Building up to toposes Toposes and their logic Toposes James Leslie University of Edinurgh [email protected] April 2018 James Leslie Toposes Building up to toposes Toposes and their logic Overview 1 Building up to toposes Limits and colimits Exponenetiation Subobject classifiers 2 Toposes and their logic Toposes Heyting algebras James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Products Z f f1 f2 X × Y π1 π2 X Y James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Coproducts i i X 1 X + Y 2 Y f f1 f2 Z James Leslie Toposes e : E ! X is an equaliser if for any commuting diagram f Z h X Y g there is a unique h such that triangle commutes: f E e X Y g h h Z Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers Suppose we have the following commutative diagram: f E e X Y g James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers Suppose we have the following commutative diagram: f E e X Y g e : E ! X is an equaliser if for any commuting diagram f Z h X Y g there is a unique h such that triangle commutes: f E e X Y g h h Z James Leslie Toposes The following is an equaliser diagram: f ker G i G H 0 Given a k such that following commutes: f K k G H ker G i G 0 k k K k : K ! ker(G), k(g) = k(g). Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers - an example Let G; H be groups, f : G ! H a group homomorphism and 0 : G ! H the constant eH map. James Leslie Toposes Given a k such that following commutes: f K k G H ker G i G 0 k k K k : K ! ker(G), k(g) = k(g). Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers - an example Let G; H be groups, f : G ! H a group homomorphism and 0 : G ! H the constant eH map. The following is an equaliser diagram: f ker G i G H 0 James Leslie Toposes k : K ! ker(G), k(g) = k(g). Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers - an example Let G; H be groups, f : G ! H a group homomorphism and 0 : G ! H the constant eH map. The following is an equaliser diagram: f ker G i G H 0 Given a k such that following commutes: f K k G H ker G i G 0 k k K James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Equalisers - an example Let G; H be groups, f : G ! H a group homomorphism and 0 : G ! H the constant eH map. The following is an equaliser diagram: f ker G i G H 0 Given a k such that following commutes: f K k G H ker G i G 0 k k K k : K ! ker(G), k(g) = k(g). James Leslie Toposes q : Y ! Q is a coequaliser if for any commuting diagram f X Y h Z g there is a unique h such that the triangle commutes: f q X Y Q g h h Z Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Coequalisers Suppose we have the following commutative diagram: f q X Y Q g James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Coequalisers Suppose we have the following commutative diagram: f q X Y Q g q : Y ! Q is a coequaliser if for any commuting diagram f X Y h Z g there is a unique h such that the triangle commutes: f q X Y Q g h h Z James Leslie Toposes Given a h such that the following commutes: 5z h q Z Z H Z Z=5Z 0 h h H h(g) = h(g). Look familiar? Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Coequalisers - an example Let Z be a group under addition. The following is a coequaliser diagram: 5z q Z Z Z=5Z 0 James Leslie Toposes h(g) = h(g). Look familiar? Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Coequalisers - an example Let Z be a group under addition. The following is a coequaliser diagram: 5z q Z Z Z=5Z 0 Given a h such that the following commutes: 5z h q Z Z H Z Z=5Z 0 h h H James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Coequalisers - an example Let Z be a group under addition. The following is a coequaliser diagram: 5z q Z Z Z=5Z 0 Given a h such that the following commutes: 5z h q Z Z H Z Z=5Z 0 h h H h(g) = h(g). Look familiar? James Leslie Toposes As a diagram: π P 2 Y π1 g X Z f Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Pullbacks Take functions f : X ! Z, g : Y ! Z. The pullback of f and g the following set: P = f(x; y) 2 X × Y : f (x) = g(y)g James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Pullbacks Take functions f : X ! Z, g : Y ! Z. The pullback of f and g the following set: P = f(x; y) 2 X × Y : f (x) = g(y)g As a diagram: π P 2 Y π1 g X Z f James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Pullbacks The universal property: S h2 h π P 2 Y h1 π1 g X Z f James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Pullbacks - Intersection Take two subsets i : 2Z ,−! Z and j : 3Z ,−! Z. Their pullback is their intersection: i ∗ 6Z 3Z j∗ j 2 Z i Z James Leslie Toposes Q is (Y + Z)= ∼, where ∼ is generated by ∼= hf(f (x); g(x)) : x 2 X gi. Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Pushouts Given two functions with same domain f : X ! Y , g : X ! Z, we can from their pushout square: X f Z X f Z g g q2 Y Y Q q1 James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Pushouts Given two functions with same domain f : X ! Y , g : X ! Z, we can from their pushout square: X f Z X f Z g g q2 Y Y Q q1 Q is (Y + Z)= ∼, where ∼ is generated by ∼= hf(f (x); g(x)) : x 2 X gi. James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Pushouts The universal property: X f Z g q2 h2 Y Q q1 h h1 S James Leslie Toposes fI A cone on D is an object A 2 A with maps (A −! D(I ))I 2I, such that the following commutes for all u : I ! J in I. A fJ fI D(J) D(I ) D(u) Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Limits Let I be a small category. A diagram in A is a functor D : I ! A . James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Limits Let I be a small category. A diagram in A is a functor D : I ! A . fI A cone on D is an object A 2 A with maps (A −! D(I ))I 2I, such that the following commutes for all u : I ! J in I. A fJ fI D(J) D(I ) D(u) James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Limits πI A limit of D is a universal cone (L −! D(I ))I 2I such that the following commutes for all I ; J 2 I: A f fJ fI L πJ πI D(J) D(I ) D(u) James Leslie Toposes If I consists the following objects and maps, a limit of D : I ! A is an equaliser: · · If I consists the following objects and maps, a limit of D : I ! A is an pullback: · · · Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Limit examples If I is a discrete, 2 object category, a limit of D : I ! A is a product. James Leslie Toposes If I consists the following objects and maps, a limit of D : I ! A is an pullback: · · · Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Limit examples If I is a discrete, 2 object category, a limit of D : I ! A is a product. If I consists the following objects and maps, a limit of D : I ! A is an equaliser: · · James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classifiers Limit examples If I is a discrete, 2 object category, a limit of D : I ! A is a product.