Toposes Toposes and Their Logic

Home , Equaliser (mathematics), Subobject, Subobject classifier

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

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 = {(x, y) ∈ X × Y : f (x) = g(y)}

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 = {(x, y) ∈ X × Y : f (x) = g(y)}

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 ∼= h{(f (x), g(x)) : x ∈ X }i.

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 ∼= h{(f (x), g(x)) : x ∈ X }i.

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 ∈ A with maps (A −→ D(I ))I ∈I, 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 ∈ A with maps (A −→ D(I ))I ∈I, 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 ∈I such that the following commutes for all I , J ∈ I:

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. 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:

· ·

James Leslie Toposes A category has all limits if it has limits of shape I for all small I. A category has all finite limits if it has limits of shape I for all finite I. All limits is equivalent to all products and equalisers. All finite limits is equivalent to all binary products, equalisers and a terminal object. Dualising everything gives co-limits. Set, Top, Grp, [A op, Set] etc all have all limits and colimits. FinSet has finite limits and finite colimits.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Limits

A category has limits of shape I if for every diagram D of shape I in A , a limit of D exists.

James Leslie Toposes A category has all finite limits if it has limits of shape I for all finite I. All limits is equivalent to all products and equalisers. All finite limits is equivalent to all binary products, equalisers and a terminal object. Dualising everything gives co-limits. Set, Top, Grp, [A op, Set] etc all have all limits and colimits. FinSet has finite limits and finite colimits.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Limits

A category has limits of shape I if for every diagram D of shape I in A , a limit of D exists. A category has all limits if it has limits of shape I for all small I.

James Leslie Toposes All limits is equivalent to all products and equalisers. All finite limits is equivalent to all binary products, equalisers and a terminal object. Dualising everything gives co-limits. Set, Top, Grp, [A op, Set] etc all have all limits and colimits. FinSet has finite limits and finite colimits.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Limits

James Leslie Toposes All finite limits is equivalent to all binary products, equalisers and a terminal object. Dualising everything gives co-limits. Set, Top, Grp, [A op, Set] etc all have all limits and colimits. FinSet has finite limits and finite colimits.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Limits

James Leslie Toposes Dualising everything gives co-limits. Set, Top, Grp, [A op, Set] etc all have all limits and colimits. FinSet has ﬁnite limits and ﬁnite colimits.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Limits

A category has limits of shape I if for every diagram D of shape I in A , a limit of D exists. A category has all limits if it has limits of shape I for all small I. A category has all finite limits if it has limits of shape I for all finite I. All limits is equivalent to all products and equalisers. All finite limits is equivalent to all binary products, equalisers and a terminal object.

James Leslie Toposes Set, Top, Grp, [A op, Set] etc all have all limits and colimits. FinSet has ﬁnite limits and ﬁnite colimits.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Limits

James Leslie Toposes FinSet has ﬁnite limits and ﬁnite colimits.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Limits

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Limits

James Leslie Toposes A function f : X × Y → Z corresponds to a function Y fX : X → Z , where f (x) = f (x, −): Y → Z. This correspondence is a bijection and natural in the following sense: For all X ∈ Set, there are functors (−) × X , (−)X : Set → Set with

Set(Y × X , Z) =∼ Set(Y , Z X ), naturally in X , Y , Z ∈ Set. (−) × X a (−)X .

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Sets of maps

For sets X and Y , there is a set of maps from X to Y denoted Set(X , Y ), or Y X .

James Leslie Toposes This correspondence is a bijection and natural in the following sense: For all X ∈ Set, there are functors (−) × X , (−)X : Set → Set with

Set(Y × X , Z) =∼ Set(Y , Z X ), naturally in X , Y , Z ∈ Set. (−) × X a (−)X .

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Sets of maps

For sets X and Y , there is a set of maps from X to Y denoted Set(X , Y ), or Y X . A function f : X × Y → Z corresponds to a function Y fX : X → Z , where f (x) = f (x, −): Y → Z.

James Leslie Toposes For all X ∈ Set, there are functors (−) × X , (−)X : Set → Set with

Set(Y × X , Z) =∼ Set(Y , Z X ), naturally in X , Y , Z ∈ Set. (−) × X a (−)X .

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Sets of maps

For sets X and Y , there is a set of maps from X to Y denoted Set(X , Y ), or Y X . A function f : X × Y → Z corresponds to a function Y fX : X → Z , where f (x) = f (x, −): Y → Z. This correspondence is a bijection and natural in the following sense:

James Leslie Toposes (−) × X a (−)X .

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Sets of maps

Set(Y × X , Z) =∼ Set(Y , Z X ), naturally in X , Y , Z ∈ Set.

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Sets of maps

Set(Y × X , Z) =∼ Set(Y , Z X ), naturally in X , Y , Z ∈ Set. (−) × X a (−)X .

James Leslie Toposes Let X ∈ A . A has an exponential for X if (−) × X : A → A has a right adjoint, (−)X . A has exponentials if A has exponentials for all X ∈ A .

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Exponentials

Let A be a category with ﬁnite products.

James Leslie Toposes A has exponentials if A has exponentials for all X ∈ A .

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Exponentials

Let A be a category with ﬁnite products. Let X ∈ A . A has an exponential for X if (−) × X : A → A has a right adjoint, (−)X .

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Exponentials

Let A be a category with ﬁnite products. Let X ∈ A . A has an exponential for X if (−) × X : A → A has a right adjoint, (−)X . A has exponentials if A has exponentials for all X ∈ A .

James Leslie Toposes : Z X × X → Z is the counit of the adjunction. In Set it is an evaluation map.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Universal property of exponentials

A category having exponentials is equivalent to the following universal property:

Y Y × X f f f ×1X X X Z Z × X Z

James Leslie Toposes In Set it is an evaluation map.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Universal property of exponentials

A category having exponentials is equivalent to the following universal property:

Y Y × X f f f ×1X X X Z Z × X Z

: Z X × X → Z is the counit of the adjunction.

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Universal property of exponentials

A category having exponentials is equivalent to the following universal property:

Y Y × X f f f ×1X X X Z Z × X Z

: Z X × X → Z is the counit of the adjunction. In Set it is an evaluation map.

James Leslie Toposes A subset U ⊆ X is ‘the same as’ the inclusion map U ,−→ X . What is an inclusion between algebraic structures? 2 We can embed R into R (as vector spaces) with the injective linear map f (x) = (x, 0). The map g(x) = (2x, 0) is ‘the same’ embedding.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

An element x ∈ X can be thought of as a map x : 1 → X , which is the unique map selecting x ∈ X .

James Leslie Toposes What is an inclusion between algebraic structures? 2 We can embed R into R (as vector spaces) with the injective linear map f (x) = (x, 0). The map g(x) = (2x, 0) is ‘the same’ embedding.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

An element x ∈ X can be thought of as a map x : 1 → X , which is the unique map selecting x ∈ X . A subset U ⊆ X is ‘the same as’ the inclusion map U ,−→ X .

James Leslie Toposes 2 We can embed R into R (as vector spaces) with the injective linear map f (x) = (x, 0). The map g(x) = (2x, 0) is ‘the same’ embedding.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

James Leslie Toposes The map g(x) = (2x, 0) is ‘the same’ embedding.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

An element x ∈ X can be thought of as a map x : 1 → X , which is the unique map selecting x ∈ X . A subset U ⊆ X is ‘the same as’ the inclusion map U ,−→ X . What is an inclusion between algebraic structures? 2 We can embed R into R (as vector spaces) with the injective linear map f (x) = (x, 0).

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

James Leslie Toposes Two ‘inclusion’ maps i, j are isomorphic if there is an invertible map f such that:

A i E

f ∼ j B

Subobjects of groups, rings, ﬁelds and modules are exactly subgroups, subrings, subﬁelds and submodules. What about subobjects of topological spaces?

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

Let A be a category and E ∈ A . A subobject of E is an isomorphism class of monics into E.

James Leslie Toposes Subobjects of groups, rings, ﬁelds and modules are exactly subgroups, subrings, subﬁelds and submodules. What about subobjects of topological spaces?

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

Let A be a category and E ∈ A . A subobject of E is an isomorphism class of monics into E. Two ‘inclusion’ maps i, j are isomorphic if there is an invertible map f such that:

A i E

f ∼ j B

James Leslie Toposes What about subobjects of topological spaces?

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

Let A be a category and E ∈ A . A subobject of E is an isomorphism class of monics into E. Two ‘inclusion’ maps i, j are isomorphic if there is an invertible map f such that:

A i E

f ∼ j B

Subobjects of groups, rings, ﬁelds and modules are exactly subgroups, subrings, subﬁelds and submodules.

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobjects

Let A be a category and E ∈ A . A subobject of E is an isomorphism class of monics into E. Two ‘inclusion’ maps i, j are isomorphic if there is an invertible map f such that:

A i E

f ∼ j B

Subobjects of groups, rings, ﬁelds and modules are exactly subgroups, subrings, subﬁelds and submodules. What about subobjects of topological spaces?

James Leslie Toposes P(X ) =∼ Sub(X ) =∼ Set(X , 2). This isomorphism lets us classify subobjects with maps into 2.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Characteristic maps

A subset i : U ,−→ X corresponds with map χi : X → 2, where ( 1 if x ∈ U χi (x) = 0 otherwise.

This correspondence is bijective.

James Leslie Toposes This isomorphism lets us classify subobjects with maps into 2.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Characteristic maps

A subset i : U ,−→ X corresponds with map χi : X → 2, where ( 1 if x ∈ U χi (x) = 0 otherwise.

This correspondence is bijective. P(X ) =∼ Sub(X ) =∼ Set(X , 2).

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Characteristic maps

A subset i : U ,−→ X corresponds with map χi : X → 2, where ( 1 if x ∈ U χi (x) = 0 otherwise.

This correspondence is bijective. P(X ) =∼ Sub(X ) =∼ Set(X , 2). This isomorphism lets us classify subobjects with maps into 2.

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobject classiﬁers

Let A be a category with terminal object 1. A subobject classiﬁer is a map True : 1 → Ω such that for any subobject m : U X , there exists a unique map χm : X → Ω such that the following is a pullback diagram:

U m X

! χm 1 Ω True

James Leslie Toposes For a fixed group G, True : 1 → 2 is a subobject classifier for G − Set. (2 has trivial group action) For a small category A ,[A op, Set] has a subobject classifier. A sieve on A ∈ A is a set of arrows with codomain A, that is closed under right composition. Let Ω : A op → Set be the functor sending A to the set of sieves on A.

(TrueA : 1(A) → Ω(A))A∈A is a natural transformation that selects the maximal sieve on A. G − Set is a special case of [A op, Set], where A is G regarded as a category.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobject classiﬁers

True : 1 → 2 in Set is a subobject classiﬁer.

James Leslie Toposes For a small category A ,[A op, Set] has a subobject classiﬁer. A sieve on A ∈ A is a set of arrows with codomain A, that is closed under right composition. Let Ω : A op → Set be the functor sending A to the set of sieves on A.

(TrueA : 1(A) → Ω(A))A∈A is a natural transformation that selects the maximal sieve on A. G − Set is a special case of [A op, Set], where A is G regarded as a category.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobject classiﬁers

True : 1 → 2 in Set is a subobject classifier. For a fixed group G, True : 1 → 2 is a subobject classifier for G − Set. (2 has trivial group action)

James Leslie Toposes A sieve on A ∈ A is a set of arrows with codomain A, that is closed under right composition. Let Ω : A op → Set be the functor sending A to the set of sieves on A.

(TrueA : 1(A) → Ω(A))A∈A is a natural transformation that selects the maximal sieve on A. G − Set is a special case of [A op, Set], where A is G regarded as a category.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobject classiﬁers

James Leslie Toposes Let Ω : A op → Set be the functor sending A to the set of sieves on A.

(TrueA : 1(A) → Ω(A))A∈A is a natural transformation that selects the maximal sieve on A. G − Set is a special case of [A op, Set], where A is G regarded as a category.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobject classiﬁers

True : 1 → 2 in Set is a subobject classifier. For a fixed group G, True : 1 → 2 is a subobject classifier for G − Set. (2 has trivial group action) For a small category A ,[A op, Set] has a subobject classifier. A sieve on A ∈ A is a set of arrows with codomain A, that is closed under right composition.

James Leslie Toposes (TrueA : 1(A) → Ω(A))A∈A is a natural transformation that selects the maximal sieve on A. G − Set is a special case of [A op, Set], where A is G regarded as a category.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobject classiﬁers

James Leslie Toposes G − Set is a special case of [A op, Set], where A is G regarded as a category.

Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobject classiﬁers

(TrueA : 1(A) → Ω(A))A∈A is a natural transformation that selects the maximal sieve on A.

James Leslie Toposes Limits and colimits Building up to toposes Exponenetiation Toposes and their logic Subobject classiﬁers Subobject classiﬁers

(TrueA : 1(A) → Ω(A))A∈A is a natural transformation that selects the maximal sieve on A. G − Set is a special case of [A op, Set], where A is G regarded as a category.

James Leslie Toposes 1 Finite limits and colimits, 2 Exponentiation, 3 A subobject classiﬁer. Set is the archetypal topos. Set2, Setn, SetN, [A op, Set], FinSet are all toposes. For any topos E and A in E , E /A is a topos.

Building up to toposes Toposes Toposes and their logic Heyting algebras Toposes

A topos E is a category with:

James Leslie Toposes 2 Exponentiation, 3 A subobject classiﬁer. Set is the archetypal topos. Set2, Setn, SetN, [A op, Set], FinSet are all toposes. For any topos E and A in E , E /A is a topos.

Building up to toposes Toposes Toposes and their logic Heyting algebras Toposes

A topos E is a category with: 1 Finite limits and colimits,

James Leslie Toposes 3 A subobject classiﬁer. Set is the archetypal topos. Set2, Setn, SetN, [A op, Set], FinSet are all toposes. For any topos E and A in E , E /A is a topos.

Building up to toposes Toposes Toposes and their logic Heyting algebras Toposes

A topos E is a category with: 1 Finite limits and colimits, 2 Exponentiation,

James Leslie Toposes Set is the archetypal topos. Set2, Setn, SetN, [A op, Set], FinSet are all toposes. For any topos E and A in E , E /A is a topos.

Building up to toposes Toposes Toposes and their logic Heyting algebras Toposes

A topos E is a category with: 1 Finite limits and colimits, 2 Exponentiation, 3 A subobject classiﬁer.

James Leslie Toposes Set2, Setn, SetN, [A op, Set], FinSet are all toposes. For any topos E and A in E , E /A is a topos.

Building up to toposes Toposes Toposes and their logic Heyting algebras Toposes

A topos E is a category with: 1 Finite limits and colimits, 2 Exponentiation, 3 A subobject classiﬁer. Set is the archetypal topos.

James Leslie Toposes For any topos E and A in E , E /A is a topos.

Building up to toposes Toposes Toposes and their logic Heyting algebras Toposes

A topos E is a category with: 1 Finite limits and colimits, 2 Exponentiation, 3 A subobject classiﬁer. Set is the archetypal topos. Set2, Setn, SetN, [A op, Set], FinSet are all toposes.

James Leslie Toposes Building up to toposes Toposes Toposes and their logic Heyting algebras Toposes

A topos E is a category with: 1 Finite limits and colimits, 2 Exponentiation, 3 A subobject classiﬁer. Set is the archetypal topos. Set2, Setn, SetN, [A op, Set], FinSet are all toposes. For any topos E and A in E , E /A is a topos.

James Leslie Toposes In Set they form a Boolean algebra. In an arbitrary topos, they form a Heyting algebra. Their logic is intuitionistic - proof by contradiction isn’t always valid.

Building up to toposes Toposes Toposes and their logic Heyting algebras Lattice of Subobjects

In every category, subobjects form a partially ordered set.

James Leslie Toposes In an arbitrary topos, they form a Heyting algebra. Their logic is intuitionistic - proof by contradiction isn’t always valid.

Building up to toposes Toposes Toposes and their logic Heyting algebras Lattice of Subobjects

In every category, subobjects form a partially ordered set. In Set they form a Boolean algebra.

James Leslie Toposes Their logic is intuitionistic - proof by contradiction isn’t always valid.

Building up to toposes Toposes Toposes and their logic Heyting algebras Lattice of Subobjects

In every category, subobjects form a partially ordered set. In Set they form a Boolean algebra. In an arbitrary topos, they form a Heyting algebra.

James Leslie Toposes Building up to toposes Toposes Toposes and their logic Heyting algebras Lattice of Subobjects

In every category, subobjects form a partially ordered set. In Set they form a Boolean algebra. In an arbitrary topos, they form a Heyting algebra. Their logic is intuitionistic - proof by contradiction isn’t always valid.

James Leslie Toposes Building up to toposes Toposes Toposes and their logic Heyting algebras

The End

James Leslie Toposes