deﬁnition Grothendieck topologies A basis (for a Grothendieck topology) on a category C with Grothendieck toposes pullbacks is a function K assigning to each object c of C a Basic properties of Grothendieck collection K (c) of presieves on c in such a way that the following toposes properties hold: Subobject lattices Balancedness (i) {1 : c → c} ∈ K (c) The epi-mono c factorization (ii) if {f : c → c | i ∈ I} ∈ K (c) then for any arrow g : d → c in C, The closure i i operation on ∗ subobjects the family of pullbacks {g (fi ): ci ×c d → d | i ∈ I} lies in K (d). Monomorphisms and epimorphisms (iii) if {fi : ci → c | i ∈ I} ∈ K (c) and for each i ∈ I we have a Exponentials presieve {gij : dij → ci | j ∈ Ii } ∈ K (ci ) then the family of The subobject classiﬁer composites { fi ◦ gij : dij → c | i ∈ I, j ∈ Ii } belongs to K (c). Local operators N.B. If C does not have pullbacks then condition (ii) can be replaced For further reading by the following requirement: if {fi : ci → c | i ∈ I} ∈ K (c) then for any arrow g : d → c in C, there is a presieve {hj : dj → d | j ∈ J} ∈ K (d) such that for each j ∈ J, g ◦ hj factors through some fi . Every basis K generates a Grothendieck topology J given by:

R ∈ J(c) if and only if R ⊇ S for some S ∈ K (c)

5 / 43 Topos Theory Grothendieck topology generated by a coverage Olivia Caramello

Sheaves on a site As we shall also see when we talk about sheaves, the axioms for Grothendieck topologies Grothendieck topologies do not have all the same status: the Grothendieck toposes most important one is the stability axiom. This motivates the

Basic properties following definition. of Grothendieck toposes Definition Subobject lattices Balancedness A (sifted) coverage on a category C is a collection of sieves which The epi-mono factorization is stable under pullback. The closure operation on subobjects Fact Monomorphisms and epimorphisms The Grothendieck topology generated by a coverage is the Exponentials The subobject smallest collection of sieves containing it which is closed under classifier

Local operators maximality and transitivity.

For further reading Theorem Let C be a small category and D a coverage on D. Then the Grothendieck topology GD generated by D is given by f GD(c) = {S sieve on c | for any arrow d → c and sieve T on d, g [(for any arrow e → d and sieve Z on e (Z ∈ D(e) and Z ⊆ g∗(T )) implies g ∈ T ) and ∗ (f (S) ⊆ T )] implies T = Md } 6 / 43 for any object c ∈ C. Topos Theory Examples of Grothendieck topologies I Olivia Caramello

Sheaves on a site • For any (small) category C, the trivial topology on C is the Grothendieck topologies Grothendieck topology in which the only sieve covering an Grothendieck toposes object c is the maximal sieve Mc. Basic properties • The dense topology D on a category C is defined by: for a of Grothendieck toposes sieve S, Subobject lattices Balancedness The epi-mono factorization S ∈ D(c) if and only if for any f : d → c there exists The closure operation on g : e → d such that f ◦ g ∈ S . subobjects Monomorphisms and epimorphisms If C satisfies the right Ore condition i.e. the property that any Exponentials The subobject two arrows f : d → c and g : e → c with a common codomain classifier c can be completed to a commutative square Local operators

For further reading • / d

f g e / c then the dense topology on C specializes to the atomic topology on C i.e. the topology Jat deﬁned by: for a sieve S,

S ∈ Jat (c) if and only if S 6= ∅ . 7 / 43 Topos Theory Examples of Grothendieck topologies II Olivia Caramello

Sheaves on a site • If X is a topological space, the usual notion of covering in Grothendieck topologies Topology gives rise to the following Grothendieck topology Grothendieck toposes JO(X) on the poset category O(X): for a sieve Basic properties of Grothendieck S = {Ui ,→ U | i ∈ I} on U ∈ Ob(O(X)), toposes Subobject lattices Balancedness S ∈ JO(X)(U) if and only if ∪ Ui = U . The epi-mono factorization i∈I The closure operation on subobjects • More generally, given a frame (or complete Heyting algebra) Monomorphisms and epimorphisms H, we can deﬁne a Grothendieck topology JH , called the Exponentials The subobject canonical topology on H, by: classiﬁer

Local operators For further {ai | i ∈ I} ∈ JH (a) if and only if ∨ ai = a . reading i∈I

• Given a small category of topological spaces which is closed under ﬁnite limits and under taking open subspaces, one may deﬁne the open cover topology on it by specifying as basis the collection of open embeddings {Yi ,→ X | i ∈ I} such that

∪Yi = X. i∈I

8 / 43 Topos Theory Topologies with smallest covering sieves Olivia Caramello

Sheaves on a site Grothendieck Deﬁnition topologies Let A be a collection of arrows in a category C which is closed Grothendieck toposes under composition on the left and which is interpolative in the Basic properties of Grothendieck sense that every arrow in A can be factored as the composition of toposes Subobject lattices two arrows in A. Then there is a Grothendieck topology JA on C Balancedness given by: The epi-mono factorization The closure operation on S ∈ J (c) if and only if ∀f ∈ A, cod(f ) = c implies f ∈ S . subobjects A Monomorphisms and epimorphisms Exponentials The subobject classiﬁer Example Local operators Given a full subcategory D of C, one can take A to be the For further reading collection of arrows whose domains lie in D. Fact The Grothendieck topologies on C of the form JA are precisely as those which have a smallest covering sieve on every object. N.B. If A is also closed under composition on the right then it can be recovered from the associated Grothendieck topology as the collection of arrows which belong to a smallest covering sieve. 9 / 43 Topos Theory The Zariski site I Olivia Caramello • Given a commutative ring with unit A, we can endow the Sheaves on a site Grothendieck collection Spec(A) of its prime ideals with the Zariski topologies Grothendieck topology, whose basis of open sets is given by the subsets toposes

Basic properties of Grothendieck Spec(A)f := {P ∈ Spec(A) | f ∈/ P} toposes Subobject lattices (for f ∈ A). Balancedness The epi-mono • One can prove that Spec(A) = Spec(A)f1 ∪ ... ∪ Spec(A)fn if factorization The closure and only if A = (f1,..., fn). operation on subobjects • We have a structure sheaf O on Spec(A) such that Monomorphisms and epimorphisms O(Spec(A)f ) = Af for each f ∈ A. The fact that it is a sheaf Exponentials The subobject results from the fact that if A = (f1,..., fn) then the canonical classiﬁer map Local operators Y For further A → Afi reading i∈{1,...n} is the equalizer of the two canonical maps Y Y Afi → Afi fj . i∈I i,j∈{1,...,n}

• The stalk OP of O at a prime ideal P is the localization AP = colimf ∈/P Af . 10 / 43 Topos Theory The Zariski site II Olivia Caramello Notice that Spec(A) identiﬁes with Spec(A ) under the Sheaves on a site f f Grothendieck embedding topologies Grothendieck Spec(A ) ,→ Spec(A) toposes f

Basic properties of Grothendieck induced by the canonical homomorphism A → Af . toposes This motivates the following definition. Subobject lattices Balancedness The epi-mono Definition factorization The closure The Zariski site (over ) is obtained by equipping the opposite of operation on Z subobjects the category Rng of finitely generated commutative rings with Monomorphisms and f.g. epimorphisms unit with the Grothendieck topology Z given by: for any cosieve S Exponentials The subobject in Rng on an object A, S ∈ Z(A) if and only if S contains a finite classifier f.g.

Local operators family {ξi : A → Afi | 1 ≤ i ≤ n} of canonical maps ξi : A → Afi in For further Rngf.g. where {f1,..., fn} is a set of elements of A which is not reading contained in any proper ideal of A. This definition can be generalized to an arbitrary (commutative) base ring k, by considering the category of finitely presented (equivalently, finitely generated) k-algebras and homomorphisms between them. Notice that pushouts exist in this category (whence pullbacks exist in the opposite category) as they are given by tensor products of k-algebras. 11 / 43 Topos Theory Sheaves on a site I Olivia Caramello

Sheaves on a site Grothendieck topologies Grothendieck toposes Deﬁnition Basic properties • op of Grothendieck A presheaf on a (small) category C is a functor P : C → Set. toposes op Subobject lattices • Let P : C → Set be a presheaf on C and S be a sieve on an Balancedness The epi-mono object c of C. factorization The closure operation on A matching family for S of elements of P is a function which subobjects Monomorphisms and assigns to each arrow f : d → c in S an element xf ∈ P(d) in epimorphisms Exponentials such a way that The subobject classiﬁer

Local operators P(g)(xf ) = xf ◦g for all g : e → d .

For further reading An amalgamation for such a family is a single element x ∈ P(c) such that

P(f )(x) = xf for all f in S .

12 / 43 Topos Theory Sheaves on a site II Olivia Caramello

Sheaves on a site Grothendieck topologies • Given a site (C, J), a presheaf on C is a J-sheaf if every Grothendieck toposes matching family for any J-covering sieve on any object of C Basic properties of Grothendieck has a unique amalgamation. toposes Subobject lattices • The J-sheaf condition can be expressed as the requirement Balancedness The epi-mono that for every J-covering sieve S the canonical arrow factorization The closure operation on Y subobjects P(c) → P(dom(f )) Monomorphisms and epimorphisms f ∈S Exponentials The subobject classiﬁer given by x → (P(f )(x) | f ∈ S) should be the equalizer of the Local operators two arrows For further reading Y Y P(dom(f )) → P(dom(g)) f ∈S f,g, f ∈ S cod(g)=dom(f )

given by (xf → (xf ◦g)) and (xf → (P(g)(xf ))).

13 / 43 Topos Theory The notion of Grothendieck topos Olivia Caramello

Sheaves on a site • The J-sheaf condition can also be expressed as the Grothendieck requirement that for every J-covering sieve S (regarded as a topologies op Grothendieck subobject of HomC(−, c) in [C , Set]), every natural toposes transformation α : S → P admits a unique extension α˜ along Basic properties the embedding S HomC(−, c): of Grothendieck toposes α Subobject lattices S / P Balancedness 9 The epi-mono factorization α˜ The closure operation on (−, ) subobjects HomC c Monomorphisms and epimorphisms Exponentials (notice that a matching family for R of elements of P is The subobject precisely a natural transformation R → P) classiﬁer

Local operators • It can also be expressed as the condition

For further ( ) = ( ) reading P c ←−lim P d f :d→c∈S

for each J-covering sieve S on an object c. • The category Sh(C, J) of sheaves on the site (C, J) is the full subcategory of [Cop, Set] on the presheaves which are J-sheaves. • A Grothendieck topos is any category equivalent to the category of sheaves on a site. 14 / 43 Topos Theory Examples of toposes Olivia Caramello

Sheaves on a site Grothendieck The following examples show that toposes can be naturally topologies Grothendieck attached to mathematical notions as different as (small) toposes

Basic properties categories, topological spaces, or groups. In fact, as we shall see of Grothendieck toposes later in the course, toposes can also be naturally attached to Subobject lattices many other kinds of mathematical objects. Balancedness The epi-mono factorization Examples The closure operation on subobjects • For any (small) category C, [Cop, Set] is the category of Monomorphisms and epimorphisms sheaves Sh(C, T ) where T is the trivial topology on C. Exponentials The subobject classiﬁer • For any topological space X, Sh(O(X), JO(X)) is equivalent to Local operators the usual category Sh(X) of sheaves on X. For further reading • For any (topological) group G, the category BG = Cont(G) of continuous actions of G on discrete sets is a Grothendieck topos (equivalent, as we shall see, to the category Sh(Contt(G), Jat) of sheaves on the full subcategory Contt(G) on the non-empty transitive actions with respect to the atomic topology).

15 / 43 Topos Theory The sheaf condition for presieves Olivia Caramello It is sometimes convenient to check the sheaf condition for the sieve Sheaves on a site Grothendieck generated by a presieve directly in terms of the presieve. topologies Grothendieck toposes Definition op Basic properties A presheaf F : C → Set satisfies the sheaf condition with respect of Grothendieck toposes to a presieve P = {fi : ci → c | i ∈ I} if for any family of elements Subobject lattices {x ∈ P(c ) | i ∈ I} such that for any arrows h and k with Balancedness i i The epi-mono fi ◦ h = fj ◦ k, F(h)(xi ) = F(k)(xj ) there exists a unique element factorization The closure x ∈ P(c) such that F(fi )(x) = xi for all i. operation on subobjects Clearly, F satisfies the sheaf condition with respect to the presieve Monomorphisms and epimorphisms P if and only if it satisfies it with respect to the sieve generated by P. Exponentials The subobject classifier The sheaf condition for the presieve P can be expressed as the Local operators requirement that the canonical diagram For further reading Q / Q F(c) / F(ci ) / F(e) i∈I h : e → ci , k : e → cj

fi ◦ h = fj ◦ k

is an equalizer. N.B. If C has pullbacks then the product on the right-hand side can be simply indexed by the pairs (i, j) (e = ci ×c cj and h and k being

equal to the pullback projections). 16 / 43 Topos Theory Some remarks I Olivia Caramello

Sheaves on a site Grothendieck topologies Grothendieck The following facts show that the notion of sheaf behaves very toposes

Basic properties naturally with respect to the notions of coverage and of of Grothendieck toposes Grothendieck topology: Subobject lattices Balancedness (i) For any presheaf P, the collection LP of sieves R such that P The epi-mono factorization satisﬁes the sheaf axiom with respect to all the pullbacks The closure ∗ operation on sieves f (R) is a Grothendieck topology, and the largest one subobjects Monomorphisms and for which P is a sheaf. epimorphisms Exponentials The subobject classiﬁer (ii) By intersecting such topologies, we can deduce that for any

Local operators given collection of presheaves there is a largest For further Grothendieck topology for which all of them are sheaves. reading (iii) By (i), if a presheaf satisﬁes the sheaf condition with respect to a coverage then it satisﬁes the sheaf condition with respect to the Grothendieck topology generated by it.

17 / 43 Topos Theory Some remarks II Olivia Caramello

Sheaves on a site Proposition Grothendieck op topologies Let P be a presheaf C → Set. Then Grothendieck toposes (i) If P satisfies the sheaf condition with respect to a sieve S and Basic properties of Grothendieck to each of the sieves in a family {Rf | f ∈ S} (where Rf is a toposes Subobject lattices sieve on dom(f )) and all their pullbacks then P satisfies the Balancedness sheaf condition with respect to the composite sieve The epi-mono factorization S ∗ {Rf | f ∈ S}. The closure operation on (ii) If P satisfies the sheaf condition with respect to all the subobjects Monomorphisms and pullbacks of a sieve S then it satisfies the sheaf condition with epimorphisms Exponentials respect to each sieve T ⊇ S. The subobject classifier (iii) If P satisfies the sheaf condition with respect to all the

Local operators pullbacks of a sieve S on an object c and all the pullbacks of ∗ For further sieves of the form g (R) for a sieve R on c indexed by arrows reading g in S then it satisﬁes the sheaf condition with respect to R.

The fact that LP satisfies the transitivity axiom for Grothendieck topologies follows from (iii) (the sheaf condition for the pullbacks f ∗(R) of R follows from (iii) applied to f ∗(R) and f ∗(S) in place of R and S), which in turn can be proved by using (i) to deduce that P satisfies the sheaf condition for S ∗ {g∗(R) | g ∈ S} ⊆ R, and by an argument similar to that used for establishing (ii) to deduce from this that P satisfies the sheaf condition also with respect to R. 18 / 43 Topos Theory Subcanonical sites Olivia Caramello

Sheaves on a site Definition Grothendieck C topologies A Grothendieck topology J on a (small) category is said to be op Grothendieck subcanonical if every representable functor HomC(−, c): C → Set is a toposes J-sheaf. Basic properties of Grothendieck toposes Fact Subobject lattices Balancedness For any locally small category C, there exists the largest Grothendieck The epi-mono factorization topology J on C for which all representables on C are J-sheaves. It is The closure operation on called the canonical topology on C. subobjects Monomorphisms and epimorphisms Definition Exponentials The subobject • A sieve R on an object c of a locally small category C is said to be classifier effective-epimorphic if it forms a colimit cone under the (large!) Local operators diagram consisting of the domains of all the morphisms in R, and For further reading all the morphisms over c between them. • It is said to be universally effective-epimorphic if its pullback along every arrow to c is effective-epimorphic.

The covering sieves for the canonical topology on a locally small category are precisely the universally effective-epimorphic ones. It follows that a Grothendieck topology is subcanonical if and only if it is contained in the canonical topology, that is if and only if all its covering

sieves are effective-epimorphic. 19 / 43 Topos Theory Basic properties of Grothendieck toposes Olivia Caramello In the next lectures, we shall prove the following result, showing that Sheaves on a site Grothendieck Grothendieck toposes satisfy all the categorical properties that one topologies Grothendieck might hope for. toposes Basic properties Theorem of Grothendieck toposes Let (C, J) be a site. Then Subobject lattices op Balancedness • the inclusion Sh(C, J) ,→ [C , Set] has a left adjoint The epi-mono op factorization a :[C , Set] → Sh(C, J) (called the associated sheaf functor), The closure which preserves finite limits. operation on subobjects • The category Sh(C, J) has all (small) limits, which are preserved Monomorphisms and op epimorphisms by the inclusion functor Sh(C, J) ,→ [C , Set]; in particular, limits Exponentials The subobject are computed pointwise and the terminal object 1Sh(C,J) of classifier Sh(C, J) is the functor T : Cop → Set sending each object Local operators c ∈ Ob(C) to the singleton {∗}. For further op reading • The associated sheaf functor a :[C , Set] → Sh(C, J) preserves colimits; in particular, Sh(C, J) has all (small) colimits. • The category Sh(C, J) has exponentials, which are constructed as in the topos [Cop, Set]. • The category Sh(C, J) has a subobject classifier.

Corollary Every Grothendieck topos is an elementary topos. 20 / 43 Topos Theory The plus construction Olivia Caramello Let us start by establishing the following fundamental theorem. Sheaves on a site Grothendieck topologies Theorem Grothendieck op toposes For any site (C, J), the inclusion Sh(C, J) ,→ [C , Set] has a left op Basic properties adjoint a :[C , Set] → Sh(C, J), called the associated sheaf of Grothendieck functor, which preserves finite limits. toposes Subobject lattices The associated sheaf functor can be constructed as the functor Balancedness + The epi-mono obtained by applying twice the plus construction P → P . The plus factorization functor is defined as follows: The closure operation on subobjects + Monomorphisms and P (c) = colimR∈J(c)Match(R, P) epimorphisms Exponentials The subobject where Match(R, P) is the set of matching families for R of elements classifier of P (the action of P+ on arrows being given by reindexing of the Local operators matching family along the pullback sieve). For further reading Notice that this is a filtered (actually, directed) colimit, so the elements of P+(c) are equivalence classes [x] of matching families x with respect to the equivalence relation ∼ given by equality on a common refinement, that is

x = {xf | f ∈ R} ∼ y = {yg | g ∈ S} if and only if

there exists T ⊆ R ∩ S in J(c) such that xh = yh for all h ∈ T .

21 / 43 Topos Theory Properties of the plus construction Olivia Caramello

Sheaves on a site The following properties of the plus construction will be Grothendieck ++ topologies instrumental for proving that the functor P → P satisfies the Grothendieck toposes universal property of the associated sheaf functor. + Basic properties Notice that we have a natural transformation ηP : P → P given of Grothendieck toposes by: Subobject lattices η ( )( ) = [{ ( )( ) | ∈ }] . Balancedness P c x P f x f Mc The epi-mono factorization A presheaf is said to be separated if it satisfies the uniqueness The closure operation on (but not necessarily the existence) requirement in the definition of subobjects Monomorphisms and a sheaf. epimorphisms Exponentials The subobject Theorem classifier + Local operators (i) A presheaf P is separated if and only if ηP : P → P is a For further monomorphism. reading + (ii) A presheaf P is a sheaf if and only if ηP : P → P is an isomorphism. (iii) Every morphism P → F of a presheaf P to a sheaf F factors + uniquely through ηP : P → P . (iv) For any presheaf P, P+ is a separated presheaf. (v) For any separated presheaf P, P+ is a sheaf. 22 / 43 Topos Theory The associated sheaf functor Olivia Caramello Applying the plus construction just once is in general not enough Sheaves on a site Grothendieck for building a sheaf starting from a presheaf (unless the presheaf topologies Grothendieck is separated). Nonetheless, by the above theorem, for any toposes ++ ++ Basic properties presheaf P, P is a sheaf and the morphism P → P given by of Grothendieck + + ++ + toposes the composite of ηP : P → P and ηP : P → P satisfies the Subobject lattices universal property of the associated sheaf of the presheaf P; that Balancedness The epi-mono is, every morphism φ : P → F of a presheaf P to a sheaf F factors factorization ++ The closure uniquely through ηP+ ◦ ηP : P → P : operation on subobjects Monomorphisms and epimorphisms φ Exponentials P / F The subobject = classifier ηP+ ◦ηP Local operators ˜ φ For further ++ reading P

In other words, the associated sheaf functor op aJ :[C , Set] → Sh(C, J) is left adjoint to the inclusion functor op iJ : Sh(C, J) ,→ [C , Set]. This implies in particular that aJ preserves all (small) colimits. On the other hand, the plus construction preserves finite limits and filtered colimits commute with finite limits in Set, so aJ also preserves finite limits. 23 / 43 Topos Theory Description in terms of locally matching families Olivia Caramello

Sheaves on a site A more compact description of the associated sheaf functor Grothendieck topologies op Grothendieck aJ :[C , Set] → Sh(C, J) toposes

Basic properties of Grothendieck is available. toposes Subobject lattices Deﬁnition Balancedness op The epi-mono Let P : C → Set be a presheaf and J a Grothendieck topology factorization The closure on C. Then operation on subobjects Monomorphisms and • We say that two elements x, y ∈ P(c) of P are locally equal if epimorphisms Exponentials there exists a J-covering sieve R on c such that The subobject classiﬁer P(f )(x) = P(f )(y) for each f ∈ R. Local operators • Given a sieve S on an object c, a locally matching family for For further reading S of elements of P is a function assigning to each arrow f : d → c in S an element xf ∈ P(d) in such a way that, whenever g is composable with f , P(g)(xf ) and P(f ◦ g)(x) are locally equal.

Then aJ (P)(c) consists of equivalence classes of locally matching families for J-covering sieves on c of elements P modulo local equality on a common reﬁnement. 24 / 43 Topos Theory Subobjects in a Grothendieck topos Olivia Caramello

Sheaves on a site Grothendieck topologies Grothendieck Since limits in a topos Sh(C, J) are computed as in the presheaf toposes topos [Cop, Set], a subobject of a sheaf F in Sh(C, J) is just a Basic properties of Grothendieck subsheaf, that is a subfunctor which is a sheaf. toposes Subobject lattices 0 Balancedness Notice that a subfunctor F ⊆ F is a sheaf if and only if for every The epi-mono 0 factorization J-covering sieve S and every element x ∈ F(c), x ∈ F (c) if and The closure 0 operation on only if F(f )(x) ∈ F (dom(f )) for every f ∈ S. subobjects Monomorphisms and epimorphisms Theorem Exponentials The subobject • classiﬁer For any Grothendieck topos E and any object a of E, the

Local operators poset SubE (a) of all subobjects of a in E is a complete For further Heyting algebra. reading • For any arrow f : a → b in a Grothendieck topos E, the ∗ pullback functor f : SubE (b) → SubE (a) has both a left adjoint ∃f : SubE (a) → SubE (b) and a right adjoint ∀f : SubE (a) → SubE (b).

25 / 43 Topos Theory The Heyting operations on subobjects Olivia Caramello

Sheaves on a site Proposition Grothendieck topologies The collection (E) of subobjects of an object E in Grothendieck SubSh(C,J) toposes Sh(C, J) has the structure of a complete Heyting algebra with Basic properties of Grothendieck respect to the natural ordering A ≤ B if and only if for every c ∈ C, toposes A(c) ⊆ B(c). We have that Subobject lattices Balancedness • The epi-mono (A ∧ B)(c) = A(c) ∩ B(c) for any c ∈ C; factorization • (A ∨ B)(c) = {x ∈ E(c) | {f : d → c | E(f )(x) ∈ A(d) ∪ B(d)} The closure operation on subobjects ∈ J(c)} for any c ∈ C; Monomorphisms and epimorphisms (the inﬁnitary analogue of this holds) Exponentials • (A⇒B)(c) = {x ∈ E(c) | for every f : d → c, E(f )(x) ∈ A(d) The subobject classiﬁer implies E(f )(x) ∈ B(d)} for any c ∈ C. Local operators • the bottom subobject 0 E is given by the embedding into E For further reading of the initial object 0 of Sh(C, J) (given by: 0(c) = ∅ if ∅ ∈/ J(c) and 0(c) = {∗} if ∅ ∈ J(c)); • the top subobject is the identity one.

Remark From the Yoneda Lemma it follows that the subobject classiﬁer Ω in Sh(C, J) (see below) has the structure of an internal Heyting algebra in Sh(C, J). 26 / 43 Topos Theory The interpretation of quantiﬁers Olivia Caramello

Sheaves on a site Let φ : E → F be a morphism in Sh(C, J). Grothendieck topologies • The pullback functor Grothendieck toposes ∗ Basic properties φ : SubSh(C,J)(F) → SubSh(C,J)(E) of Grothendieck toposes ∗ −1 Subobject lattices is given by: φ (B)(c) = φ(c) (B(c)) for any subobject Balancedness The epi-mono B F and any c ∈ C. factorization The closure • The left adjoint operation on subobjects Monomorphisms and epimorphisms ∃φ : SubSh(C,J)(E) → SubSh(C,J)(F) Exponentials The subobject classiﬁer is given by: ∃φ(A)(c) = {y ∈ E(c) | {f : d → c | (∃a ∈ Local operators A(d))(φ(d)(a) = E(f )(y))} ∈ J(c)} For further reading for any subobject A E and any c ∈ C. • The right adjoint

∀φ : SubSh(C,J)(E) → SubSh(C,J)(F)

is given by ∀φ(A)(c) = {y ∈ E(c) | for all f : d → c, φ(d)−1(E(f )(y)) ⊆ A(d)} for any subobject A E and any c ∈ C. 27 / 43 Topos Theory Toposes are balanced Olivia Caramello

Sheaves on a site Grothendieck topologies Grothendieck toposes Deﬁnition Basic properties of Grothendieck A category is said to be balanced if every arrow which is both a toposes Subobject lattices monomorphism and an epimorphism is an isomorphism. Balancedness The epi-mono factorization Remark The closure operation on subobjects If in a category a monomorphism is regular (that is, occurs as the Monomorphisms and epimorphisms equalizer of a pair of arrows) then it is an isomorphism if and only Exponentials it it is an epimorphism. The subobject classiﬁer Local operators Proposition For further reading In a Grothendieck topos E, every monomorphism is regular (that is, it is the equalizer of its cokernel pair). In particular, E is balanced. In fact, also epimorphisms in E are all regular.

Recall that regular epimorphisms are stable under pullbacks.

28 / 43 Topos Theory The epi-mono factorization Olivia Caramello

Sheaves on a site Grothendieck topologies Deﬁnition Grothendieck toposes The image Im(f ) of an arrow f : A → B in a category C is, if it Basic properties exists, the smallest subobject of B through which f factors. of Grothendieck toposes Subobject lattices Balancedness Remark The epi-mono Images exist in every Grothendieck topos (and are stable under factorization The closure operation on pullback). In fact, they are obtained from the images calculated in subobjects Monomorphisms and the presheaf topos by applying the associated sheaf functor. epimorphisms Exponentials By recalling that a topos is balanced, we can immediately prove The subobject classiﬁer the following Local operators

For further Proposition reading In every Grothendieck topos, every arrow f can be uniquely (up to a unique isomorphism) factored as an epimorphism followed by a monomorphism; the monic part of the factorization of f is given by its image. The proposition implies in particular that epimorphisms in E can be characterized as the arrows whose image is an isomorphism.

29 / 43 Topos Theory The closure operation on subobjects I Olivia Caramello op Sheaves on a site The associated sheaf functor aJ :[C , Set] → Sh(C, J) induces a Grothendieck op topologies closure operation cJ (m) on subobjects m of [C , Set] (compatible Grothendieck toposes with pullbacks of subobjects), deﬁned by taking the pullback of the Basic properties image a (m) of m : A0 A under a along the unit η of the of Grothendieck J J J toposes adjunction between iJ and aJ : Subobject lattices Balancedness The epi-mono 0 0 factorization cJ (A ) / aJ (A ) The closure operation on subobjects cJ (m) aJ (m) Monomorphisms and epimorphisms η (A) Exponentials J The subobject A / aJ (A) classiﬁer Local operators Concretely, we have For further reading 0 0 cJ (A )(c) = {x ∈ A(c) | {f : d → c | A(f )(x) ∈ A (d)} ∈ J(c)} . Remarks 0 0 • If A is a J-sheaf then aJ (A ) is isomorphic to cJ (A ).

• m is cJ -dense (that is, cJ (m) = 1A) if and only if aJ (m) is an isomorphism.

30 / 43 Topos Theory The closure operation on subobjects II Olivia Caramello

Sheaves on a site Proposition Grothendieck topologies Grothendieck Given a sieve S on an object c, regarded as a subobject toposes op mS : S HomC(−, c) in [C , Set], the following conditions are Basic properties of Grothendieck equivalent: toposes Subobject lattices (a) aJ sends mS to an isomorphism; Balancedness The epi-mono (b) the collection of arrows aJ (yC(f )) for f ∈ S is jointly epimorphic; factorization The closure (c) S is J-covering. operation on subobjects Monomorphisms and We have previously remarked that the sheaf condition for a epimorphisms Exponentials presheaf P with respect to a sieve S could be reformulated as the The subobject classiﬁer requirement that every morphism S → P admits a unique Local operators extension along the canonical embedding S HomC(−, c). In 0 op For further fact, for any cJ -dense subobject A A in [C , Set], if P is a reading J-sheaf then every morphism α : A0 → P admits a unique 0 extension α˜ : A → P along the embedding A A: α A0 / P ?

α˜ A

31 / 43 Topos Theory Monomorphisms and epimorphisms in Sh(C, J) Olivia Caramello

Sheaves on a site Grothendieck topologies Grothendieck op toposes • Since limits in Sh(C, J) are computed as in [C , Set], and the Basic properties of Grothendieck latter are computed pointwise, we have that a morphism toposes α : P → Q in Sh(C, J) is a monomorphism if and only if for Subobject lattices Balancedness every c ∈ C, The epi-mono factorization α(c): P(c) → Q(c) The closure operation on subobjects is an injective function. Monomorphisms and epimorphisms Exponentials The subobject • Since the epimorphisms in Sh(C, J) are precisely the classiﬁer

Local operators morphisms whose image is an isomorphism, we have that a

For further morphism α : P → Q in Sh(C, J) is an epimorphism if and reading only if it is locally surjective in the sense that for every c ∈ C and every x ∈ Q(c),

{f : d → c | Q(f )(x) ∈ Im(α(d))} ∈ J(c) .

32 / 43 Topos Theory Exponentials in Sh(C, J) Olivia Caramello

Sheaves on a site Grothendieck topologies Grothendieck • We preliminarily remark that if exponentials exist in Sh(C, J) toposes then they are computed as in [Cop, Set], by using the Basic properties of Grothendieck adjunction between aJ and iJ and the fact that aJ preserves toposes Subobject lattices ﬁnite products. Balancedness The epi-mono factorization • Next, we use the characterization of the J-sheaves on C as The closure operation on the presheaves P such that for every cJ -dense subobject subobjects 0 0 Monomorphisms and A A, every morphism A → P admits a unique extension epimorphisms 0 Exponentials A → P along the embedding A A to conclude that if F is a The subobject P classiﬁer sheaf then F is a sheaf for every presheaf P: Local operators

For further reading S / F P S × P / F 9 8

HomC(−, c) HomC(−, c) × P

33 / 43 Topos Theory The subobject classiﬁer in Sh(C, J) Olivia Caramello

Sheaves on a site Grothendieck topologies • Given a site (C, J) and a sieve S in C on an object c, we say Grothendieck ∗ toposes that S is J-closed if for any arrow f : d → c, f (S) ∈ J(d) Basic properties of Grothendieck implies that f ∈ S. toposes op Subobject lattices • Let us define ΩJ : C → Set by: Balancedness The epi-mono ΩJ (c) = {R | R is a J-closed sieve on c} (for an object c ∈ C), factorization ∗ The closure ΩJ (f ) = f (−) (for an arrow f in C), operation on ∗ subobjects where f (−) denotes the operation of pullback of sieves in C Monomorphisms and epimorphisms along f . Exponentials The subobject Then the arrow true : 1Sh(C,J) → ΩJ defined by: classifier true(∗)(c) = Mc for each c ∈ Ob(C) Local operators is a subobject classifier for Sh(C, J). For further reading 0 • The classifying arrow χA0 : A → ΩJ of a subobject A ⊆ A in Sh(C, J) is given by:

0 χA0 (c)(x) = {f : d → c | A(f )(x) ∈ A (d)}

where c ∈ Ob(C) and x ∈ A(c).

34 / 43 Topos Theory Closure operations Olivia Caramello

Sheaves on a site Deﬁnition Grothendieck topologies Grothendieck (a)A closure operation c on a partially ordered set (A, ≤) is a toposes function c : A → A satisfying the following properties: Basic properties of Grothendieck • (extensivity) a ≤ c(a) for any a ∈ A; toposes • (order preservation) if a ≤ b then c(a) ≤ c(b); Subobject lattices Balancedness • (idempotency) c(c(a)) = c(a) for any a ∈ A. The epi-mono factorization (b) A closure operation c on subobjects in a topos E is said to be The closure operation on universal if it commutes with pullback, that is if subobjects ∗ ∗ 0 Monomorphisms and c(f (m)) = f (c(m)) for any subobject m : A → A and any epimorphisms arrow f : B → A in E. Exponentials The subobject classiﬁer

Local operators Proposition For further Every universal closure operation c on subobjects in an reading elementary topos preserves ﬁnite intersections of subobjects; that is, c(m ∩ n) = c(m) ∩ c(n). Remark Given a Grothendieck topology J on a small category C, the op operation cJ on subobjects in [C , Set] induced by the associated sheaf functor aJ (as described above) is a universal closure operation in the sense of this deﬁnition. 35 / 43 Topos Theory The concept of local operator Olivia Caramello

Sheaves on a site Deﬁnition Grothendieck topologies Let E be a topos, with subobject classiﬁer > : 1 → Ω.A local Grothendieck toposes operator (or Lawvere-Tierney topology) on E is an arrow j :Ω → Ω

Basic properties in E such that the diagrams of Grothendieck toposes ∧ Subobject lattices 1 Ω Ω × Ω / Ω Balancedness The epi-mono > j factorization > j j×j j The closure operation on subobjects Ω / ΩΩ / ΩΩ × Ω / Ω j j ∧ Monomorphisms and epimorphisms Exponentials commute (where ∧ :Ω × Ω → Ω is the meet operation of the The subobject classiﬁer internal Heyting algebra Ω). Local operators

For further Theorem reading For any elementary topos E, there is a bijection between universal closure operations on E and local operators on E. Sketch of proof. The bijection sends a universal closure operation c on E to the local operator jc :Ω → Ω given by the classifying arrow of the > subobject c(1 >), and a local operator j to the closure operation cj induced by composing classifying arrows with j. 36 / 43 Topos Theory Abstract sheaves Olivia Caramello

Sheaves on a site Deﬁnition Grothendieck topologies Let c be a universal closure operation on an elementary topos E. Grothendieck toposes • A subobject m : a0 → a in E is said to be c-dense if Basic properties of Grothendieck c(m) = ida, and c-closed if c(m) = m. toposes Subobject lattices • An object a of E is said to be a c-sheaf if whenever we have a Balancedness The epi-mono diagram factorization 0 The closure 0 f operation on b / a subobjects Monomorphisms and epimorphisms m Exponentials The subobject classiﬁer b Local operators

For further where m is a c-dense subobject, there exists exactly one reading arrow f : b → a such that f ◦ m = f 0. • The full subcategory of E on the objects which are c-sheaves will be denoted by shc(E).

Fact A subobject of a c-sheaf is c-closed if and only if its domain is a c-sheaf. 37 / 43 Topos Theory The dense-closed factorization Olivia Caramello

Sheaves on a site Grothendieck topologies • Grothendieck For any universal closure operation c, we have the following toposes orthogonality property: for any commutative square Basic properties of Grothendieck toposes 0 0 f 0 Subobject lattices A / B Balancedness The epi-mono factorization m n The closure operation on subobjects Monomorphisms and A / B epimorphisms f Exponentials The subobject classiﬁer where m is c-dense and n is c-closed, there exists a unique 0 Local operators arrow g : A → B such that n ◦ g = f and g ◦ m = f . For further reading • The factorization of a monomorphism m : A0 → A as the canonical monomorphism A0 → c(A0) followed by the subobject c(m) is the unique factorization (up to isomorphism) of m as a c-dense subobject followed by a c-closed one.

38 / 43 Topos Theory Cartesian reflectors Olivia Caramello Definition Sheaves on a site Grothendieck (a) A subcategory F of a category E is said to reflective it is topologies Grothendieck replete (that is, every object isomorphic in E to an object of F toposes also lies in F) and the inclusion functor F ,→ E is full and has Basic properties a left adjoint. of Grothendieck toposes (b) A reflective subcategory of a cartesian category is said to be a Subobject lattices localization if the left adjoint to the inclusion functor preserves Balancedness finite limits. The epi-mono factorization (c) The reflector associated with a localization of a topos C is the The closure operation on functor E → E given by the composite of the inclusion functor subobjects with its left adjoint. (Notice that such a functor is always Monomorphisms and epimorphisms cartesian.) Exponentials The subobject classifier Proposition Local operators Every (cartesian) reflector L : E → E associated with a localization For further of E induces a universal closure operation cL on C defined as reading 0 follows: for any subobject m : A A in E, cL(m) is the subobject of A obtained by taking the pullback of the image L(m) of 0 m : A A under L along the unit η of the localization:

0 0 cL(A ) / L(A )

cL(m) L(m) η(A) A / L(A)

39 / 43 Topos Theory Three equivalent points of view I Olivia Caramello

Sheaves on a site Theorem Grothendieck topologies Grothendieck (i) For any local operator j on an elementary (resp. toposes Grothendieck) topos E, shcj (E) is an elementary (resp. Basic properties of Grothendieck Grothendieck) topos, and the inclusion shcj (E) ,→ E has a left toposes adjoint a : E → sh (E) which preserves finite limits (and Subobject lattices j cj Balancedness satisfies the property that the monomorphisms which it sends The epi-mono factorization to isomorphisms are precisely the cj -dense ones). The closure operation on (ii) Conversely, a localization of E defines, as specified above, a subobjects Monomorphisms and universal closure operation on E and hence a local operator epimorphisms on E. Exponentials The subobject (iii) In fact, these assignments define a bijection between the classifier localizations of E and the local operators (equivalently, the Local operators universal closure operations) on E: For further reading Localizations

Local operators

Universal closure operations

40 / 43 Topos Theory Three equivalent points of view II Olivia Caramello

Sheaves on a site Grothendieck • The proof of the fact that a localization can be recovered from topologies Grothendieck the corresponding closure operation relies on the following toposes result: for any localization L of E with associated reﬂector L Basic properties of Grothendieck and closure operation cL, the following conditions are toposes equivalent for an object A of C: Subobject lattices Balancedness (i) A is cL-separated; The epi-mono factorization (ii) ηA : A → LA is a monomorphism. The closure operation on subobjects Also, the following conditions are equivalent: Monomorphisms and epimorphisms (i) A is a cL-sheaf; Exponentials (ii) ηA : A → LA is an isomorphism; The subobject classiﬁer (iii) A lies in L. Local operators

For further reading • The fact that a closure operation c can be recovered from the associated localization follows from the fact that, for a monomorphism m : A0 → A, m factors as the canonical 0 ∗ monomophism A → dom(ηA(Lm)), which is c-dense (since it is sent by L to an isomorphism), followed by the ∗ monomorphism ηA(Lm), which is c-closed (as it is the pullback of Lm, which is closed).

41 / 43 Topos Theory Local operators and Grothendieck topologies Olivia Caramello

Sheaves on a site Theorem Grothendieck topologies If C is a small category, the Grothendieck topologies J on C Grothendieck toposes correspond exactly to the local operators on the presheaf topos Basic properties op 0 of Grothendieck [C , Set]. (More generally, the Grothendieck topologies J which toposes Subobject lattices contain a given Grothendieck topology J on C correspond exactly Balancedness to the local operators on the topos Sh(C, J).) The epi-mono factorization The closure operation on Sketch of proof. subobjects Monomorphisms and epimorphisms The correspondence sends a local operator j :Ω → Ω to the Exponentials subobject J Ω which it classifies, that is to the Grothendieck The subobject classifier topology J on C defined by: Local operators

For further reading S ∈ J(c) if and only if j(c)(S) = Mc

Conversely, it sends a Grothendieck topology J, regarded as a subobject J Ω, to the arrow j :Ω → Ω that classiﬁes it. In fact, if J is the Grothendieck topology corresponding to a local operator j, an object of [Cop, Set] is a J-sheaf (in the sense of Grothendieck toposes) if and only if it is a cj -sheaf (in the sense of universal closure operations). 42 / 43 Topos Theory For further reading Olivia Caramello

Sheaves on a site Grothendieck topologies Grothendieck toposes Basic properties O. Caramello. of Grothendieck toposes Theories, Sites, Toposes: Relating and studying Subobject lattices Balancedness mathematical theories through topos-theoretic ‘bridges’ The epi-mono factorization Oxford University Press, 2017. The closure operation on subobjects P. T. Johnstone. Monomorphisms and epimorphisms Sketches of an Elephant: a topos theory compendium, vols. 1 Exponentials The subobject and 2 classiﬁer Oxford University Press, 2002. Local operators For further S. Mac Lane and I. Moerdijk. reading Sheaves in geometry and logic: a ﬁrst introduction to topos theory Springer-Verlag, 1992.

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