A Constructive View of Weak Topologies on a Topos Zeinab - - PowerPoint PPT Presentation
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
Zeinab Khanjanzadeh, Ali Madanshekaf
Semnan University
BLAST 2018 University of Denver, Colorado, USA August 6-10 2018
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 1 / 29
Outline In this talk:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 2 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 2 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 2 / 29
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
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 2 / 29
Definition An (elementary) topos is a category E with finite limits, provided that the following conditions are satisfied:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 3 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 3 / 29
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: S
Ω.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 3 / 29
In fact, for each object B of E we have a natural isomorphism in B as follows SubE(B) ∼ = HomE(B, Ω) The subobject classifier Ω on a topos E has an internal Heyting algebra
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 4 / 29
In fact, for each object B of E we have a natural isomorphism in B as follows SubE(B) ∼ = HomE(B, Ω) The subobject classifier Ω on a topos E has an internal Heyting algebra
1 The meet operation ∩ : SubE(B) × SubE(B) → SubE(B) is natural in
= SubE(B), which is again natural in B, we obtain an operation ∧B such that the following diagram is commutative:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 4 / 29
SubE(B) × SubE(B)
∼ =
SubE(B)
∼ =
∼ =
∼ =
∧B
HomE(B, Ω)
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 5 / 29
SubE(B) × SubE(B)
∼ =
SubE(B)
∼ =
∼ =
∼ =
∧B
HomE(B, 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(idΩ×Ω). The arrow ∧ is the internal meet operation on Ω.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 5 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 6 / 29
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 SubE(1) ∼
= HomE(1, Ω), the top and bottom elements of SubE(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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 6 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 7 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 7 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 7 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 7 / 29
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
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 7 / 29
Example
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 8 / 29
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 (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 8 / 29
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
will be a topology on E if α2
Ω = αΩ.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 8 / 29
Definition An operator on the subobjects of each object E of E A → A, SubE(E) → SubE(E), is a modal closure operator if and only if it has, for all A, B ∈ SubE(E), the properties:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 9 / 29
Definition An operator on the subobjects of each object E of E A → A, SubE(E) → SubE(E), is a modal closure operator if and only if it has, for all A, B ∈ SubE(E), the properties:
1 (Extension) A ⊆ A; Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 9 / 29
Definition An operator on the subobjects of each object E of E A → A, SubE(E) → SubE(E), is a modal closure operator if and only if it has, for all A, B ∈ SubE(E), the properties:
1 (Extension) A ⊆ A; 2 (Monotonicity) A ⊆ B yields that A ⊆ B; Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 9 / 29
Definition An operator on the subobjects of each object E of E A → A, SubE(E) → SubE(E), is a modal closure operator if and only if it has, for all A, B ∈ SubE(E), the properties:
1 (Extension) A ⊆ A; 2 (Monotonicity) A ⊆ B yields that A ⊆ B; 3 (Modal) For each arrow f : F → E in E, we have f −1(A) = f −1(A),
where f −1 is the pullback functor.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 9 / 29
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 10 / 29
Any weak topology j on E, determines a modal closure operator A → A on the subobjects A E of each object E, in such a way that for any subobject A
ι
E, the j-closure of A is the subobject A of E with the characteristic map jchar(ι), shown as in the diagram below A
true
true
E char(ι)
Ω
j
Ω
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 10 / 29
Conversely, any modal closure operator on a topos E always gives a unique weak topology j as indicated in the following pullback diagram: 1
j
Ω
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 11 / 29
Conversely, any modal closure operator on a topos E always gives a unique weak topology j as indicated in the following pullback diagram: 1
j
Ω
One can prove that
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 11 / 29
Conversely, any modal closure operator on a topos E always gives a unique weak topology j as indicated in the following pullback diagram: 1
j
Ω
One can prove that Lemma On a topos E, weak topologies j are in one-to-one correspondence with modal closure operators (·).
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 11 / 29
For a weak topology j on E, it can be easily checked that j ◦ ∧ = char(true × true) and ∧ ◦ (j × j) = char(true × true).
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 12 / 29
For a weak topology j on E, it can be easily checked that j ◦ ∧ = char(true × true) and ∧ ◦ (j × j) = char(true × true). So two subobjects 1 × 1 and 1 × 1 of Ω × Ω, are not equal. This means that the modal closure operator associated to j, is not productive; that is the closure does not commute with products.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 12 / 29
For a weak topology j on E, it can be easily checked that j ◦ ∧ = char(true × true) and ∧ ◦ (j × j) = char(true × true). So two subobjects 1 × 1 and 1 × 1 of Ω × Ω, are not equal. This means that the modal closure operator associated to j, is not productive; that is the closure does not commute with products. We can prove that for a weak topology j on E, the modal closure operator associated to j, is productive if and only if one has j ◦ ∧ = ∧ ◦ (j × j) if and only if the modal closure operator associated to j, commutes with binary intersections.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 12 / 29
For a weak topology j on E, it can be easily checked that j ◦ ∧ = char(true × true) and ∧ ◦ (j × j) = char(true × true). So two subobjects 1 × 1 and 1 × 1 of Ω × Ω, are not equal. This means that the modal closure operator associated to j, is not productive; that is the closure does not commute with products. We can prove that for a weak topology j on E, the modal closure operator associated to j, is productive if and only if one has j ◦ ∧ = ∧ ◦ (j × j) if and only if the modal closure operator associated to j, commutes with binary intersections. For this reason, we call a weak topology j with the property j ◦ ∧ = ∧ ◦ (j × j) a productive weak topology.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 12 / 29
Definition Let j be a weak topology on E and (·) the modal closure operator associated to j. A monomorphism k : A C in E is
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 13 / 29
Definition Let j be a weak topology on E and (·) the modal closure operator associated to j. A monomorphism k : A C in E is j-dense whenever A = C, as subobjects of C;
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 13 / 29
Definition Let j be a weak topology on E and (·) the modal closure operator associated to j. A monomorphism k : A C in E is j-dense whenever A = C, as subobjects of C; j-closed if A = A, as subobjects of C.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 13 / 29
Definition Let j be a weak topology on E and (·) the modal closure operator associated to j. A monomorphism k : A C in E is j-dense whenever A = C, as subobjects of C; j-closed if A = A, as subobjects of C. Morover, an object C is called a j-sheaf whenever for any j-dense monomorphism m : B A, one can uniquely extend any arrow h : B → C in E to A as follows
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 13 / 29
Definition Let j be a weak topology on E and (·) the modal closure operator associated to j. A monomorphism k : A C in E is j-dense whenever A = C, as subobjects of C; j-closed if A = A, as subobjects of C. Morover, an object C is called a j-sheaf whenever for any j-dense monomorphism m : B A, one can uniquely extend any arrow h : B → C in E to A as follows B
h
A
g
(Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 13 / 29
Definition Let j be a weak topology on E and (·) the modal closure operator associated to j. A monomorphism k : A C in E is j-dense whenever A = C, as subobjects of C; j-closed if A = A, as subobjects of C. Morover, an object C is called a j-sheaf whenever for any j-dense monomorphism m : B A, one can uniquely extend any arrow h : B → C in E to A as follows B
h
A
g
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 13 / 29
For a weak topology j on a topos E,
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 14 / 29
For a weak topology j on a topos E, We will denote the equalizer of two arrows j, idΩ : Ω → Ω by Ωj as follows Ωj
m
Ω
j
Ω.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 14 / 29
For a weak topology j on a topos E, We will denote the equalizer of two arrows j, idΩ : Ω → Ω by Ωj as follows Ωj
m
Ω
j
Ω.
The object Ωj classifies closed subobjects, in the sense that, for each
HomE(E, Ωj)
∼
− → ClSubE(E); which is natural in E. Here ClSubE(E) is the set of all closed subobjects of E.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 14 / 29
For a weak topology j on a topos E,
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 15 / 29
For a weak topology j on a topos E, We will denote the full subcategories of E consisting of all j-separated
check that for a productive weak topology j on E, Shj(E) is a topos with the subobject classifier Ωj.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 15 / 29
For a weak topology j on a topos E, We will denote the full subcategories of E consisting of all j-separated
check that for a productive weak topology j on E, Shj(E) is a topos with the subobject classifier Ωj. We will denote the image of the weak topology j by im(j), that is the smallest subobject k : im(j) Ω which j can factor through it.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 15 / 29
The following proposition is the basic difference between weak topologies and topologies on E. It shows that we are unable to construct the associated sheaf functor to a weak topology j on E as usual.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 16 / 29
The following proposition is the basic difference between weak topologies and topologies on E. It shows that we are unable to construct the associated sheaf functor to a weak topology j on E as usual. Proposition For a weak topology j on a topos E, we factor j through its image as Ω
r
։ im(j)
k
Ω. Then j is idempotent (or equivalently, is a topology on E) if and only if Ωj = im(j), as subobjects of Ω.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 16 / 29
Remark The set of (productive) weak topologies on a topos E has a natural partial order given by j ≤ k if and only if j = j ∧ k, for any (productive) weak topologies j, k : Ω → Ω, where j ∧ k is the composite arrow Ω
(j,k)
− → Ω × Ω
∧
− → Ω.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 17 / 29
Remark The set of (productive) weak topologies on a topos E has a natural partial order given by j ≤ k if and only if j = j ∧ k, for any (productive) weak topologies j, k : Ω → Ω, where j ∧ k is the composite arrow Ω
(j,k)
− → Ω × Ω
∧
− → Ω. We denote by WTop(E), PWTop(E) and Top(E) for the posets of weak topologies, productive weak topologies and topologies on E,
Top(E) ⊆ PWTop(E) ⊆ WTop(E). Notice that all these posets have the same binary meets which is pointwise, and also they have the top and bottom elements which are true◦!Ω and idΩ, respectively.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 17 / 29
It is clear that (WTop(E), ◦, idΩ) is a monoid in which ◦ is the ordinary composition of weak topologies on E.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 18 / 29
It is clear that (WTop(E), ◦, idΩ) is a monoid in which ◦ is the ordinary composition of weak topologies on E. Let E be a (co)complete topos. We define two binary operations \ and /
j\k =
and k/j =
for weak topologies j and k on E.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 18 / 29
It is clear that (WTop(E), ◦, idΩ) is a monoid in which ◦ is the ordinary composition of weak topologies on E. Let E be a (co)complete topos. We define two binary operations \ and /
j\k =
and k/j =
for weak topologies j and k on E. It is easily seen that we have j ◦ j′ ≥ k ⇐ ⇒ j ≥ k/j′ ⇐ ⇒ j′ ≥ j\k. The desired result now is:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 18 / 29
It is clear that (WTop(E), ◦, idΩ) is a monoid in which ◦ is the ordinary composition of weak topologies on E. Let E be a (co)complete topos. We define two binary operations \ and /
j\k =
and k/j =
for weak topologies j and k on E. It is easily seen that we have j ◦ j′ ≥ k ⇐ ⇒ j ≥ k/j′ ⇐ ⇒ j′ ≥ j\k. The desired result now is: Theorem Let E be a (co)complete topos. Then, (WTop(E), ∧, ∨, ◦, idΩ, \, /) is a complete resituated lattice.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 18 / 29
Let E be a cocomplete topos and j a (productive) weak topology on E. We define the ascending extended ordinal chain of (productive) weak topologies j ≤ j2 ≤ j3 ≤ . . . ≤ jα ≤ jα+1 ≤ . . . ≤ j∞ ≤ j∞+1 in which: jα+1 = j ◦ jα, jβ =
jγ for every (small) ordinal number α and for α = ∞, and for every limit
∞ + 1 > ∞ > α for all α ∈ Ord, the class of small ordinals. Then,
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 19 / 29
Let E be a cocomplete topos and j a (productive) weak topology on E. We define the ascending extended ordinal chain of (productive) weak topologies j ≤ j2 ≤ j3 ≤ . . . ≤ jα ≤ jα+1 ≤ . . . ≤ j∞ ≤ j∞+1 in which: jα+1 = j ◦ jα, jβ =
jγ for every (small) ordinal number α and for α = ∞, and for every limit
∞ + 1 > ∞ > α for all α ∈ Ord, the class of small ordinals. Then, Proposition Let j be a (productive) weak topology on a cocomplete topos E. Then j∞ is the idempotent hull of j, i.e. the smallest topology containing j. Moreover, one has Shj∞(E) =
γ<∞ Shjγ(E), as full subcategories of E.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 19 / 29
For a cocomplete topos E, the inclusion functor U : Top(E) WTop(E) (or U : Top(E) PWTop(E)) has a left adjoint, F : WTop(E) → Top(E) which, as any left adjoint to an inclusion, assigns to each (productive) weak topology j the least topology j∞ with j ≤ j∞, we call it the topological reflection of j.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 20 / 29
For a cocomplete topos E, the inclusion functor U : Top(E) WTop(E) (or U : Top(E) PWTop(E)) has a left adjoint, F : WTop(E) → Top(E) which, as any left adjoint to an inclusion, assigns to each (productive) weak topology j the least topology j∞ with j ≤ j∞, we call it the topological reflection of j. Thus, the join of a set of topologies {jα}α∈Λ on E is the topological reflection of its join in WTop(E), i.e. (
α∈Λ U(jα))∞.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 20 / 29
For a cocomplete topos E, the inclusion functor U : Top(E) WTop(E) (or U : Top(E) PWTop(E)) has a left adjoint, F : WTop(E) → Top(E) which, as any left adjoint to an inclusion, assigns to each (productive) weak topology j the least topology j∞ with j ≤ j∞, we call it the topological reflection of j. Thus, the join of a set of topologies {jα}α∈Λ on E is the topological reflection of its join in WTop(E), i.e. (
α∈Λ U(jα))∞.
Note that for a cocomplete topos E, the subobject classifier Ω has arbitrary joins.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 20 / 29
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 21 / 29
Let j be a weak topology on E and E an object of E. We assume that Ω
r
։ im(j)
k
Ω be the image factorization of j and E
θE
։ SE
ωE
im(j)E the image factorization of the compound arrow rE{·}E : E → im(j)E in which {·}E : E ΩE stands for the transpose of the characteristic map of the diagonal △E : E → E × E which is the arrow (idE, idE).
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 21 / 29
Let j be a weak topology on E and E an object of E. We assume that Ω
r
։ im(j)
k
Ω be the image factorization of j and E
θE
։ SE
ωE
im(j)E the image factorization of the compound arrow rE{·}E : E → im(j)E in which {·}E : E ΩE stands for the transpose of the characteristic map of the diagonal △E : E → E × E which is the arrow (idE, idE). For a productive weak topology j on a topos E, we write Cj for the full subcategory of E consisting of all objects E of E for which the subobject △E of E × E is closed.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 21 / 29
The following characterizes the objects of Cj for a weak topology j on E.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 22 / 29
The following characterizes the objects of Cj for a weak topology j on E. Lemma For a weak topology j on a topos E and for any object E of E the following are equivalent:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 22 / 29
The following characterizes the objects of Cj for a weak topology j on E. Lemma For a weak topology j on a topos E and for any object E of E the following are equivalent: (i) E is separated;
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 22 / 29
The following characterizes the objects of Cj for a weak topology j on E. Lemma For a weak topology j on a topos E and for any object E of E the following are equivalent: (i) E is separated; (ii) the diagonal △E ∈ SubE(E × E) is a closed subobject of E × E;
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 22 / 29
The following characterizes the objects of Cj for a weak topology j on E. Lemma For a weak topology j on a topos E and for any object E of E the following are equivalent: (i) E is separated; (ii) the diagonal △E ∈ SubE(E × E) is a closed subobject of E × E; (iii) jE ◦ {·}E = {·}E, as in the commutative diagram E
{·}E
jE
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 22 / 29
The following characterizes the objects of Cj for a weak topology j on E. Lemma For a weak topology j on a topos E and for any object E of E the following are equivalent: (i) E is separated; (ii) the diagonal △E ∈ SubE(E × E) is a closed subobject of E × E; (iii) jE ◦ {·}E = {·}E, as in the commutative diagram E
{·}E
jE
(iv) for any f : A → E, the graph of f which is (idA, f ) : A A × E, is a closed subobject of A × E.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 22 / 29
Now, we will give an explicit description of a restricted associated sheaf functor on E in two steps: First, one can deduce that:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 23 / 29
Now, we will give an explicit description of a restricted associated sheaf functor on E in two steps: First, one can deduce that: Theorem For any productive weak topology j on a topos E, the inclusion functor Sepj(E) Cj has a left adjoint L : Cj − → Sepj(E) defined by E → SE.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 23 / 29
Now, we will give an explicit description of a restricted associated sheaf functor on E in two steps: First, one can deduce that: Theorem For any productive weak topology j on a topos E, the inclusion functor Sepj(E) Cj has a left adjoint L : Cj − → Sepj(E) defined by E → SE. Let E be a complete, cocomplete and well-copowered topos and j a productive weak topology on E. Then, it is well known that the inclusion functor Sepj(E) E has a left adjoint R : E − → Sepj(E). One can construct the functor R via the adjoint functor theorem.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 23 / 29
Let j be a weak topology on a topos E. For any separated object E of E the diagonal △E is a closed subobject of E × E. In this case, the characteristic map of △E denoted by δE : E × E → Ω satisfies jδE = δE. Since Ωj is the equalizer of j and idΩ, so there is a unique arrow αE : E × E → Ωj s.t. mαE = δE. We denote the exponential transpose of αE by αE : E → ΩE
j . Then we
have:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 24 / 29
Let j be a weak topology on a topos E. For any separated object E of E the diagonal △E is a closed subobject of E × E. In this case, the characteristic map of △E denoted by δE : E × E → Ω satisfies jδE = δE. Since Ωj is the equalizer of j and idΩ, so there is a unique arrow αE : E × E → Ωj s.t. mαE = δE. We denote the exponential transpose of αE by αE : E → ΩE
j . Then we
have: Lemma Let j be a weak topology on a topos E and E a separated object of E. Then the arrow αE as defined before, is a monomorphism.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 24 / 29
In what follows we provide the sheaf associated to a separated object in a topos E.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 25 / 29
In what follows we provide the sheaf associated to a separated object in a topos E. Theorem Let j be a productive weak topology on a topos E and E a separated
j via the arrow
j . Then E is a j-sheaf in E.
Moreover, the inclusion functor Shj(E) Sepj(E) has a left adjoint S : Sepj(E) − → Shj(E) defined by E → E as a subobject of ΩE
j .
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 25 / 29
In what follows we provide the associated sheaf functor with respect to the weak topology j on E.
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 26 / 29
In what follows we provide the associated sheaf functor with respect to the weak topology j on E. Corollary We can constitute the compound left adjoint SL : Cj − → Shj(E) to the inclusion functor Shj(E) Cj which assigns to any E of Cj the sheaf SE as a subobject of ΩSE
j .
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 26 / 29
In what follows we provide the associated sheaf functor with respect to the weak topology j on E. Corollary We can constitute the compound left adjoint SL : Cj − → Shj(E) to the inclusion functor Shj(E) Cj which assigns to any E of Cj the sheaf SE as a subobject of ΩSE
j .
In the case of a complete, cocomplete and well-copowered topos E the inclusion functor Shj(E) E has the compound left adjoint SR : E − → Shj(E).
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 26 / 29
We can define a weak topology (modal closure operator) on a category with finite limits. Hence, for a weak topology j on a topos E, the notion of j-sheaves can be defined in the finite complete category Sepj(E).
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 27 / 29
We can define a weak topology (modal closure operator) on a category with finite limits. Hence, for a weak topology j on a topos E, the notion of j-sheaves can be defined in the finite complete category Sepj(E). The following determines j-sheaves in Sepj(E).
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 27 / 29
We can define a weak topology (modal closure operator) on a category with finite limits. Hence, for a weak topology j on a topos E, the notion of j-sheaves can be defined in the finite complete category Sepj(E). The following determines j-sheaves in Sepj(E). Proposition Let j be a productive weak topology on E and E a separated object of E. Then the following conditions are equivalent:
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 27 / 29
We can define a weak topology (modal closure operator) on a category with finite limits. Hence, for a weak topology j on a topos E, the notion of j-sheaves can be defined in the finite complete category Sepj(E). The following determines j-sheaves in Sepj(E). Proposition Let j be a productive weak topology on E and E a separated object of E. Then the following conditions are equivalent: (i) E is a j-sheaf in Sepj(E);
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 27 / 29
We can define a weak topology (modal closure operator) on a category with finite limits. Hence, for a weak topology j on a topos E, the notion of j-sheaves can be defined in the finite complete category Sepj(E). The following determines j-sheaves in Sepj(E). Proposition Let j be a productive weak topology on E and E a separated object of E. Then the following conditions are equivalent: (i) E is a j-sheaf in Sepj(E); (ii) E is a j-sheaf in E;
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 27 / 29
We can define a weak topology (modal closure operator) on a category with finite limits. Hence, for a weak topology j on a topos E, the notion of j-sheaves can be defined in the finite complete category Sepj(E). The following determines j-sheaves in Sepj(E). Proposition Let j be a productive weak topology on E and E a separated object of E. Then the following conditions are equivalent: (i) E is a j-sheaf in Sepj(E); (ii) E is a j-sheaf in E; (iii) E is closed in ΩE
j , by the monic
αE : E ΩE
j .
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 27 / 29
Handbook of Categorical Algebra, Vol. I and III, Cambridge University Press, (1994).
Categorical Structure of Closure Operators, Kluwer, Netherlands, (1995).
A Relation Between Closure Operators on a Small Category and Its Category of Presheaves,
Sketches of an Elephant: a Topos Theory Compendium, Vol. 1, Clarendon Press, Oxford, (2002).
Sheaves in Geometry and Logic, Springer-Verlag, New York, (1992).
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 28 / 29
Zeinab Khanjanzadeh, Ali Madanshekaf (Semnan University) A Constructive View of Weak Topologies ... August 6-10 2018 29 / 29