Some Aspects of Weak Ideal Topology on the Topos of Right Acts over - PowerPoint PPT Presentation

Preliminaries Weak Ideal Topology Results References Some Aspects of Weak Ideal Topology on the Topos of Right Acts over a Monoid Ali Madanshekaf Semnan University amadanshekaf@semnan.ac.ir BLAST 2018 University of Denver, Colorado, USA August 6-10 2018 Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Outline Overview of talk In this talk we Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Outline Overview of talk In this talk we introduce the (weak) ideal topology j I on the topos Act - S of right S -acts. Afterwards, we observe that some known categories of separated acts over a monoid are special cases of j I -separated categories. Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Results References Outline Overview of talk In this talk we introduce the (weak) ideal topology j I on the topos Act - S of right S -acts. Afterwards, we observe that some known categories of separated acts over a monoid are special cases of j I -separated categories. give a simple description of the associated sheaf functor with respect to the ideal topology j I where I is a central band of S . Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Definition Let S be a monoid with the identity 1. A (right) S -act is a set A equipped with an action µ : A × S → A , ( a , s ) � as , such that one has a 1 = a and a ( st ) = ( as ) t , for all a ∈ A and s , t ∈ S . Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Definition Let S be a monoid with the identity 1. A (right) S -act is a set A equipped with an action µ : A × S → A , ( a , s ) � as , such that one has a 1 = a and a ( st ) = ( as ) t , for all a ∈ A and s , t ∈ S . We denote the category of right S -acts with equivariant maps between them by Act - S . Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Subobject classifier of Act - S It is well known that the set Idl ( S ) of all right ideals K of S endowed with the action Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Subobject classifier of Act - S It is well known that the set Idl ( S ) of all right ideals K of S endowed with the action K · s = { t ∈ S | st ∈ K } , Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Subobject classifier of Act - S It is well known that the set Idl ( S ) of all right ideals K of S endowed with the action K · s = { t ∈ S | st ∈ K } , and the monomorphism true : 1 = { θ } ֌ Idl ( S ) , given by θ � S , is the subobject classifier of Act - S . Ali Madanshekaf Semnan University

� � � � Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Subobject classifier of Act - S It is well known that the set Idl ( S ) of all right ideals K of S endowed with the action K · s = { t ∈ S | st ∈ K } , and the monomorphism true : 1 = { θ } ֌ Idl ( S ) , given by θ � S , is the subobject classifier of Act - S . Indeed, associated to any subact B ⊆ A there is a unique map φ B : A → Idl ( S ), so-called characteristic map of B ⊆ A , given by φ B ( a ) = { s ∈ S | as ∈ B } such that the following diagram is a pullback 1 B � � true φ B � Ω . A Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Modal closure operator A family C = ( C A ) A ∈ Act − S , where C A : Sub Act − S ( A ) → Sub Act − S ( A ) , is called a modal closure operator on Act - S whenever for any S -act A we have Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Modal closure operator A family C = ( C A ) A ∈ Act − S , where C A : Sub Act − S ( A ) → Sub Act − S ( A ) , is called a modal closure operator on Act - S whenever for any S -act A we have (Extensive) E ⊆ C A ( E ), for all subacts E of A ; 1 Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Modal closure operator A family C = ( C A ) A ∈ Act − S , where C A : Sub Act − S ( A ) → Sub Act − S ( A ) , is called a modal closure operator on Act - S whenever for any S -act A we have (Extensive) E ⊆ C A ( E ), for all subacts E of A ; 1 (Monotonicity) D ⊆ E implies C A ( D ) ⊆ C A ( E ), for all subacts 2 D and E of A ; Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Modal closure operator A family C = ( C A ) A ∈ Act − S , where C A : Sub Act − S ( A ) → Sub Act − S ( A ) , is called a modal closure operator on Act - S whenever for any S -act A we have (Extensive) E ⊆ C A ( E ), for all subacts E of A ; 1 (Monotonicity) D ⊆ E implies C A ( D ) ⊆ C A ( E ), for all subacts 2 D and E of A ; (Modal) f − 1 ( C B ( E )) = C A ( f − 1 ( E )), for any equivariant map 3 f : A → B and any subact E of B . Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Modal closure operator A family C = ( C A ) A ∈ Act − S , where C A : Sub Act − S ( A ) → Sub Act − S ( A ) , is called a modal closure operator on Act - S whenever for any S -act A we have (Extensive) E ⊆ C A ( E ), for all subacts E of A ; 1 (Monotonicity) D ⊆ E implies C A ( D ) ⊆ C A ( E ), for all subacts 2 D and E of A ; (Modal) f − 1 ( C B ( E )) = C A ( f − 1 ( E )), for any equivariant map 3 f : A → B and any subact E of B . For simplicity the subact C A ( B ) of A is often denoted by B . Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak (Lawvere-Tierney) topology An equivariant map j : Idl ( S ) → Idl ( S ) is said to be a weak (Lawvere-Tierney) topology on Act - S whenever the following hold: Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak (Lawvere-Tierney) topology An equivariant map j : Idl ( S ) → Idl ( S ) is said to be a weak (Lawvere-Tierney) topology on Act - S whenever the following hold: j ( S ) = S ; 1 Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak (Lawvere-Tierney) topology An equivariant map j : Idl ( S ) → Idl ( S ) is said to be a weak (Lawvere-Tierney) topology on Act - S whenever the following hold: j ( S ) = S ; 1 j ( K 1 ∩ K 2 ) ⊆ j ( K 1 ) ∩ j ( K 2 ) , for all right ideals K 1 and K 2 of S . 2 Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak (Lawvere-Tierney) topology An equivariant map j : Idl ( S ) → Idl ( S ) is said to be a weak (Lawvere-Tierney) topology on Act - S whenever the following hold: j ( S ) = S ; 1 j ( K 1 ∩ K 2 ) ⊆ j ( K 1 ) ∩ j ( K 2 ) , for all right ideals K 1 and K 2 of S . 2 Furthermore, j is called productive whenever the inclusion in (2) is an equality. Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak (Lawvere-Tierney) topology An equivariant map j : Idl ( S ) → Idl ( S ) is said to be a weak (Lawvere-Tierney) topology on Act - S whenever the following hold: j ( S ) = S ; 1 j ( K 1 ∩ K 2 ) ⊆ j ( K 1 ) ∩ j ( K 2 ) , for all right ideals K 1 and K 2 of S . 2 Furthermore, j is called productive whenever the inclusion in (2) is an equality. An idempotent weak topology on Act - S is called a topology on Act - S . Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak Topologies and Modal Closure Operators Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak Topologies and Modal Closure Operators Corresponding to any weak topology j on Act - S , there is a modal closure operator on Act - S given by Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak Topologies and Modal Closure Operators Corresponding to any weak topology j on Act - S , there is a modal closure operator on Act - S given by C A ( B ) = { a ∈ A | j ( φ B ( a )) = S } , for any subact B ⊆ A . Ali Madanshekaf Semnan University

Preliminaries Weak Ideal Topology Weak Topologies and Modal Closure Operators Results References Weak Topologies and Modal Closure Operators Corresponding to any weak topology j on Act - S , there is a modal closure operator on Act - S given by C A ( B ) = { a ∈ A | j ( φ B ( a )) = S } , for any subact B ⊆ A . Conversely, any modal closure operator C on Act - S induces a weak topology j : Idl ( S ) → Idl ( S ) on Act - S given by Ali Madanshekaf Semnan University