Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid S. Sepahani M. Mahmoudi Department of Mathematics Shahid Beheshti University Tehran CT, 13 July 2018 S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 1 / 20
Previously Done Boolean Algebras in a localic topos Banaschewski, Bhutani; 1986 Borceux, Peddicchio, Rossi; 1990 S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 2 / 20
The Category MSet MSet ≃ Set M Limits as in Set The subobject classifier Ω = { K | K is a left ideal of M } mK = { x ∈ M | xm ∈ K } Exponentiation B A = { f | f : M × A → B : f is equivariant } = { f | f = ( f s ) : ∀ s , t ∈ M , f s : A → B , tf s = f ts t } Free functor F : Set → MSet : F ( X ) = M × X m ( n , x ) = ( mn , x ) Cofree functor H : Set → MSet : H ( X ) = { f : M → X } ( mf )( n ) = f ( nm ) H (2) = P ( M ) , mX = { x ∈ M | xm ∈ X } H : Boo → MBoo Monomorphisms in MSet are equivariant one-one maps S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 3 / 20
Closure Operator in a Category Definition A family C = ( C X ) X ∈ MSet , with C X : Sub ( X ) → Sub ( X ) taking Y ≤ X to C X ( Y ), is called a closure operator on M Set if it satisfies the following: 1 (Extension) Y ≤ C X ( Y ) 2 (Monotonicity) Y 1 ≤ Y 2 ⇒ C X ( Y 1 ) ≤ C X ( Y 2 ) 3 (Continuity) f ( C X ( Y )) ≤ C Z ( f ( X )) for all morphisms f : X → Z and we say that C is idempotetnt if additionally we have C X ( C X ( Y )) = Y for every Y ≤ X for Y ≤ X , Y is said to be closed in X if C X ( Y ) = Y dense in X if C X ( Y ) = X S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 4 / 20
Ideal Closure Operator for I � R M Definition → B . C I ( A ) = { b ∈ B |∀ s ∈ I , sb ∈ A } Let A ֒ C I is idempotent iff I is idempotent S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 5 / 20
Ideal Closure Operator for I � R M Definition → B . C I ( A ) = { b ∈ B |∀ s ∈ I , sb ∈ A } Let A ֒ C I is idempotent iff I is idempotent j I ( K ) = { x ∈ M |∀ s ∈ I , sx ∈ K } S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 5 / 20
Ideal Closure Operator for I � R M Definition → B . C I ( A ) = { b ∈ B |∀ s ∈ I , sb ∈ A } Let A ֒ C I is idempotent iff I is idempotent j I ( K ) = { x ∈ M |∀ s ∈ I , sx ∈ K } m : Y X is I -dense if ∀ s ∈ I , ∀ x ∈ X , sx ∈ Y S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 5 / 20
I-Separated Objects and I-Sheaves A ∈ MSet is an I -separated object if for every dense monomorphism m , any two equivariant maps from C to A making the diagram commutative are equivalent. A is an I -sheaf if this map uniquely exists for every I -dense m and every f . S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 6 / 20
� � � I-Separated Objects and I-Sheaves A ∈ MSet is an I -separated object if for every dense monomorphism m , any two equivariant maps from C to A making the diagram commutative are equivalent. A is an I -sheaf if this map uniquely exists for every I -dense m and every f . m B C f ¯ f A S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 6 / 20
� � � I-Separated Objects and I-Sheaves A ∈ MSet is an I -separated object if for every dense monomorphism m , any two equivariant maps from C to A making the diagram commutative are equivalent. A is an I -sheaf if this map uniquely exists for every I -dense m and every f . m B C f ¯ f A Remark A is I -separated iff ∀ a , b ∈ A , ( ∀ s ∈ I , sa = sb ⇒ a = b ) S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 6 / 20
The Category S h j I MSet S h j I MSet is closed under limits in MSet . S h j I MSet is closed under exponentiation in MSet . Ω j I = Eq ( j I , id Ω ) is the subobject classifier of S h j I MSet Ω j I ≤ im ( j I ) S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 7 / 20
The Category S h j I MSet S h j I MSet is closed under limits in MSet . S h j I MSet is closed under exponentiation in MSet . Ω j I = Eq ( j I , id Ω ) is the subobject classifier of S h j I MSet Ω j I ≤ im ( j I ) Sh j I MSet is a topos. S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 7 / 20
S h j I MSet is a subtopos of MSet Theorem (Adamek, Herrlich, Strecker) If E is strongly complete and co-wellpowered, then the following conditions are equivalent for any functor G : E → F : G is adjoint G preserves small limits and is cowellpowered. Proposition (Johnstone) Let E be a cartesian closed category, and L be a reflective subcategory of E , corresponding to a reflector L on E . Then L preserves finite products iff L is an exponential ideal of E . S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 8 / 20
Boolean Algebras in a Topos MBoo S h j I Boo H : Set → MSet can be lifted to H : Boo → MBoo An internal counterpart for Ult ( A ) for a Boolean algebra A . S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 9 / 20
� � � Internal hom Object B A × A ev � B � B A × A n λ [ A , B ] × A n λ ev ( B A × A ) n λ ev n λ � B A × A S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 10 / 20
Internal hom Object in MBoo and in Boo S h j I MSet Definition In Boo S h j I MSet we have the following explicit definition for [ A , B ] [ A , B ] = { ( f s ) s ∈ M | for every s ∈ M , f s : A → B is a Boolean homomorphism , ∀ s , t ∈ M , tf s = f ts t } Example f : A → B Boolean homomorphism for A , B ∈ MSet . Let f e = f and for every s ∈ M , f s = sfs − 1 . Then ( f s ) s ∈ M ∈ [ A , B ]. S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 11 / 20
Initial Boolean Algebras In Set The initial Boolean algebra is 2, the two-element Boolean algebra. In MSet The initial Boolean algebra is 2 . i.e. The two-element Boolean algebra with identity action of M. in BooSh j I MSet The initial Boolean algebra is the sheaf reflection of 2 which is the I -closure of 2 in Ω 2 j I : ¯ 2 = { f ∈ Ω 2 j I : ∀ s ∈ I , sf ∈ 2 } S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 12 / 20
� � � � Example * e a b e e a b a a a b b b a b I = { a , b } 1 x 1 x 2 0 S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 13 / 20
Ore-like Conditions Lemma If for the monoid M and its right ideal I we have that ∃ s ∈ I ∀ t ∈ M , Ms ∩ Mst � = ∅ then 2 is injective with respect to all I -dense monomorphisms and ¯ 2 = 2 Lemma If for the monoid M and its right ideal I we have that 2 = ¯ 2 then ∀ t ∈ M , Mt ∩ MI � = ∅ S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 14 / 20
� � Stone Map in Set Lemma The functor U lt ( − ) : Boo → Set is left adjoint to the functor P ( − ) : Set → Boo . s : A → P ( U lt ( A )) is the unit of the adjunction at A . s ( a )( α ) = α ( a ). � 2 P ( U lt ( A )) × U lt ( A ) f s ( a ) × id U lt ( A ) A × U lt ( A ) f ( a , α ) = α ( a ) S. Sepahani, M. Mahmoudi (Shahid Beheshti University) Stone Representation Theorem for Boolean Algebras in the Topos of (Pre)Sheaves on a Monoid CT, 13 July 2018 15 / 20
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