A topos-theoretic approach to Stone-type dualities Olivia Caramello The abstract machinery The general A topos-theoretic approach to method Equivalences with categories of frames Stone-type dualities The subterminal topology Dualities with topological spaces New dualities Olivia Caramello Other applications For further University of Cambridge reading Workshop on Duality Theory Oxford, 3 August 2011
A topos-theoretic Stone-type dualities approach to Stone-type dualities Olivia Caramello Consider the following ‘Stone-type dualities’: The abstract machinery • Stone duality for distributive lattices (and Boolean algebras) The general method • Lindenbaum-Tarski duality for atomic complete Boolean Equivalences with categories of frames algebras The subterminal topology • The duality between spatial frames and sober spaces Dualities with topological spaces • M. A. Moshier and P . Jipsen’s topological duality for New dualities meet-semilattices Other applications • Alexandrov equivalence between preorders and Alexandrov For further reading spaces • Birkhoff duality for finite distributive lattices • The duality between algebraic lattices and sup-semilattices • The duality between completely distributive algebraic lattices and posets They are all instances of just one topos-theoretic phenomenon. 2 / 27
A topos-theoretic Stone-type dualities approach to Stone-type dualities Olivia Caramello Consider the following ‘Stone-type dualities’: The abstract machinery • Stone duality for distributive lattices (and Boolean algebras) The general method • Lindenbaum-Tarski duality for atomic complete Boolean Equivalences with categories of frames algebras The subterminal topology • The duality between spatial frames and sober spaces Dualities with topological spaces • M. A. Moshier and P . Jipsen’s topological duality for New dualities meet-semilattices Other applications • Alexandrov equivalence between preorders and Alexandrov For further reading spaces • Birkhoff duality for finite distributive lattices • The duality between algebraic lattices and sup-semilattices • The duality between completely distributive algebraic lattices and posets They are all instances of just one topos-theoretic phenomenon. 3 / 27
A topos-theoretic A machinery for generating dualities approach to Stone-type dualities Olivia Caramello The abstract machinery • In this talk we present a general topos-theoretic machinery The general method for generating dualites or equivalences between categories Equivalences with categories of frames of preordered structures and categories of posets, locales or The subterminal topology topological spaces. Dualities with topological spaces • All of the above-mentioned dualities are recovered as the New dualities result of applying the machinery to particular sets of Other applications ‘ingredients’, and new dualities are established. For further reading • In fact, infinitely many new dualities can be generated through the machinery in an essentially automatic way. • The machinery is interesting because of its inherent technical flexibility; there are essentially four degrees of freedom in choosing the ingredients. 4 / 27
A topos-theoretic Grothendieck topologies on preorders approach to Stone-type dualities Olivia Caramello The abstract machinery Definition The general method Let C be a preorder. Equivalences with categories of frames (i) A (basis for a) Grothendieck topology on C is a function J The subterminal topology which assigns to every element c ∈ C a family J ( c ) of lower Dualities with topological spaces subsets of ( c ) ↓ , called the J -covers on c , such that for any New dualities S ∈ J ( c ) and any c ′ ≤ c the subset S c ′ = { d ≤ c ′ | d ∈ S } Other belongs to J ( c ′ ) . applications For further (ii) A preorder site is a pair ( C , J ) , where C is a preorder and J reading is a Grothendieck topology on C . (iii) A Grothendiek topology J on C is subcanonical if for every c ∈ C and any subset S ∈ J ( c ) , c is the supremum in C of the elements d ∈ S (i.e., for any element c ′ in C such that for every d ∈ S d ≤ c ′ , we have c ≤ c ′ ). 5 / 27
A topos-theoretic Examples of Grothendieck topologies approach to Stone-type dualities Olivia Caramello • If P is a preorder, the trivial topology on P is the one in which The abstract machinery the only covers are the maximal ones. The general method • If D is a distributive lattice, the coherent topology on D is the Equivalences with categories of frames one in which the covers are exactly those which contain finite The subterminal topology families whose join is the given element. Dualities with topological spaces • If F is a frame, the canonical topology on F is the one in New dualities which the covers are exactly the families whose join is the Other applications given element. For further reading • If D is a disjunctively distributive lattice, the disjunctive topology on D is the one in which the covers are exactly those which contain finite families of pairwise disjoint elements whose join is the given element. • If U is a k -frame, the k -covering topology on U is the one in which the covers are the those which contain families of less than k elements whose join is the given element. 6 / 27
A topos-theoretic J -ideals approach to Stone-type dualities Olivia Caramello Definition The abstract machinery Given a preorder site ( C , J ) , a J -ideal on C is a subset I ⊆ C such The general that method Equivalences with categories of frames • for any a , b ∈ C such that b ≤ a in C , a ∈ I implies b ∈ I , and The subterminal topology • for any J -cover R on an element c of C , if a ∈ I for every Dualities with topological spaces a ∈ R then c ∈ I . New dualities Other We denote by Id J ( C ) the set of all the J -ideals on C . applications For further Theorem reading Let C be a preorder and J be a Grothendieck topology on C . Then ( Id J ( C ) , ⊆ ) is a frame. Remark If J is subcanonical (i.e. all the principal ideals on C are J-ideals) and C is a poset then we have an embedding C ֒ → Id J ( C ) , which identifies C with the set of principal ideals on C . 7 / 27
� � A topos-theoretic The underlying philosophy approach to Stone-type dualities • In my paper Olivia Caramello The unification of Mathematics via Topos Theory The abstract machinery I give a set of principles and methodologies which justify a view The general of Grothendieck toposes as ‘bridges’ for transferring information method between distinct mathematical theories. Equivalences with categories of frames • This work represents a faithful implementation of this philosophy The subterminal topology in a particular context. Dualities with topological spaces • In fact, we establish our dualities precisely by ‘functorializing’ New dualities different representations of a given topos, which thus acts as a Other applications ‘bridge’ connecting the two sites: For further reading ( C , J ) Id J ( C ) � � � � � � � � � � � � � � Sh ( C , J ) ≃ Sh ( Id J ( C )) Remark For any preorder site ( C , J ) , the J-ideals on C correspond precisely to the subterminal objects of the topos Sh ( C , J ) . 8 / 27
A topos-theoretic Functorialization I approach to Stone-type dualities We can generate covariant or controvariant equivalences with Olivia Caramello categories of posets by appropriately functorializing the The abstract assignments above. machinery The general Definition method Equivalences with A morphism of sites ( C , J ) → ( D , K ) , where C and D are categories of frames The subterminal meet-semilattices, is a meet-semilattice homomorphism C → D topology Dualities with which sends J -covers to K -covers. topological spaces New dualities Theorem Other applications 1 A morphism of sites f : ( C , J ) → ( D , K ) induces, naturally in f, For further reading a frame homomorphism ˙ f : Id J ( C ) → Id K ( D ) . This homomorphism sends a J-ideal I on C to the smallest K-ideal on D containing the image of I under f. 2 If J and K are subcanonical then a frame homomorphism Id J ( C ) → Id K ( D ) is of the form ˙ f for some f if and only if it sends principal ideals to principal ideals; if this is the case then f is isomorphic to the restriction of ˙ f to the principal ideals. 9 / 27
A topos-theoretic Functorialization II approach to Stone-type dualities Olivia Caramello The abstract Theorem machinery The general Let C and D be two preorders. Then method Equivalences with 1 For any monotone map f : C → D , the map categories of frames The subterminal B f : Id ( D ) → Id ( C ) sending an ideal I on D to the inverse topology Dualities with image f − 1 ( I ) of I under f is a frame homomorphism. topological spaces New dualities 2 A frame homomorphism F : Id ( D ) → Id ( C ) is of the form B f Other for some monotone map f : C → D if and only if F preserves applications arbitrary infima, equivalently if and only if it has a left adjoint For further F ! : Id ( C ) → Id ( D ) , given by the formula F ! ( I ) = ∩ reading I ⊆ F ( I ′ ) I ′ (for any I ∈ Id ( C ) ). 3 If C and D are posets then any monotone map f : C → D can be recovered from B f as the restriction of its left adjoint ( B f ) ! to the subsets of principal ideals. 10 / 27
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