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Topos Theory Topos Theory 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 2 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 2 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 2 S ) f ◦ g 2 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) := fg | cod(g) = d; h ◦ g 2 Sg 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 = ff | cod(f ) = cg is Balancedness The epi-mono in J(c); factorization 2 ( ) ∗( ) 2 ( ) 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 2 J(c) and R is any sieve on c such that ∗ Exponentials f (R) 2 J(d) for all f : d ! c in S, then R 2 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 2 J(c) then R \ S 2 J(c); • If R and R’ are sieves on an object c such that R0 ⊇ R then R 2 J(c) implies R0 2 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 2 J(c) and epimorphisms Exponentials S ⊇ T , S 2 J(c); The subobject classifier (c) if R 2 J(c) then for any arrow g : d ! c there exists a sieve Local operators S 2 J(d) such that for each arrow f in S, g ◦ f 2 R; For further reading (d) if the sieve S generated by a presieve ffi : ci ! c j i 2 Ig belongs to J(c) and for each i 2 I we have a presieve fgij : dij ! ci j j 2 Ii g such that the sieve Ti generated by it belongs to J(ci ), then the sieve R generated by the family of composites f fi ◦ gij : dij ! c j i 2 I; j 2 Ii g 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 2 I and denoted by S ∗ fTi j i 2 Ig. 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) f1 : c ! cg 2 K (c) The epi-mono c factorization (ii) if ff : c ! c j i 2 Ig 2 K (c) then for any arrow g : d ! c in C, The closure i i operation on ∗ subobjects the family of pullbacks fg (fi ): ci ×c d ! d j i 2 Ig lies in K (d). Monomorphisms and epimorphisms (iii) if ffi : ci ! c j i 2 Ig 2 K (c) and for each i 2 I we have a Exponentials presieve fgij : dij ! ci j j 2 Ii g 2 K (ci ) then the family of The subobject classifier composites f fi ◦ gij : dij ! c j i 2 I; j 2 Ii g 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 ffi : ci ! c j i 2 Ig 2 K (c) then for any arrow g : d ! c in C, there is a presieve fhj : dj ! d j j 2 Jg 2 K (d) such that for each j 2 J, g ◦ hj factors through some fi . Every basis K generates a Grothendieck topology J given by: R 2 J(c) if and only if R ⊇ S for some S 2 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) = fS sieve on c j for any arrow d ! c and sieve T on d, g [(for any arrow e ! d and sieve Z on e (Z 2 D(e) and Z ⊆ g∗(T )) implies g 2 T ) and ∗ (f (S) ⊆ T )] implies T = Md g 6 / 43 for any object c 2 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 2 D(c) if and only if for any f : d ! c there exists The closure operation on g : e ! d such that f ◦ g 2 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 2 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 = fUi ,! U | i 2 Ig on U 2 Ob(O(X)), toposes Subobject lattices Balancedness S 2 JO(X)(U) if and only if [ Ui = U : The epi-mono factorization i2I 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 fai | i 2 Ig 2 JH (a) if and only if _ ai = a : reading i2I • 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 fYi ,! X j i 2 Ig such that [Yi = X.
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