Central extensions and closure operators in the category of - - PowerPoint PPT Presentation

▶
central extensions and closure operators in the category
SMART_READER_LITE
LIVE PREVIEW

Central extensions and closure operators in the category of - - PowerPoint PPT Presentation

Dec 03, 2022 288 likes •1.07k views

Central extensions and closure operators in the category of quandles Valrian Even Universit catholique de Louvain joint work with M. Gran and A. Montoli Category Theory 2015 University of Aveiro June 16, 2015 Table of contents Quandles

slide-1
SLIDE 1

Central extensions and closure

  • perators in the category of

quandles

Valérian Even Université catholique de Louvain joint work with M. Gran and A. Montoli

Category Theory 2015 University of Aveiro

June 16, 2015

slide-2
SLIDE 2

Quandles Covering theory Closure operator Central extension for another adjunction

Table of contents

1 Quandles 2 Covering theory 3 Closure operator

Connected and separated objects Connectedness and disconnectedness

4 Central extension for another adjunction

slide-3
SLIDE 3

Quandles Covering theory Closure operator Central extension for another adjunction

Outline

1 Quandles 2 Covering theory 3 Closure operator 4 Central extension for another adjunction

slide-4
SLIDE 4

Quandles Covering theory Closure operator Central extension for another adjunction

What are quandles?

Definition (Joyce, 1982)

A quandle is a set X with two binary operations ⊳ and ⊳−1 satisfying :

slide-5
SLIDE 5

Quandles Covering theory Closure operator Central extension for another adjunction

What are quandles?

Definition (Joyce, 1982)

A quandle is a set X with two binary operations ⊳ and ⊳−1 satisfying : x ⊳ x = x (idempotency);

slide-6
SLIDE 6

Quandles Covering theory Closure operator Central extension for another adjunction

What are quandles?

Definition (Joyce, 1982)

A quandle is a set X with two binary operations ⊳ and ⊳−1 satisfying : x ⊳ x = x (idempotency); (x ⊳ y) ⊳−1 y = x = (x ⊳−1 y) ⊳ y (right invertibility);

slide-7
SLIDE 7

Quandles Covering theory Closure operator Central extension for another adjunction

What are quandles?

Definition (Joyce, 1982)

A quandle is a set X with two binary operations ⊳ and ⊳−1 satisfying : x ⊳ x = x (idempotency); (x ⊳ y) ⊳−1 y = x = (x ⊳−1 y) ⊳ y (right invertibility); (x ⊳ y) ⊳ z = (x ⊳ z) ⊳ (y ⊳ z) (self-distributivity).

slide-8
SLIDE 8

Quandles Covering theory Closure operator Central extension for another adjunction

What are quandles?

Definition (Joyce, 1982)

A quandle is a set X with two binary operations ⊳ and ⊳−1 satisfying : x ⊳ x = x (idempotency); (x ⊳ y) ⊳−1 y = x = (x ⊳−1 y) ⊳ y (right invertibility); (x ⊳ y) ⊳ z = (x ⊳ z) ⊳ (y ⊳ z) (self-distributivity). Denote Qnd the corresponding category.

slide-9
SLIDE 9

Quandles Covering theory Closure operator Central extension for another adjunction

Examples

Examples

Let X be a set, define x ⊳ y = x = x ⊳−1 y.

slide-10
SLIDE 10

Quandles Covering theory Closure operator Central extension for another adjunction

Examples

Examples

Let X be a set, define x ⊳ y = x = x ⊳−1 y. It is a trivial

  • quandle. The corresponding category is denoted Qnd∗.
slide-11
SLIDE 11

Quandles Covering theory Closure operator Central extension for another adjunction

Examples

Examples

Let X be a set, define x ⊳ y = x = x ⊳−1 y. It is a trivial

  • quandle. The corresponding category is denoted Qnd∗.

Let G be a group, define g ⊳ h = h−1gh and g ⊳−1 h = hgh−1. It defines the conjugation quandle.

slide-12
SLIDE 12

Quandles Covering theory Closure operator Central extension for another adjunction

Examples

Examples

Let X be a set, define x ⊳ y = x = x ⊳−1 y. It is a trivial

  • quandle. The corresponding category is denoted Qnd∗.

Let G be a group, define g ⊳ h = h−1gh and g ⊳−1 h = hgh−1. It defines the conjugation quandle. Let G be a group and φ an automorphism of G, define g ⊳ h = φ(gh−1)h and g ⊳−1 h = φ−1(gh−1)h.

slide-13
SLIDE 13

Quandles Covering theory Closure operator Central extension for another adjunction

Some properties

Right translations ρy : X → X : x → x ⊳ y are automorphisms.

slide-14
SLIDE 14

Quandles Covering theory Closure operator Central extension for another adjunction

Some properties

Right translations ρy : X → X : x → x ⊳ y are automorphisms. The subgroup of inner automorphisms of X : Inn(X) = {ρy | y ∈ X}.

slide-15
SLIDE 15

Quandles Covering theory Closure operator Central extension for another adjunction

Some properties

Right translations ρy : X → X : x → x ⊳ y are automorphisms. The subgroup of inner automorphisms of X : Inn(X) = {ρy | y ∈ X}.

Definition

A connected component of X is an orbit under the action

  • f Inn(X) : [x]Inn(X).
slide-16
SLIDE 16

Quandles Covering theory Closure operator Central extension for another adjunction

Some properties

Right translations ρy : X → X : x → x ⊳ y are automorphisms. The subgroup of inner automorphisms of X : Inn(X) = {ρy | y ∈ X}.

Definition

A connected component of X is an orbit under the action

  • f Inn(X) : [x]Inn(X).

π0(X) = {connected components of X}.

slide-17
SLIDE 17

Quandles Covering theory Closure operator Central extension for another adjunction

Some properties

Right translations ρy : X → X : x → x ⊳ y are automorphisms. The subgroup of inner automorphisms of X : Inn(X) = {ρy | y ∈ X}.

Definition

A connected component of X is an orbit under the action

  • f Inn(X) : [x]Inn(X).

π0(X) = {connected components of X}. A quandle X is algebraically connected if π0(X) = {⋆}.

slide-18
SLIDE 18

Quandles Covering theory Closure operator Central extension for another adjunction

Adjunction

Qnd Qnd∗ ⊥ π0 U

slide-19
SLIDE 19

Quandles Covering theory Closure operator Central extension for another adjunction

Outline

1 Quandles 2 Covering theory 3 Closure operator 4 Central extension for another adjunction

slide-20
SLIDE 20

Quandles Covering theory Closure operator Central extension for another adjunction

Admissibility

Proposition

The previous adjunction is admissible for Categorical Galois Theory.

slide-21
SLIDE 21

Quandles Covering theory Closure operator Central extension for another adjunction

Admissibility

Proposition

The previous adjunction is admissible for Categorical Galois Theory. The functor π0 preserves B ×UY UX UX B UY π2 π1 Uφ f

slide-22
SLIDE 22

Quandles Covering theory Closure operator Central extension for another adjunction

Admissibility

Proposition

The previous adjunction is admissible for Categorical Galois Theory. The functor π0 preserves B ×UY UX UX B UY π2 π1 Uφ f where φ: X → Y is a regular epimorphism in Qnd∗.

slide-23
SLIDE 23

Quandles Covering theory Closure operator Central extension for another adjunction

Mal’tsev?

Definition (Bunch, Lofgren, Rapp, Yetter, 2010)

Normal subgroups N ⊂ Inn X define congruences on X x ∼N y ⇐ ⇒ [x]N = [y]N.

slide-24
SLIDE 24

Quandles Covering theory Closure operator Central extension for another adjunction

Mal’tsev?

Definition (Bunch, Lofgren, Rapp, Yetter, 2010)

Normal subgroups N ⊂ Inn X define congruences on X x ∼N y ⇐ ⇒ [x]N = [y]N.

Lemma

For any reflexive relation R on X in Qnd R◦ ∼N=∼N ◦R.

slide-25
SLIDE 25

Quandles Covering theory Closure operator Central extension for another adjunction

Mal’tsev?

Definition (Bunch, Lofgren, Rapp, Yetter, 2010)

Normal subgroups N ⊂ Inn X define congruences on X x ∼N y ⇐ ⇒ [x]N = [y]N.

Lemma

For any reflexive relation R on X in Qnd R◦ ∼N=∼N ◦R.

Remark

Eq(ηX) =∼Inn(X).

slide-26
SLIDE 26

Quandles Covering theory Closure operator Central extension for another adjunction

Trivial Extension

Definition

A regular epimorphism f : A → B is a trivial extension when A Uπ0(A) B Uπ0(B) ηA f Uπ0(f ) ηB is a pullback.

slide-27
SLIDE 27

Quandles Covering theory Closure operator Central extension for another adjunction

Central Extension

Definition

A regular epimorphism f : A → B is a central extension if there exists a regular epimorphism p : E → B such that the pullback π1

  • f f along p is a trivial extension.

E ×B A A E B π2 π1 f p

slide-28
SLIDE 28

Quandles Covering theory Closure operator Central extension for another adjunction

Results

Definition (Eisermann, 2014)

A quandle homomorphism f : X → Y is a quandle covering if it is surjective and f (x) = f (x′) implies z ⊳ x = z ⊳ x′ for all z ∈ X.

slide-29
SLIDE 29

Quandles Covering theory Closure operator Central extension for another adjunction

Results

Definition (Eisermann, 2014)

A quandle homomorphism f : X → Y is a quandle covering if it is surjective and f (x) = f (x′) implies z ⊳ x = z ⊳ x′ for all z ∈ X.

Theorem (E., 2014)

f : X → Y is a central extension if and only if f : X → Y is a quandle covering.

slide-30
SLIDE 30

Quandles Covering theory Closure operator Central extension for another adjunction

Outline

1 Quandles 2 Covering theory 3 Closure operator Connected and separated objects Connectedness and disconnectedness 4 Central extension for another adjunction

slide-31
SLIDE 31

Quandles Covering theory Closure operator Central extension for another adjunction

Closure operators

Definition

A closure operator c in C associates, with any subobject M

m

− → X, another subobject cX(M)

cX(m)

− − − − → X M cX(M) X m/cX(m) m cX(m)

slide-32
SLIDE 32

Quandles Covering theory Closure operator Central extension for another adjunction

A closure operator c satisfies the following properties :

1 m ≤ cX(m) (extension); 2 if m ≤ n, then cX(m) ≤ cX(n) (order-preserving); 3 f (cX(m)) ≤ cY (f (m)), f : X → Y (continuity).

slide-33
SLIDE 33

Quandles Covering theory Closure operator Central extension for another adjunction

A closure operator c satisfies the following properties :

1 m ≤ cX(m) (extension); 2 if m ≤ n, then cX(m) ≤ cX(n) (order-preserving); 3 f (cX(m)) ≤ cY (f (m)), f : X → Y (continuity).

It is idempotent if, moreover,

4 cX(M) cX(m)

− − − − → X is closed : cX(cX(m)) = cX(m).

slide-34
SLIDE 34

Quandles Covering theory Closure operator Central extension for another adjunction

Pullback closure operator

Pullback closure operator (Holgate, 1996): Given a reflective subcategory X of a regular category C C X ⊥ I H

slide-35
SLIDE 35

Quandles Covering theory Closure operator Central extension for another adjunction

Construction

M X HI(M) HI(X) m ηM ηX HI(m)

slide-36
SLIDE 36

Quandles Covering theory Closure operator Central extension for another adjunction

Construction

M X N HI(M) HI(X) m ηM ηX HI(m) e i

slide-37
SLIDE 37

Quandles Covering theory Closure operator Central extension for another adjunction

Construction

M X N cX(M) HI(M) HI(X) m ηM ηX HI(m) e i m/cX(m) cX(m)

slide-38
SLIDE 38

Quandles Covering theory Closure operator Central extension for another adjunction

Special case

Lemma

In the case of the adjunction Qnd Qnd∗ ⊥ π0 U For a subobject M

m

− → X ∈ Qnd, cX(M) = {x ∈ X | x ∈ [a]Inn(X) for some a ∈ M}.

slide-39
SLIDE 39

Quandles Covering theory Closure operator Central extension for another adjunction

Illustration

The closure of a subquandle M can be represented as follows : X M

slide-40
SLIDE 40

Quandles Covering theory Closure operator Central extension for another adjunction

Illustration

The closure of a subquandle M can be represented as follows : X M It is idempotent.

slide-41
SLIDE 41

Quandles Covering theory Closure operator Central extension for another adjunction

Properties

Proposition

It has the following properties :

slide-42
SLIDE 42

Quandles Covering theory Closure operator Central extension for another adjunction

Properties

Proposition

It has the following properties :

1 cX( i∈I si) = i∈I cX(si) (fully additive);

slide-43
SLIDE 43

Quandles Covering theory Closure operator Central extension for another adjunction

Properties

Proposition

It has the following properties :

1 cX( i∈I si) = i∈I cX(si) (fully additive); 2 cX( 1≤i≤n mi) = 1≤i≤n cXi(mi), where X = 1≤i≤n Xi

(finitely productive);

slide-44
SLIDE 44

Quandles Covering theory Closure operator Central extension for another adjunction

Properties

Proposition

It has the following properties :

1 cX( i∈I si) = i∈I cX(si) (fully additive); 2 cX( 1≤i≤n mi) = 1≤i≤n cXi(mi), where X = 1≤i≤n Xi

(finitely productive);

3 f (cX(m)) = cY (f (m)) for any surjective

homomorphism f : X → Y .

slide-45
SLIDE 45

Quandles Covering theory Closure operator Central extension for another adjunction

Properties

Proposition

It has the following properties :

1 cX( i∈I si) = i∈I cX(si) (fully additive); 2 cX( 1≤i≤n mi) = 1≤i≤n cXi(mi), where X = 1≤i≤n Xi

(finitely productive);

3 f (cX(m)) = cY (f (m)) for any surjective

homomorphism f : X → Y .

Remark

It is not weakly hereditary.

slide-46
SLIDE 46

Quandles Covering theory Closure operator Central extension for another adjunction

Outline

1 Quandles 2 Covering theory 3 Closure operator

Connected and separated objects Connectedness and disconnectedness

4 Central extension for another adjunction

slide-47
SLIDE 47

Quandles Covering theory Closure operator Central extension for another adjunction

Connected and separated

Definition

An object X is c-connected if ∆X : X → X × X is dense.

slide-48
SLIDE 48

Quandles Covering theory Closure operator Central extension for another adjunction

Connected and separated

Definition

An object X is c-connected if ∆X : X → X × X is dense. An object X is c-separated if ∆X : X → X × X is closed.

slide-49
SLIDE 49

Quandles Covering theory Closure operator Central extension for another adjunction

Connected and separated

Definition

An object X is c-connected if ∆X : X → X × X is dense. An object X is c-separated if ∆X : X → X × X is closed.

Proposition

A quandle X is c-connected if and only if it is algebraically connected.

slide-50
SLIDE 50

Quandles Covering theory Closure operator Central extension for another adjunction

Connected and separated

Definition

An object X is c-connected if ∆X : X → X × X is dense. An object X is c-separated if ∆X : X → X × X is closed.

Proposition

A quandle X is c-connected if and only if it is algebraically connected. A quandle X is c-separated if and only if it is a trivial quandle.

slide-51
SLIDE 51

Quandles Covering theory Closure operator Central extension for another adjunction

Outline

1 Quandles 2 Covering theory 3 Closure operator

Connected and separated objects Connectedness and disconnectedness

4 Central extension for another adjunction

slide-52
SLIDE 52

Quandles Covering theory Closure operator Central extension for another adjunction

Constant morphism

Definition

A morphism f : X → Y is said to be constant if f ◦ u = f ◦ v for all u, v : Z → X.

slide-53
SLIDE 53

Quandles Covering theory Closure operator Central extension for another adjunction

Constant morphism

Definition

A morphism f : X → Y is said to be constant if f ◦ u = f ◦ v for all u, v : Z → X. In the category of Qnd, f : X → Y is constant if X Y {⋆} f

slide-54
SLIDE 54

Quandles Covering theory Closure operator Central extension for another adjunction

Connectedness and disconnectedness

Definition

For a full subcategory H of C, r(H) := {C ∈ C | every f : X → C is constant for all X ∈ H} is called a disconnectedness.

slide-55
SLIDE 55

Quandles Covering theory Closure operator Central extension for another adjunction

Connectedness and disconnectedness

Definition

For a full subcategory H of C, r(H) := {C ∈ C | every f : X → C is constant for all X ∈ H} is called a disconnectedness. l(H) := {C ∈ C | every f : C → X is constant for all X ∈ H} is called a connectedness.

slide-56
SLIDE 56

Quandles Covering theory Closure operator Central extension for another adjunction

Correspondence

Let Sub(C) be the class of all full subcategories of C ordered by inclusion:

slide-57
SLIDE 57

Quandles Covering theory Closure operator Central extension for another adjunction

Correspondence

Let Sub(C) be the class of all full subcategories of C ordered by inclusion: Sub(C) Sub(C)op ⊥ r l

slide-58
SLIDE 58

Quandles Covering theory Closure operator Central extension for another adjunction

Correspondence

Let Sub(C) be the class of all full subcategories of C ordered by inclusion: Sub(C) Sub(C)op ⊥ r l

Example

In the category Top of topological spaces, Y := {connected spaces} = l(r(Y)) Z := {hereditarily disconnected spaces} = r(Y)

slide-59
SLIDE 59

Quandles Covering theory Closure operator Central extension for another adjunction

Correspondence

Proposition

In the category Qnd, given X = Qnd∗

slide-60
SLIDE 60

Quandles Covering theory Closure operator Central extension for another adjunction

Correspondence

Proposition

In the category Qnd, given X = Qnd∗ Y := {connected quandles} = l(X) = l(r(Y))

slide-61
SLIDE 61

Quandles Covering theory Closure operator Central extension for another adjunction

Correspondence

Proposition

In the category Qnd, given X = Qnd∗ Y := {connected quandles} = l(X) = l(r(Y)) {X ∈ Qnd | X has no (non-trivial) connected subquandles} = r(Y).

slide-62
SLIDE 62

Quandles Covering theory Closure operator Central extension for another adjunction

Outline

1 Quandles 2 Covering theory 3 Closure operator 4 Central extension for another adjunction

slide-63
SLIDE 63

Quandles Covering theory Closure operator Central extension for another adjunction

Symmetric quandles

Definition

A quandle X is symmetric if x ⊳ y = y ⊳ x.

slide-64
SLIDE 64

Quandles Covering theory Closure operator Central extension for another adjunction

Symmetric quandles

Definition

A quandle X is symmetric if x ⊳ y = y ⊳ x.

Remark

The variety of SymQnd is a Mal’tsev variety. p(x, y, z) = (x ⊳ z) ⊳−1 y.

slide-65
SLIDE 65

Quandles Covering theory Closure operator Central extension for another adjunction

Abelian symmetric quandles

Definition

A quandle X is abelian if (x ⊳ y) ⊳ (z ⊳ w) = (x ⊳ z) ⊳ (y ⊳ w).

slide-66
SLIDE 66

Quandles Covering theory Closure operator Central extension for another adjunction

Abelian symmetric quandles

Definition

A quandle X is abelian if (x ⊳ y) ⊳ (z ⊳ w) = (x ⊳ z) ⊳ (y ⊳ w).

Lemma

The variety AbSymQnd of abelian symmetric quandles is a naturally Mal’tsev category (in the sense of Johnstone, 1989).

slide-67
SLIDE 67

Quandles Covering theory Closure operator Central extension for another adjunction

Adjunction

Qnd SymQnd AbSymQnd ⊥ ⊥ sym V ab U

slide-68
SLIDE 68

Quandles Covering theory Closure operator Central extension for another adjunction

Adjunction

Qnd SymQnd AbSymQnd ⊥ ⊥ sym V ab U

Proposition

The adjunction is admissible for Categorical Galois Theory.

slide-69
SLIDE 69

Quandles Covering theory Closure operator Central extension for another adjunction

Adjunction

Qnd SymQnd AbSymQnd ⊥ ⊥ sym V ab U

Proposition

The adjunction is admissible for Categorical Galois Theory. There is a special class of congruences (Bourn, 2014) which permute with any other congruence in Qnd.

slide-70
SLIDE 70

Quandles Covering theory Closure operator Central extension for another adjunction

Fibers

In Qnd, the fiber f −1(y) is a subquandle f −1(y) {y} X Y f

slide-71
SLIDE 71

Quandles Covering theory Closure operator Central extension for another adjunction

Fibers

In Qnd, the fiber f −1(y) is a subquandle f −1(y) {y} X Y f f : X → Y has abelian symmetric fibers if each f −1(y) belongs to AbSymQnd.

slide-72
SLIDE 72

Quandles Covering theory Closure operator Central extension for another adjunction

Centralizing relation

Definition

A double equivalence relation C on R and S C R S X p1 p2 s1 s2 π2 π1 r2 r1 is called centralizing when C R S X π2 p1 r2 s1

slide-73
SLIDE 73

Quandles Covering theory Closure operator Central extension for another adjunction

Algebraically central extension

Definition

A surjective quandle homomorphism f : X → Y is an algebraically central extension if there is a centralizing double relation C on Eq(f ) and A × A.

slide-74
SLIDE 74

Quandles Covering theory Closure operator Central extension for another adjunction

Algebraically central extension

Definition

A surjective quandle homomorphism f : X → Y is an algebraically central extension if there is a centralizing double relation C on Eq(f ) and A × A.

Lemma

For a surjective quandle homomorphism f : X → Y with abelian symmetric fibers such a centralizing double relation is unique when it exists;

slide-75
SLIDE 75

Quandles Covering theory Closure operator Central extension for another adjunction

Algebraically central extension

Definition

A surjective quandle homomorphism f : X → Y is an algebraically central extension if there is a centralizing double relation C on Eq(f ) and A × A.

Lemma

For a surjective quandle homomorphism f : X → Y with abelian symmetric fibers such a centralizing double relation is unique when it exists; Eq(f ) ≃ X × Q where Q is an abelian symmetric quandle.

slide-76
SLIDE 76

Quandles Covering theory Closure operator Central extension for another adjunction

Central extension

Theorem (E., Gran, Montoli, 2015)

For a surjective homomorphism f : X → Y in Qnd, tfcae :

1 f is a central extension; 2 f is an algebraically central extension with abelian symmetric

fibers.