[PPT] - Hereditary coreflective subcategories in categories of PowerPoint Presentation

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Hereditary coreflective subcategories in categories of semitopological groups

Veronika Pitrov´ a

Jan Evangelista Purkynˇ e University in ´ Ust´ ı nad Labem

Veronika Pitrov´ a (UJEP) 1 / 17

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Introduction

Structure Group operation Inverse semitopological group separately continuous – quasitopological group separately continuous continuous paratopological group continuous – topological group continuous continuous

Veronika Pitrov´ a (UJEP) 2 / 17

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Introduction

Reflective subcategories

Definition A subcategory A of C is reflective in C provided that for every X ∈ C there exists an A-reflection: XA ∈ A and a C-morphism rX : X → XA such that for every C-morphism f : X → Y where Y ∈ A there exists a unique A-morphism ¯ f : XA → Y , such that the following diagram commutes: X XA Y f rX f

Veronika Pitrov´ a (UJEP) 3 / 17

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Introduction

Reflective subcategories in STopGr

epireflective ⇔ closed under the formation of subgroups and

products

Veronika Pitrov´ a (UJEP) 4 / 17

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Introduction

Reflective subcategories in STopGr

epireflective ⇔ closed under the formation of subgroups and

products

extremal epimorphisms are precisely the open surjective

homomorphisms:

Veronika Pitrov´ a (UJEP) 4 / 17

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Introduction

Reflective subcategories in STopGr

epireflective ⇔ closed under the formation of subgroups and

products

extremal epimorphisms are precisely the open surjective

homomorphisms:

extremal epireflective ⇔ closed under the formation of subgroups,

products and semitopological groups with finer topologies e.g. STopAb, the category of all torsion-free semitopological groups

Veronika Pitrov´ a (UJEP) 4 / 17

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Introduction

Reflective subcategories in STopGr

epireflective ⇔ closed under the formation of subgroups and

products

extremal epimorphisms are precisely the open surjective

homomorphisms:

extremal epireflective ⇔ closed under the formation of subgroups,

products and semitopological groups with finer topologies e.g. STopAb, the category of all torsion-free semitopological groups

epireflective, closed under the formation of (usual) quotients

e.g. QTopGr, PTopGr, TopGr, TopAb

Veronika Pitrov´ a (UJEP) 4 / 17

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Introduction

Coreflective subcategories

Definition A subcategory B of A is coreflective in A provided that for every X ∈ A there exists a B-coreflection: XB ∈ B and an A-morphism cX : XB → X such that for every A-morphism f : Y → X where Y ∈ B there exists a unique B-morphism ¯ f : Y → XB, such that the following diagram commutes: X XB Y f cX f

Veronika Pitrov´ a (UJEP) 5 / 17

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Introduction

Coreflective subcategories in A

A denotes an epireflective subcategory of STopGr

Veronika Pitrov´ a (UJEP) 6 / 17

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Introduction

Coreflective subcategories in A

A denotes an epireflective subcategory of STopGr
hereditary coreflective ⇒ monocoreflective

Veronika Pitrov´ a (UJEP) 6 / 17

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Introduction

Coreflective subcategories in A

A denotes an epireflective subcategory of STopGr
hereditary coreflective ⇒ monocoreflective
monocoreflective ⇔ closed under the formation of coproducts and

extremal quotients

Veronika Pitrov´ a (UJEP) 6 / 17

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Introduction

Coreflective subcategories in A

A denotes an epireflective subcategory of STopGr
hereditary coreflective ⇒ monocoreflective
monocoreflective ⇔ closed under the formation of coproducts and

extremal quotients

bicoreflective ⇔ monocoreflective, contains rA(Z)

e.g. QTopGr in STopGr, TopGr in PTopGr

Veronika Pitrov´ a (UJEP) 6 / 17

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Introduction

The coproduct A

i∈I Gi

the ”most general” group from A that contains each Gi as a

subgroup

Veronika Pitrov´ a (UJEP) 7 / 17

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Introduction

The coproduct A

i∈I Gi

the ”most general” group from A that contains each Gi as a

subgroup

in STopGr: the free product with the finest topology such that
i∈I Gi is a semitopological group and all mj : Gj →

i∈I Gi are

continuous

Veronika Pitrov´ a (UJEP) 7 / 17

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Introduction

The coproduct A

i∈I Gi

the ”most general” group from A that contains each Gi as a

subgroup

in STopGr: the free product with the finest topology such that
i∈I Gi is a semitopological group and all mj : Gj →

i∈I Gi are

continuous

in STopAb, QTopAb: the direct sum with the cross topology

Veronika Pitrov´ a (UJEP) 7 / 17

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Introduction

The coproduct A

i∈I Gi

the ”most general” group from A that contains each Gi as a

subgroup

in STopGr: the free product with the finest topology such that
i∈I Gi is a semitopological group and all mj : Gj →

i∈I Gi are

continuous

in STopAb, QTopAb: the direct sum with the cross topology
in PTopAb, TopAb: the direct sum with the usual topology

Veronika Pitrov´ a (UJEP) 7 / 17

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Introduction

Questions

1. What is the hereditary coreflective hull of subcategories of A?
2. Which hereditary coreflective subcategories of A are bicoreflective

in A?

Veronika Pitrov´ a (UJEP) 8 / 17

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The hereditary coreflective hull

in general:

Proposition

Let A be an epireflective subcategory of STopGr and B be a subcategory of A. Moreover, let

1. B0 = B,
2. Bα+1 = MCHA(SBα) for every ordinal α,
3. Bβ =

α<β Bα for every limit ordinal β.

Then the hereditary coreflective hull of B in A is the subcategory B∗ =

α∈On Bα.

Veronika Pitrov´ a (UJEP) 9 / 17

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The hereditary coreflective hull

if extremal epimorphisms in A are precisely the surjective open

homomorphisms:

Proposition

Let A be an epireflective subcategory of STopGr such that the extremal epimorphisms in A are precisely the surjective open homomorphisms and B be a subcategory of A. Moreover let

1. B0 = B,
2. B1 be the subcategory consisting of all coproducts of groups from

B0,

3. B2 be the subcategory consisting of all subgroups of groups from

B1,

4. B3 be the subcategory consisting of all extremal quotients of groups

from B2. Then the hereditary coreflective hull of B in A is the subcategory B3.

Veronika Pitrov´ a (UJEP) 10 / 17

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The hereditary coreflective hull

The hereditary coreflective hull in A ⊆ STopAb

if A ⊆ STopAb is closed under the formation extremal quotients:

Proposition

Let A be an epireflective subcategory of STopAb that is closed under the formation of extremal quotients and B be a coreflective subcategory

f A. Then the hereditary coreflective hull of B in A is the subcategory

consisting of all subgroups of groups from B.

Veronika Pitrov´ a (UJEP) 11 / 17

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Hereditary bicoreflective subcategories

Proposition

Let A be an extremal epireflective subcategory of STopGr or QTopGr and B be a hereditary coreflective subcategory of A that contains the group Z of integers with a T0-topology. Then B contains the discrete group Z, therefore it is bicoreflective in A.

Veronika Pitrov´ a (UJEP) 12 / 17

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Hereditary bicoreflective subcategories

Proposition

Let A be an extremal epireflective subcategory of STopGr or QTopGr and B be a hereditary coreflective subcategory of A that contains the group Z of integers with a T0-topology. Then B contains the discrete group Z, therefore it is bicoreflective in A. Outline of proof:

B is closed under the formation of finite products with the cross

topology, since they are quotients of finite coproducts

Veronika Pitrov´ a (UJEP) 12 / 17

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Hereditary bicoreflective subcategories

Proposition

Let A be an extremal epireflective subcategory of STopGr or QTopGr and B be a hereditary coreflective subcategory of A that contains the group Z of integers with a T0-topology. Then B contains the discrete group Z, therefore it is bicoreflective in A. Outline of proof:

B is closed under the formation of finite products with the cross

topology, since they are quotients of finite coproducts

B contains the group Z′ of integers with a T1-topology

Veronika Pitrov´ a (UJEP) 12 / 17

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Hereditary bicoreflective subcategories

Proposition

Let A be an extremal epireflective subcategory of STopGr or QTopGr and B be a hereditary coreflective subcategory of A that contains the group Z of integers with a T0-topology. Then B contains the discrete group Z, therefore it is bicoreflective in A. Outline of proof:

B is closed under the formation of finite products with the cross

topology, since they are quotients of finite coproducts

B contains the group Z′ of integers with a T1-topology
the subset V = U × U \ {(n, n) : n is a non-zero integer} of

Z′ ×∗ Z′ is open (U is a neighborhood of 0 in Z′)

Veronika Pitrov´ a (UJEP) 12 / 17

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Hereditary bicoreflective subcategories

Proposition

Let A be an extremal epireflective subcategory of STopGr or QTopGr and B be a hereditary coreflective subcategory of A that contains the group Z of integers with a T0-topology. Then B contains the discrete group Z, therefore it is bicoreflective in A. Outline of proof:

B is closed under the formation of finite products with the cross

topology, since they are quotients of finite coproducts

B contains the group Z′ of integers with a T1-topology
the subset V = U × U \ {(n, n) : n is a non-zero integer} of

Z′ ×∗ Z′ is open (U is a neighborhood of 0 in Z′)

[(1, 1)] ∩ V = {(0, 0)}

Veronika Pitrov´ a (UJEP) 12 / 17

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Hereditary bicoreflective subcategories

Proposition

Let A be an extremal epireflective subcategory of STopGr or QTopGr and B be a hereditary coreflective subcategory of A that contains the group Z of integers with a T0-topology. Then B contains the discrete group Z, therefore it is bicoreflective in A. Outline of proof:

B is closed under the formation of finite products with the cross

topology, since they are quotients of finite coproducts

B contains the group Z′ of integers with a T1-topology
the subset V = U × U \ {(n, n) : n is a non-zero integer} of

Z′ ×∗ Z′ is open (U is a neighborhood of 0 in Z′)

[(1, 1)] ∩ V = {(0, 0)}

! the proof fails in PTopGr and TopGr, since Z ×∗ Z does not need to be a paratopological group

Veronika Pitrov´ a (UJEP) 12 / 17

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Hereditary bicoreflective subcategories

Proposition

Let A be an epireflective subcategory of STopGr that satisfies one of the following conditions:

1. A is closed under the formation of finite coproducts,
2. A contains the group Zn ⊔ Zn for every n ∈ N,

and B be a hereditary coreflective subcategory of A that contains a group with a proper open subgroup. Then B contains the discrete group Z, therefore it is bicoreflective in A.

Veronika Pitrov´ a (UJEP) 13 / 17

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Hereditary bicoreflective subcategories

Proposition

Let A be an epireflective subcategory of STopGr that satisfies one of the following conditions:

1. A is closed under the formation of finite coproducts,
2. A contains the group Zn ⊔ Zn for every n ∈ N,

and B be a hereditary coreflective subcategory of A that contains a group with a proper open subgroup. Then B contains the discrete group Z, therefore it is bicoreflective in A.

Corollary

If A satisfies the conditions of the preceding proposition, B is a hereditary coreflective subcategory of A that contains a cyclic group with a non-indiscrete topology that is not T0, then B is bicoreflective in A.

Veronika Pitrov´ a (UJEP) 13 / 17

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Hereditary bicoreflective subcategories

Corollary

Let A be an extremal epireflective subcategory of STopGr or QTopGr that satisfies the conditions of the preceding proposition. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

Veronika Pitrov´ a (UJEP) 14 / 17

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Hereditary bicoreflective subcategories

Corollary

Let A be an extremal epireflective subcategory of STopGr or QTopGr that satisfies the conditions of the preceding proposition. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

holds particularly for STopGr and QTopGr

Veronika Pitrov´ a (UJEP) 14 / 17

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Hereditary bicoreflective subcategories

Corollary

Let A be an extremal epireflective subcategory of STopGr or QTopGr that satisfies the conditions of the preceding proposition. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

holds particularly for STopGr and QTopGr
hereditary coreflective subcategories that are not bicoreflective:
1. the subcategory containing only the trivial group
2. the subcategory of all indiscrete groups

Veronika Pitrov´ a (UJEP) 14 / 17

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Hereditary bicoreflective subcategories

Corollary

Let A be an extremal epireflective subcategory of STopGr or QTopGr that satisfies the conditions of the preceding proposition. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

holds particularly for STopGr and QTopGr
hereditary coreflective subcategories that are not bicoreflective:
1. the subcategory containing only the trivial group
2. the subcategory of all indiscrete groups
PTopGr, TopGr:

Veronika Pitrov´ a (UJEP) 14 / 17

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Hereditary bicoreflective subcategories

Corollary

Let A be an extremal epireflective subcategory of STopGr or QTopGr that satisfies the conditions of the preceding proposition. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

holds particularly for STopGr and QTopGr
hereditary coreflective subcategories that are not bicoreflective:
1. the subcategory containing only the trivial group
2. the subcategory of all indiscrete groups
PTopGr, TopGr:
Z – the subgroup of the unit circle group {z ∈ C : |z| = 1}

generated by eiπr for some irrational r

Veronika Pitrov´ a (UJEP) 14 / 17

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Hereditary bicoreflective subcategories

Corollary

Let A be an extremal epireflective subcategory of STopGr or QTopGr that satisfies the conditions of the preceding proposition. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

holds particularly for STopGr and QTopGr
hereditary coreflective subcategories that are not bicoreflective:
1. the subcategory containing only the trivial group
2. the subcategory of all indiscrete groups
PTopGr, TopGr:
Z – the subgroup of the unit circle group {z ∈ C : |z| = 1}

generated by eiπr for some irrational r

every non-trivial subgroup of Z is dense

Veronika Pitrov´ a (UJEP) 14 / 17

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Hereditary bicoreflective subcategories

Corollary

Let A be an extremal epireflective subcategory of STopGr or QTopGr that satisfies the conditions of the preceding proposition. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

holds particularly for STopGr and QTopGr
hereditary coreflective subcategories that are not bicoreflective:
1. the subcategory containing only the trivial group
2. the subcategory of all indiscrete groups
PTopGr, TopGr:
Z – the subgroup of the unit circle group {z ∈ C : |z| = 1}

generated by eiπr for some irrational r

every non-trivial subgroup of Z is dense
is the hereditary coreflective hull of Z bicoreflective in PTopGr

(TopGr)?

Veronika Pitrov´ a (UJEP) 14 / 17

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Hereditary bicoreflective subcategories

Hereditary bicoreflective subcategories in A ⊆ STopAb

Proposition

Let A be an extremal epireflective subcategory of STopAb or QTopAb such that Z ∈ A and B be the subcategory of A consisting precisely of such groups G ∈ A that no infinite cyclic subgroup of G is

T0. Then B is the largest hereditary coreflective subcategory of A that

is not bicoreflective in A.

Veronika Pitrov´ a (UJEP) 15 / 17

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Hereditary bicoreflective subcategories

Hereditary bicoreflective subcategories in A ⊆ STopAb

Proposition

Let A be an extremal epireflective subcategory of STopAb or QTopAb such that Z ∈ A and B be the subcategory of A consisting precisely of such groups G ∈ A that no infinite cyclic subgroup of G is

T0. Then B is the largest hereditary coreflective subcategory of A that

is not bicoreflective in A.

Example

A – extremal epireflective in PTopAb or TopAb, Z ∈ A
B – such groups G ∈ A that every infinite cyclic subgroup of G

that is T0 has a neighborhood base at 0 consisting only of its non-trivial subgroups ⇒ B is hereditary and coreflective in A, but not bicoreflective

Veronika Pitrov´ a (UJEP) 15 / 17

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Hereditary bicoreflective subcategories

If rA(Z) = Zn

Proposition

Let n ∈ N and A be the subcategory of STopGr (QTopGr, PTopGr

r TopGr) consisting of all groups G such that the order of every

element of G is a divisor of n. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

Veronika Pitrov´ a (UJEP) 16 / 17

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Hereditary bicoreflective subcategories

If rA(Z) = Zn

Proposition

Let n ∈ N and A be the subcategory of STopGr (QTopGr, PTopGr

r TopGr) consisting of all groups G such that the order of every

element of G is a divisor of n. Then every hereditary coreflective subcategory of A that contains a non-indiscrete group is bicoreflective in A.

Proposition

Let A be an epireflective subcategory of STopAb such that rA(Z) = Zn, n = pα1

1 · . . . · pαk k

be the prime factorization of n and Bi be the subcategory of A consisting precisely of such groups G ∈ A that no cyclic subgroup of G of order pαi

i

is discrete. Then each Bi is a maximal hereditary coreflective subcategory of A that is not bicoreflective in A.

Veronika Pitrov´ a (UJEP) 16 / 17

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Thank you for your attention.

Veronika Pitrov´ a (UJEP) 17 / 17