A Constructive View of Weak Topologies on a Topos Zeinab Khanjanzadeh, Ali Madanshekaf Semnan University BLAST 2018 University of Denver, Colorado, USA August 6-10 2018 Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 1 / 29
Outline Outline In this talk: Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 2 / 29
Outline Outline In this talk: we introduce the notion of (productive) weak topology on a topos E and investigate some of its basic properties. Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 2 / 29
Outline Outline In this talk: we introduce the notion of (productive) weak topology on a topos E and investigate some of its basic properties. we show that the set of all weak topologies on a ( co ) complete topos E is a complete resituated lattice. Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 2 / 29
Outline Outline In this talk: we introduce the notion of (productive) weak topology on a topos E and investigate some of its basic properties. we show that the set of all weak topologies on a ( co ) complete topos E is a complete resituated lattice. we give an explicit description of a restricted associated sheaf functor on a topos E in two steps. Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 2 / 29
(Elementary) topos Definition An (elementary) topos is a category E with finite limits, provided that the following conditions are satisfied: Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 3 / 29
(Elementary) topos Definition An (elementary) topos is a category E with finite limits, provided that the following conditions are satisfied: 1 E is cartesian closed , i.e. all objects of E are exponentiable; Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 3 / 29
� � � � (Elementary) topos Definition An (elementary) topos is a category E with finite limits, provided that the following conditions are satisfied: 1 E is cartesian closed , i.e. all objects of E are exponentiable; 2 E has a subobject classifier, that is, an object Ω equipped with a monomorphism true : 1 Ω such that, given any monomorphism m : S B in E ; there is a unique map char ( m ) : B → Ω (sometimes denoted by char ( S )) for which the following square is a pullback: � 1 S m true � Ω . B char ( m ) Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 3 / 29
Internal Heyting algebra structure of Ω In fact, for each object B of E we have a natural isomorphism in B as follows Sub E ( B ) ∼ = Hom E ( B , Ω) The subobject classifier Ω on a topos E has an internal Heyting algebra structure. In details, Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 4 / 29
Internal Heyting algebra structure of Ω In fact, for each object B of E we have a natural isomorphism in B as follows Sub E ( B ) ∼ = Hom E ( B , Ω) The subobject classifier Ω on a topos E has an internal Heyting algebra structure. In details, 1 The meet operation ∩ : Sub E ( B ) × Sub E ( B ) → Sub E ( B ) is natural in B . Under the isomorphism Hom E ( B , Ω) ∼ = Sub E ( B ), which is again natural in B , we obtain an operation ∧ B such that the following diagram is commutative: Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 4 / 29
� � � � Internal Heyting algebra structure of Ω ∩ � Sub E ( B ) Sub E ( B ) × Sub E ( B ) ∼ ∼ = = Hom E ( B , Ω) × Hom E ( B , Ω) Hom E ( B , Ω) ∼ ∼ = = ∧ B � Hom E ( B , Ω) Hom E ( B , Ω × Ω) Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 5 / 29
� � � � Internal Heyting algebra structure of Ω ∩ � Sub E ( B ) Sub E ( B ) × Sub E ( B ) ∼ ∼ = = Hom E ( B , Ω) × Hom E ( B , Ω) Hom E ( B , Ω) ∼ ∼ = = ∧ B � Hom E ( B , Ω) Hom E ( B , Ω × Ω) Since the operation ∧ B is natural in B , so by the Yoneda lemma ∧ B comes from a uniquely determined map ∧ : Ω × Ω → Ω via composition which is ∧ = ∧ Ω × Ω ( id Ω × Ω ). The arrow ∧ is the internal meet operation on Ω. Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 5 / 29
Internal Heyting algebra structure of Ω 1 Similarly, we can define an internal join operation ∨ : Ω × Ω → Ω and an internal implication operation ⇒ : Ω × Ω → Ω on Ω. Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 6 / 29
Internal Heyting algebra structure of Ω 1 Similarly, we can define an internal join operation ∨ : Ω × Ω → Ω and an internal implication operation ⇒ : Ω × Ω → Ω on Ω. 2 Under the isomorphism Sub E (1) ∼ = Hom E (1 , Ω), the top and bottom elements of Sub E (1) which are 1 1 and 0 1, respectively, correspond to the internal top and internal bottom elements “ true = char (1 1)” and “ false = char (0 1)” of Ω. Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 6 / 29
Weak topologies Definition A weak Lawvere-Tierney topology (or a weak topology, for short) on a topos E is a morphism j : Ω → Ω such that: Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 7 / 29
Weak topologies Definition A weak Lawvere-Tierney topology (or a weak topology, for short) on a topos E is a morphism j : Ω → Ω such that: 1 j ◦ true = true ; Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 7 / 29
Weak topologies Definition A weak Lawvere-Tierney topology (or a weak topology, for short) on a topos E is a morphism j : Ω → Ω such that: 1 j ◦ true = true ; 2 j ◦ ∧ ≤ ∧ ◦ ( j × j ), in which ≤ stands for the order on Ω . Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 7 / 29
Weak topologies Definition A weak Lawvere-Tierney topology (or a weak topology, for short) on a topos E is a morphism j : Ω → Ω such that: 1 j ◦ true = true ; 2 j ◦ ∧ ≤ ∧ ◦ ( j × j ), in which ≤ stands for the order on Ω . Furthermore, j is productive whenever the non-equality in (2) is an equality. Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 7 / 29
Weak topologies Definition A weak Lawvere-Tierney topology (or a weak topology, for short) on a topos E is a morphism j : Ω → Ω such that: 1 j ◦ true = true ; 2 j ◦ ∧ ≤ ∧ ◦ ( j × j ), in which ≤ stands for the order on Ω . Furthermore, j is productive whenever the non-equality in (2) is an equality. An idempotent weak topology on E is called a (Lawvere-Tierney) topology on E . Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 7 / 29
Some Examples of Weak Topologies Example Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 8 / 29
Some Examples of Weak Topologies Example 1 The composite of any two topologies on a topos E is a productive weak topology. It is a topology on E if and only if it is idempotent. Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 8 / 29
Some Examples of Weak Topologies Example 1 The composite of any two topologies on a topos E is a productive weak topology. It is a topology on E if and only if it is idempotent. 2 It is well known that the commutative monoid of natural endomorphisms of the identity functor on a topos E is called the center of E . Let α be a natural endomorphism of the identity functor on E . It is easy to see that α Ω is a productive weak topology on E . It will be a topology on E if α 2 Ω = α Ω . Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 8 / 29
Modal closure operators Definition An operator on the subobjects of each object E of E A �→ A , Sub E ( E ) → Sub E ( E ) , is a modal closure operator if and only if it has, for all A , B ∈ Sub E ( E ), the properties: Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 9 / 29
Modal closure operators Definition An operator on the subobjects of each object E of E A �→ A , Sub E ( E ) → Sub E ( E ) , is a modal closure operator if and only if it has, for all A , B ∈ Sub E ( E ), the properties: 1 (Extension) A ⊆ A ; Zeinab Khanjanzadeh, Ali Madanshekaf A Constructive View of Weak Topologies ... (Semnan University) August 6-10 2018 9 / 29
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