Diads and their Application to Topoi Toby Kenney Mathematics, - PowerPoint PPT Presentation

Introduction Diads & Dialgebras Application to Topoi Diads and their Application to Topoi Toby Kenney Mathematics, Dalhousie University, Halifax, Canada CT2008 26-06-2008 Toby Kenney Diads and their Application to Topoi

Introduction Diads & Dialgebras Application to Topoi Theorems from Topos Theory Toby Kenney Diads and their Application to Topoi

Introduction Diads & Dialgebras Application to Topoi Theorems from Topos Theory Theorem The category of coalgebras for a finite-limit preserving comonad on a topos is again a topos. Toby Kenney Diads and their Application to Topoi

Introduction Diads & Dialgebras Application to Topoi Theorems from Topos Theory Theorem The category of coalgebras for a finite-limit preserving comonad on a topos is again a topos. Theorem The category of algebras for a finite-limit preserving idempotent monad on a topos is again a topos. Toby Kenney Diads and their Application to Topoi

Introduction Diads & Dialgebras Application to Topoi Theorems from Topos Theory Theorem The category of coalgebras for a finite-limit preserving comonad on a topos is again a topos. Theorem The category of algebras for a finite-limit preserving idempotent monad on a topos is again a topos. Theorem The full subcategory of fixed points of a finite-limit preserving idempotent endofunctor on a topos is again a topos. Toby Kenney Diads and their Application to Topoi

� � � � � � � � � � � Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Diads � C , A distributive diad on a category C is a functor T : C � T 2 and equipped with natural transformations α : T � T such that the following diagrams commute: β : T 2 β � T α T 2 T 2 T � � � α � α T α � � 1 � � � T 3 T T 2 α T β T α T T 3 T 2 T 2 T 3 T β β β T β � T � T 2 T 2 T α β Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples For a comonad ( T , ν, ǫ ) , ( T , ν, ǫ T ) is a distributive diad. Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples For a comonad ( T , ν, ǫ ) , ( T , ν, ǫ T ) is a distributive diad. For a monad ( T , η, µ ) , ( T , η T , µ ) is a distributive diad if and only if the monad is idempotent. Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples For a comonad ( T , ν, ǫ ) , ( T , ν, ǫ T ) is a distributive diad. For a monad ( T , η, µ ) , ( T , η T , µ ) is a distributive diad if and only if the monad is idempotent. Any idempotent functor is a distributive diad. Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples For a comonad ( T , ν, ǫ ) , ( T , ν, ǫ T ) is a distributive diad. For a monad ( T , η, µ ) , ( T , η T , µ ) is a distributive diad if and only if the monad is idempotent. Any idempotent functor is a distributive diad. For a monad ( T , η, µ ) , ( T , T η, µ ) is a distributive diad. Toby Kenney Diads and their Application to Topoi

� � � � � � � � � � � Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Dialgebras A distributive dialgebra for the distributive diad ( T , α, β ) is an φ � TX object X with morphisms X such that the following θ diagrams commute: φ θ � X TX X TX � � � φ � φ T φ � � 1 X � � α X � T 2 X X TX β X α X � T 2 X T 2 X TX TX T θ θ θ T θ � X � TX TX X θ φ Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples of Dialgebras For a comonad ( T , ν, ǫ ) , distributive dialgebras for ( T , ν, ǫ T ) are coalgebras for ( T , ν, ǫ ) . Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples of Dialgebras For a comonad ( T , ν, ǫ ) , distributive dialgebras for ( T , ν, ǫ T ) are coalgebras for ( T , ν, ǫ ) . For an idempotent diad, the dialgebras are exactly fixed points of T up to isomorphism. Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples of Dialgebras For a comonad ( T , ν, ǫ ) , distributive dialgebras for ( T , ν, ǫ T ) are coalgebras for ( T , ν, ǫ ) . For an idempotent diad, the dialgebras are exactly fixed points of T up to isomorphism. For a monad ( T , η, µ ) , where T is faithful, distributive dialgebras for ( T , T η, µ ) are coalgebras for the comonad induced on the category of algebras for ( T , η, µ ) . Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples of Dialgebras For a comonad ( T , ν, ǫ ) , distributive dialgebras for ( T , ν, ǫ T ) are coalgebras for ( T , ν, ǫ ) . For an idempotent diad, the dialgebras are exactly fixed points of T up to isomorphism. For a monad ( T , η, µ ) , where T is faithful, distributive dialgebras for ( T , T η, µ ) are coalgebras for the comonad induced on the category of algebras for ( T , η, µ ) . For any distributive diad ( T , α, β ) and any object X , there is a free dialgebra ( TX , α X , β X ) . Toby Kenney Diads and their Application to Topoi

� � � � � Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Dialgebra Homomorphisms A dialgebra homomorphism from ( X , φ, θ ) to ( Y , π, ρ ) is the f � Y such that obvious thing – namely a morphism X Tf Tf � TY TX TY TX and ρ φ π θ � Y � Y X X f f commute. Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples of Dialgebra Homomorphisms For the diad ( T , ν, ǫ T ) from a comonad, dialgebra homomorphisms are exactly coalgebra homomorphisms. Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples of Dialgebra Homomorphisms For the diad ( T , ν, ǫ T ) from a comonad, dialgebra homomorphisms are exactly coalgebra homomorphisms. For the diad ( T , η T , µ ) from a monad, dialgebra homomorphisms are exactly algebra homomorphisms. Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples of Dialgebra Homomorphisms For the diad ( T , ν, ǫ T ) from a comonad, dialgebra homomorphisms are exactly coalgebra homomorphisms. For the diad ( T , η T , µ ) from a monad, dialgebra homomorphisms are exactly algebra homomorphisms. f � Y , Tf For any objects X and Y , and any morphism X is a dialgebra homomorphism between the free dialgebras on X and Y . Toby Kenney Diads and their Application to Topoi

Introduction Diads Diads & Dialgebras Dialgebras Application to Topoi Examples of Dialgebra Homomorphisms For the diad ( T , ν, ǫ T ) from a comonad, dialgebra homomorphisms are exactly coalgebra homomorphisms. For the diad ( T , η T , µ ) from a monad, dialgebra homomorphisms are exactly algebra homomorphisms. f � Y , Tf For any objects X and Y , and any morphism X is a dialgebra homomorphism between the free dialgebras on X and Y . For any distributive dialgebra ( X , φ, θ ) , φ is a dialgebra homomorphism from ( X , φ, θ ) to the free dialgebra ( TX , α X , β X ) . Toby Kenney Diads and their Application to Topoi

Introduction Limits Diads & Dialgebras Exponentials Application to Topoi Subobject Classifier Main Theorem Theorem The category of distributive dialgebras for a finite-limit preserving distributive diad on a topos is again a topos. Toby Kenney Diads and their Application to Topoi

Introduction Limits Diads & Dialgebras Exponentials Application to Topoi Subobject Classifier Limits The terminal object is just the dialgebra ( 1 , 1 , 1 ) . Toby Kenney Diads and their Application to Topoi

Introduction Limits Diads & Dialgebras Exponentials Application to Topoi Subobject Classifier Limits The terminal object is just the dialgebra ( 1 , 1 , 1 ) . The product of ( X , φ, θ ) and ( Y , π, ρ ) is ( X × Y , φ × π, θ × ρ ) . Toby Kenney Diads and their Application to Topoi

Introduction Limits Diads & Dialgebras Exponentials Application to Topoi Subobject Classifier Limits The terminal object is just the dialgebra ( 1 , 1 , 1 ) . The product of ( X , φ, θ ) and ( Y , π, ρ ) is ( X × Y , φ × π, θ × ρ ) . The equaliser of dialgebra maps f and g can be given a distributive dialgebra structure using the universal property of equalisers and the fact that T preserves equalisers: Toby Kenney Diads and their Application to Topoi

� � Introduction Limits Diads & Dialgebras Exponentials Application to Topoi Subobject Classifier Equalisers Tf Te � TX � TY TE Tg f � Y � X E e g Toby Kenney Diads and their Application to Topoi