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

Central extensions and closure operators 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

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

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

What are quandles? Quandles Covering theory Closure operator Central extension for another adjunction Definition (Joyce, 1982) A quandle is a set X with two binary operations ⊳ and ⊳ − 1 satisfying :

What are quandles? Quandles Covering theory Closure operator Central extension for another adjunction Definition (Joyce, 1982) A quandle is a set X with two binary operations ⊳ and ⊳ − 1 satisfying : x ⊳ x = x (idempotency);

What are quandles? Quandles Covering theory Closure operator Central extension for another adjunction 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);

What are quandles? Quandles Covering theory Closure operator Central extension for another adjunction 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).

What are quandles? Quandles Covering theory Closure operator Central extension for another adjunction 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.

Examples Quandles Covering theory Closure operator Central extension for another adjunction Examples Let X be a set, define x ⊳ y = x = x ⊳ − 1 y .

Examples Quandles Covering theory Closure operator Central extension for another adjunction Examples Let X be a set, define x ⊳ y = x = x ⊳ − 1 y . It is a trivial quandle . The corresponding category is denoted Qnd ∗ .

Examples Quandles Covering theory Closure operator Central extension for another adjunction 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 − 1 gh and g ⊳ − 1 h = hgh − 1 . It defines the conjugation quandle .

Examples Quandles Covering theory Closure operator Central extension for another adjunction 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 − 1 gh 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 .

Some properties Quandles Covering theory Closure operator Central extension for another adjunction Right translations ρ y : X → X : x �→ x ⊳ y are automorphisms.

Some properties Quandles Covering theory Closure operator Central extension for another adjunction Right translations ρ y : X → X : x �→ x ⊳ y are automorphisms. The subgroup of inner automorphisms of X : Inn( X ) = �{ ρ y | y ∈ X }� .

Some properties Quandles Covering theory Closure operator Central extension for another adjunction Right translations ρ y : X → X : x �→ x ⊳ y are automorphisms. The subgroup of inner automorphisms of X : Inn( X ) = �{ ρ y | y ∈ X }� . Definition A connected component of X is an orbit under the action of Inn( X ) : [ x ] Inn( X ) .

Some properties Quandles Covering theory Closure operator Central extension for another adjunction Right translations ρ y : X → X : x �→ x ⊳ y are automorphisms. The subgroup of inner automorphisms of X : Inn( X ) = �{ ρ y | y ∈ X }� . Definition A connected component of X is an orbit under the action of Inn( X ) : [ x ] Inn( X ) . π 0 ( X ) = { connected components of X } .

Some properties Quandles Covering theory Closure operator Central extension for another adjunction Right translations ρ y : X → X : x �→ x ⊳ y are automorphisms. The subgroup of inner automorphisms of X : Inn( X ) = �{ ρ y | y ∈ X }� . Definition A connected component of X is an orbit under the action of Inn( X ) : [ x ] Inn( X ) . π 0 ( X ) = { connected components of X } . A quandle X is algebraically connected if π 0 ( X ) = { ⋆ } .

Adjunction Quandles Covering theory Closure operator Central extension for another adjunction π 0 Qnd ∗ Qnd ⊥ U

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

Admissibility Quandles Covering theory Closure operator Central extension for another adjunction Proposition The previous adjunction is admissible for Categorical Galois Theory.

Admissibility Quandles Covering theory Closure operator Central extension for another adjunction Proposition The previous adjunction is admissible for Categorical Galois Theory. The functor π 0 preserves π 2 B × UY UX UX π 1 U φ B UY f

Admissibility Quandles Covering theory Closure operator Central extension for another adjunction Proposition The previous adjunction is admissible for Categorical Galois Theory. The functor π 0 preserves π 2 B × UY UX UX π 1 U φ B UY f where φ : X → Y is a regular epimorphism in Qnd ∗ .

Mal’tsev? Quandles Covering theory Closure operator Central extension for another adjunction Definition (Bunch, Lofgren, Rapp, Yetter, 2010) Normal subgroups N ⊂ Inn X define congruences on X x ∼ N y ⇐ ⇒ [ x ] N = [ y ] N .

Mal’tsev? Quandles Covering theory Closure operator Central extension for another adjunction 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 .

Mal’tsev? Quandles Covering theory Closure operator Central extension for another adjunction 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( η X ) = ∼ Inn( X ) .

Trivial Extension Quandles Covering theory Closure operator Central extension for another adjunction Definition A regular epimorphism f : A → B is a trivial extension when η A U π 0 ( A ) A U π 0 ( f ) f U π 0 ( B ) B η B is a pullback.

Central Extension Quandles Covering theory Closure operator Central extension for another adjunction Definition A regular epimorphism f : A → B is a central extension if there exists a regular epimorphism p : E → B such that the pullback π 1 of f along p is a trivial extension. π 2 E × B A A π 1 f E B p

Results Quandles Covering theory Closure operator Central extension for another adjunction 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 .

Results Quandles Covering theory Closure operator Central extension for another adjunction 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.

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

Closure operators Quandles Covering theory Closure operator Central extension for another adjunction Definition m A closure operator c in C associates, with any subobject M − → X , c X ( m ) another subobject c X ( M ) − − − − → X m / c X ( m ) c X ( M ) M m c X ( m ) X

Quandles Covering theory Closure operator Central extension for another adjunction A closure operator c satisfies the following properties : 1 m ≤ c X ( m ) (extension); 2 if m ≤ n , then c X ( m ) ≤ c X ( n ) (order-preserving); 3 f ( c X ( m )) ≤ c Y ( f ( m )), f : X → Y (continuity).