The centralizer of a topos Joint work with Pieter Hofstra - PowerPoint PPT Presentation

The centralizer of a topos Joint work with Pieter Hofstra

� Symmetry: X t August 16, 2016 2 / 28

� � � Symmetry: X t Conjugation: f � X A f − 1 tf t August 16, 2016 2 / 28

� � � Symmetry: X t Conjugation: f � X A f − 1 tf t However, the morphism f may not be invertible, and typically we cannot expect it to be. August 16, 2016 2 / 28

� � � � � � We do not wish to admit any restrictions on f , but rather provide t with more structure. t g � B B m m t f � A g g A f f t = t X � X X ∀ m ( fm = g ) : mt g = t f m August 16, 2016 3 / 28

� � � � � � We do not wish to admit any restrictions on f , but rather provide t with more structure. t g � B B m m t f � A g g A f f t = t X � X X ∀ m ( fm = g ) : mt g = t f m t is an automorphism (endomorphism) of the functor f �→ A � X X / X August 16, 2016 3 / 28

� � � � � � We do not wish to admit any restrictions on f , but rather provide t with more structure. t g � B B m m t f � A g g A f f t = t X � X X ∀ m ( fm = g ) : mt g = t f m t is an automorphism (endomorphism) of the functor f �→ A � X X / X f � X by whiskering. Problem solved: move t along A m �→ fm � X / X � X X / A t August 16, 2016 3 / 28

� � Addendum: � X / X X / X t t X t is a central automorphism of X iff t X = 1 X . August 16, 2016 4 / 28

� � � � Addendum: � X / X X / X t t X t is a central automorphism of X iff t X = 1 X . Category: centralizer of X . � Y Y s f f � X X t August 16, 2016 4 / 28

Isotropy theory of toposes [2] Symmetry A symmetry of an object X of a topos E is an automorphism of the geometric morphism Σ X ⊣ X ∗ ⊣ Π X : E / X � E . August 16, 2016 5 / 28

Its an automorphism of the functor X ∗ : E � E / X � X ; t : X ∗ � X ∗ X ∗ ( E ) = E × X for every object E , we have an ‘action’ � E , t E : E × X whose pairing � E × X E × X with the projection to X is an isomorphism (as if X were a group). August 16, 2016 6 / 28

Its an automorphism of the functor X ∗ : E � E / X � X ; t : X ∗ � X ∗ X ∗ ( E ) = E × X for every object E , we have an ‘action’ � E , t E : E × X whose pairing � E × X E × X with the projection to X is an isomorphism (as if X were a group). Equivalently, its an automorphism of the left adjoint Σ X f � X ) = Y ; t : Σ X � Σ X Σ X ( Y t f � Y of automorphisms of E . This amounts to a compatible family Y August 16, 2016 6 / 28

The isotropy group Z of a topos E Z is a group internal to E that classifies its symmetries in the sense that � Z of E correspond naturally to automorphisms of morphisms X � E . E / X August 16, 2016 7 / 28

� � Z has a universal action θ X in the every object X of E : t f � Y Y θ X ( f , t ) = ft f ( f , t ) f θ X � X X × Z August 16, 2016 8 / 28

� � � � � � Z has a universal action θ X in the every object X of E : t f � Y Y θ X ( f , t ) = ft f ( f , t ) f θ X � X X × Z t � Z , Y f � X or E , we have On the other hand, for any X E × X Y t f t E (1 Y , tf ) E × t θ Y � E � Y E × Z Y × Z θ E t E ( e , x ) = θ E ( e , t ( x )) = et ( x ) ; t f ( y ) = θ Y ( y , t ( f ( y ))) = yt ( f ( y )) August 16, 2016 8 / 28

� � � � � � Z has a universal action θ X in the every object X of E : t f � Y Y θ X ( f , t ) = ft f ( f , t ) f θ X � X X × Z t � Z , Y f � X or E , we have On the other hand, for any X E × X Y t f t E (1 Y , tf ) E × t θ Y � E � Y E × Z Y × Z θ E t E ( e , x ) = θ E ( e , t ( x )) = et ( x ) ; t f ( y ) = θ Y ( y , t ( f ( y ))) = yt ( f ( y )) � Z is conjugation: θ Z ( w , z ) = z − 1 wz . θ Z : Z × Z August 16, 2016 8 / 28

� � � � � � Z has a universal action θ X in the every object X of E : t f � Y Y θ X ( f , t ) = ft f ( f , t ) f θ X � X X × Z t � Z , Y f � X or E , we have On the other hand, for any X E × X Y t f t E (1 Y , tf ) E × t θ Y � E � Y E × Z Y × Z θ E t E ( e , x ) = θ E ( e , t ( x )) = et ( x ) ; t f ( y ) = θ Y ( y , t ( f ( y ))) = yt ( f ( y )) � Z is conjugation: θ Z ( w , z ) = z − 1 wz . θ Z : Z × Z All maps of E are Z -equivariant. August 16, 2016 8 / 28

Another perspective on symmetry: crossed sheaf Definition t � Z of E , but regarded as an A crossed sheaf on E is a morphism X object of E / Z . August 16, 2016 9 / 28

Another perspective on symmetry: crossed sheaf Definition t � Z of E , but regarded as an A crossed sheaf on E is a morphism X object of E / Z . E / Z is the topos of crossed sheaves on E . August 16, 2016 9 / 28

� � � Isotropy group of E / Z What is it? Z ( Z ) � Z × Z ζ Z Z ( Z ) = { ( z , w ) | zw = wz } , ζ ( z , w ) = z . August 16, 2016 10 / 28

� � � � � � Isotropy group of E / Z What is it? Z ( Z ) � Z × Z ζ Z Z ( Z ) = { ( z , w ) | zw = wz } , ζ ( z , w ) = z . � ζ Generic element: δ : 1 Z z �→ ( z , z ) Z Z ( Z ) 1 Z ζ Z (Not to be confused with the unit element u ( z ) = ( z , u ) of ζ .) August 16, 2016 10 / 28

� � � Twist automorphism α : central automorphism of E / Z corresponding to δ What is α ? t � Z . Let α t ( x ) = θ X ( x , t ( x )) = xt ( x ) . Crossed sheaf X α t X X t t Z t ( α t ( x )) = t ( xt ( x )) = t ( x ) − 1 t ( x ) t ( x ) = t ( x ) August 16, 2016 11 / 28

� Quotient of a crossed sheaf Definition t � Z be a crossed sheaf on E . Let X Coequifier ψ � E t E / X � E ⇓ t ⇓ 1 t E � E is the E t = { E | ∀ ( e , x ) , t E ( e , x ) = et ( x ) = e , i . e ., E × X projection } August 16, 2016 12 / 28

� � � E t The geometric morphism ψ : E E t is a topos; ψ is connected atomic: ψ ∗ is inclusion; ψ ! is the orbit space of the action = coequalizer et ( x ) unit � ψ ! E ; E × X � E e ψ ∗ E = { e | ∀ x ∈ X , et ( x ) = e } = equalizer � E X (transposed maps) ψ ∗ E counit � E August 16, 2016 13 / 28

� Two special quotients � Z 1. Quotient of the identity map Z ψ � E θ E / Z � E ⇓ θ ⇓ 1 The topos E θ consists of all isotropically trivial objects: those X for which the universal action θ X is trivial. ψ ∗ ( X ) = X . ψ ! ( X ) = orbit space of universal action of θ X . August 16, 2016 14 / 28

� � ζ 2. Quotient of the generic element δ : 1 Z � Z ( E ) E / Z � E / Z ⇓ α ⇓ 1 t � Z , we have α t ( x ) = θ X ( x , t ( x )) = xt ( x ) . For any X August 16, 2016 15 / 28

� � ζ 2. Quotient of the generic element δ : 1 Z � Z ( E ) E / Z � E / Z ⇓ α ⇓ 1 t � Z , we have α t ( x ) = θ X ( x , t ( x )) = xt ( x ) . For any X Definition t � Z is unital if ∀ x ∈ X , xt ( x ) = x . A crossed sheaf X August 16, 2016 15 / 28

� � ζ 2. Quotient of the generic element δ : 1 Z � Z ( E ) E / Z � E / Z ⇓ α ⇓ 1 t � Z , we have α t ( x ) = θ X ( x , t ( x )) = xt ( x ) . For any X Definition t � Z is unital if ∀ x ∈ X , xt ( x ) = x . A crossed sheaf X Proposition The coequifier topos Z ( E ) equals the full subcategory of E / Z on the unital crossed sheaves. August 16, 2016 15 / 28

� � � � Pushout diagram ϕ � Z ( E ) E / Z � E / Z ⇓ α ⇓ 1 conn . atomic atomic ψ � E θ E / Z � E ⇓ θ ⇓ 1 conn . atomic The right-hand square is a pushout. August 16, 2016 16 / 28

� � Equivalence of unital crossed sheaves with central automorphisms Interpret the equivalence as a lifting property t � Z is unital, then there is a central automorphism If a crossed sheaf X α ( t ) of X such that the ‘whiskering’ diagram commutes. � E / X α ( t ) E / X t t � E / Z α E / Z August 16, 2016 17 / 28

� � Equivalence of unital crossed sheaves with central automorphisms Interpret the equivalence as a lifting property t � Z is unital, then there is a central automorphism If a crossed sheaf X α ( t ) of X such that the ‘whiskering’ diagram commutes. � E / X α ( t ) E / X t t � E / Z α E / Z Proof s � X , define α ( t ) s = α ts . Verify s α ( t ) s = s : For any Y s ( α ( t ) s ( y )) = s ( α ( ts ( y ))) = s ( yts ( y )) = s ( y ) ts ( y ) = s ( y ) . The second last equality holds because s is Z -equivariant. The last equality holds because t is unital. August 16, 2016 17 / 28

� � t � Z �→ ( X , α ( t )) is an equivalence X t � Z such that For every central automorphism β of X , ∃ ! unital X β = α ( t ) . � E / X E / X β = α ( t ) t t � E / Z E / Z α August 16, 2016 18 / 28

� � t � Z �→ ( X , α ( t )) is an equivalence X t � Z such that For every central automorphism β of X , ∃ ! unital X β = α ( t ) . � E / X E / X β = α ( t ) t t � E / Z E / Z α Proof t � Z . It � E corresponds to a morphism X Whiskering β with E / X follows that t is unital. August 16, 2016 18 / 28

� � � � Tensor product on crossed sheaves X × Y ( x , y ) X Y ❴ ⊗ = s s ⊗ t t s ( x ) t ( y ) Z Z Z u � Z . The identity object I is the unit 1 August 16, 2016 19 / 28