Topos Theory and the Connections between Category and Set ...people.brandeis.edu/~nstambau/GSS/CategoryVsSetTheory.pdf · Topos Theory and the Connections between Category and Set

Topos Theoryand the

Connectionsbetween

Category andSet Theory

MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples

Relations toSet Theory

ToposDefinition andExamples

Motivationand History ofTopos theory

Acknowledgements

Topos Theory and the Connections betweenCategory and Set Theory

Matthew Graham

March 13, 2008

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

1 Why Category Theory?

2 Definition

3 Basic Examples

4 Relations to Set Theory

5 Topos Definition and Examples

6 Motivation and History of Topos theory

7 Acknowledgements

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Why Category Theory?

• “The fundamental idea of representing a function by anarrow first appeared in topology around 1940” (Maclane)

• Allows the abstraction of common structure. For example,Groups, sets, topological spaces all have associativityintrinsic to them.

• Unlike Set theory specifies the codomain of a function.This allows it to distinguish the difference between anidentity map and an inclusion.

• Leads to a different foundation for mathematics via ToposTheory, which allows one to explicitly construct differenttypes of logic.

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

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Acknowledgements

• In particular there exist “Intuitionist” logics that can beconstructed that have all of the properties of classical logicexcept for the excluded middle (The statement α ∨ ¬α isnot necessarily true).

• Think about the quantum mechanical two slit experimentand try to evaluate the truth value of the statement, “ Theelectron went through the top slit.”

• In these logics, proofs by contradiction do not hold: onemust prove everything constructively.

• Some physicists are using topos theory to construct alanguage to formulate physical theories with the hope of(among other things) identifying possible unificationtheories.

• Apparently they can construct theories where the logicalstructure will demarcate which questions can be answeredtheoretically, which can be tested physically, and which areunanswerable!

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

If Category Theory generalizes set theory then all of the familiarobjects and entities in set theory must be contained in Categorytheory somewhere. Which leads to the following questions:

1 How are Cartesian product, disjoint union, equivalencerelations, inverse images, subsets, power sets, kernels,. . . represented in Category theory?

2 How are these notions and structures generalized bycategory theory?

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements

A Category is comprised of

1 a collection of C-objects and C-arrows denoted C→ withfixed domain and codomain.

2 Identity arrows for every C-object in C. That is, for any

three C-objects A,B,C s.t. , Af // B

g // C implies

Af //

f ???

????

B

1B

g

@@@

@@@@

B g// C

3 Af // B , B

g // C ∈ C→ =⇒ Af ·g // B ∈ C→

4 Associativity holds: Af // B

g // Ch // D ∈ C→

implies the commutativity of Af //

@@@

@@@@

B

g

~~~~

~~~

D Choo

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

1 vs. ∅ = 0

The category with only one object and one arrow.

Banana

Truck

.

No matter what we call the two objects it must have thisstructure. So we label the object 0 and the arrow 〈0, 0〉 = 10,which gives the number zero by the diagram

0

10

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

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2, 3, . . .

Similarly, we can define the categories 2 : 0

// 1

and 3 : 0

// 331

// 2

These are examples of partial orderings. Compare them againstthe set theoretic definitions of the natural numbers0 = ∅, 1 = ∅, 2 = 0, 1 = ∅, ∅, 3 = 0, 1, 2, . . ..

A preorder category is a category that between any twoobjects a, b ∈ C there exists at most one map a→ b. Thisallows one to define a reflexive and transitive relation. If thisrelation is also antisymmetric then it is called a partialordering and the symbol v will be used to denote the relation.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Posets and Skeletal Categories

A Poset is a pair P = 〈P,v〉 where P is a set and v is apartial ordering on P.

A Skeletal category is one in which “isomorphic” actuallymeans equal (i.e. a ∼= b =⇒ a = b).

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Discrete Categories vs. Sets

If you took any set and added an identity arrow for eachelement, you would have a discrete category. In this sensediscrete categories are sets.

board, flower, skateboard =

Board

flower

skateboard

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

A monoid M is a category with one object. A monoid isdetermined by C→(M), the identity arrow 1e and thecomposition rule .Example: Consider the category N comprised of a single objectN and an infinite collection of arrows labeled 0, 1, 2, 3, . . .which are the natural numbers.

1 Define composition law to be n m = n + m, which meansthat the bottom left diagram commutes by definition.

2 Associative law holds (by associativity of addition) Hencemiddle diagram commutes.

3 Identity arrow 1N is defined to be the number 0. Whichimplies commutativity of right diagram.

Nm //

n+m @@@

@@@@

N

n

N

Nm //

@@@

@@@@

N

n

~~~~

~~~

N Npoo

Nn //

n@

@@@@

@@N

0

m

@@@

@@@@

N m// N

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

For any category C and any a ∈ Cob, the set Hom(a, a) is amonoid.

Hom(a, b) = C(a, b) =

f : f ∈ C→ s.t. a f // b

, where

a, b ∈ Cob.

C is a Subcategory of D ( C ⊆ D) if

1 a ∈ Cob =⇒ a ∈ Dob.

2 for any two a, b ∈ Cob =⇒ C(a, b) ⊆ D(a, b).

C is a Full Subcategory of D if C ⊆ D and for any twoa, b ∈ Cob, C(a, b) = D(a, b).

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Monic vs. injectivityMonic arrows are generalized from the notions of an injectivefunction. We can get the proper definition in terms of arrowsby realizing that injective functions are left cancelable i.e.

f (x) = f (y) =⇒ x = y

orf g(x) = f h(x) =⇒ g(x) = h(x).

In arrows f is monic if for any two parallel functionsg , h : C //// A the following diagram commutes

Cg //

h

A

f

Af // B

.

That is f g = f h =⇒ g = h.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Epic vs. Surjectivity

Epic arrows are derived in the same way except these functionsare right cancelable.

In arrows it amounts to the statement that f is an epic arrow iffor any two parallel arrows g , h : B // // C the followingdiagram commutes

Cf //

f

A

g

A

h // B

.

That is g f = h f =⇒ g = h.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements

Iso arrow vs. Isomorphisms

The notion of an Iso arrow is generalized from isomorphicfunctions. These simply are the functions f where ∃ a functiong s.t. f g = 1 = g f .

In arrows it is the same: for f ∈ C→ f : a→ b is iso (orinvertible), in C if there is a g ∈ C→, g : b → a s.t.g f = 1a and f g = 1b.

a, b ∈ Cob are isomorphic in C (a ∼= b) if there is a C-arrowf : a→ b that is iso in C.

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

In the category Set and in any Topos a monic and epic arrowis an iso arrow. However, this is not true in general!! Eventhough in any category an iso arrow is monic and epic.

Example: Take the category IN, every arrow is epic and monicsince you can cancel on the left and right. And the only iso is0 : IN→ IN, since if m has an inverse n then m n = 1N orm + n = 0 =⇒ m = n = 0.

Example: In a Poset category if f : p → q has an inversef −1 : q → p then p v q andq v p =⇒ p = q =⇒ f = 1p. Thus every arrow in aPoset is monic and epic but the only iso’s are the identities.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Initial objects vs. ∅

Initial objects were abstracted from Set by asking whatproperties characterize the null set ∅? It turns out that for anyset A there exists only one function ∅ → A. In Set the initialobject is simply ∅ and it happens to be unique.

An object 0 is initial in category C if for every

a ∈ Cob, ∃!f ∈ C→ s.t. 0f // a

Any two initial C-objects must be isomorphic in CProof: Suppose ∃0, 0′ that are initial in C. then

∃!f , g ∈ C→ s.t. 0f //

0′goo . This means that

10 = g f : 0→ 0 and 10′ = f g : 0′ → 0′. Hence f has aninverse arrow and 0 ∼= 0′.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Example: of a category that has two initial objects that areisomorphic but not equal.

· ''·

gg

Example: In Grp and Mon the initial objects are any oneelement algebra M = e. Each of these categories hasinfinitely many objects (think products).

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Terminal objects vs. singletons

By reversing the direction of the arrows in the definition ofinitial object we get the idea of a terminal object. In Setterminal objects are the singleton sets (the sets with only oneelement).

An object 1 is terminal in a category C if for every

a ∈ Cob ∃!f ∈ C→ s.t. a f // 1 .

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

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Duality

Duality is the process “reverse all arrows” with the followingidentifications.

Statement Σf : a→ ba = domfi = 1a

Isomorphich = g ff is monicu is a right inverse of hf is invertiblet is a terminal objectS ,T : C → B are functorsT is fullT is faithful

Dual Statement Σ∗

f : b → aa = codfi = 1a

Isomorphich = f gf is epiu is a left inverse of hf is invertiblet is an initial objectS ,T : C → B are functorsT is fullT is faithful

Topos Theoryand the

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MatthewGraham

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Why CategoryTheory?

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The Dual category Cop is comprised of the same objects thatare in C with all of the C-arrows reversed.

Example: For any C: (Cop)op = C.

Example: If C is discrete then Cop = C

Example: If C is a pre-prder (P,R), with R ⊆ P × P, then Cop

is the pre-order (P,R−1), where pR−1q ⇐⇒ qRp. That is R−1

is the inverse relation to R.

Example: The dual statement of the theorem “any two initialC objects are isomorphic” is the theorem “any two terminal Cobjects are isomorphic”

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Products vs. Cartesian Product

Set Definition: A× B = (x , y) : x ∈ A, y ∈ BWhat structure of this definition should we use to form ourdefinition in category theory (since we can’t use elements)?

If we define the projection maps

pA : A× B → A pA(x , y) = x

pB : A× B → B pB(x , y) = y

and we are given another set C with a pair of maps f : C → Aand g : C → B define p : C → A× B by the rulep(x) = 〈f (x), g(x)〉.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements

The diagram

Cg

""FFFFFFFFF

f

||xxxxxxxxxp

A A× B

pAoo pB // B

commutes. Furthermore, the map p is the only arrow that canmake the diagram commute.Proof: Suppose that p(x) = 〈h, k〉. Since we know that thediagram commutes pA p(x) = f (x) andpA(p(x)) = pA(〈h(x), k(x)〉) = h(x) =⇒ h(x) = f (x).Similarly for k .

The uniqueness of the map p is what is generalized in categorytheory.

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

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Category theory product

A product in a category C of two objects a and b is an objecta× b ∈ Cob together with a pair of arrows(pra : a× b → a, prb : a× b → b) s.t. for any two arrows of theform f : c → a and g : c → b ∃! 〈f , g〉 : c → a× b makes thefollowing commute.

cg

""EEE

EEEE

EEf

||yyyy

yyyy

y〈f ,g〉

a a× bpraoo prb // b

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Products are not unique but areisomorphic

Suppose that d is also a product of a× b then consider

dg

""DDD

DDDD

DDf

zzzz

zzzz

z〈f ,g〉

a a× bpraoo prb //

〈pra,prb〉

b

d

g

<<zzzzzzzzzf

aaDDDDDDDDD

Hence〈pra, prb〉〈f , g〉 = 1d .

To complete the isomorphism of d ∼= a× b just interchange theroles of d and a× b to get

〈f , g〉〈pra, prb〉 = 1a×b.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Co-products vs. Disjoint Union

A Co-product of a, b ∈ C is an object a + b ∈ C together witha pair of C arrows (ia, ib) (where ia : a→ a + b andib : b → a + b) s.t. for any C arrows of the form f : a→ c andg : b → c there is a unique arrow [f , g ] : a + b → c making thefollowing commute

aia //

f ""EEE

EEEE

EE a + b

[f ,g ]

biboo

g||yy

yyyy

yyy

c

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

In Set the co-product of A and B is there disjoint union

A + B := (A× 0)⋃

(B × 1) .

The injections are given by

iA(a) = (a, 0) iB(b) = (b, 1)

In this case you need to define the function

[f , g ](x , d) = f (x)(1− d) + dg(x)

We get to use elements since we are in Set.

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Equalisers vs. Equalisers

Set Definition: Given a pair of parallel functions Af //g// B

define E = x : x ∈ A, f (x) = g(x). E equalizes f and g bydefining the inclusion E

// A we get f i = g i .

It turns out that i is a canonical equaliser of f and g . That is ifh : C → A is any other equaliser of f and g (i.e. f h = g h)then there exists a unique k : C → E s.t. i k = h (h ‘factorsuniquely through’ i).

E // A

f //g// B

C

k

[c???????

??????? h

??

Uniqueness: If i k = h then i(k(c)) = k(c) = h(c) =⇒f (h(c)) = g(h(c)) =⇒ h(c) ∈ E .

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Category Definition:An arrow i : e → a in C is an equaliser (in C) of parallelC→ f , g : a→ b if:

1 f i = g i

2 If h : c → a satisfies f h = g h then∃!k ∈ C→ k : c → e s.t. i k = h

e // af //g// b

ck

]eCCCCCCCC

CCCCCCCC h

==

• Every equaliser is monic

• In any category, an epic equaliser is iso.

• In Set (and any Topos) monics are equalisers

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements

Why are the statements ofcategory theory so complicated?

They don’t have to be.

Topos Theoryand the

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Limits and Co-limits vs.A Diagram D is simply a Subcategory of some other category(i.e. a collection of C objects and some C-arrows between theseobjects).A D-Cone (denoted fi : c → di) of a diagram D consists ofa c ∈ Cob with a C arrow fi : c → di , ∀di ∈ D s.t.

di g// dj

c

fiaaDDDDDDDD

fj==zzzzzzzz

commutes whenever g is an arrow in the diagram D.A D-limit of a diagram D is a D-cone fi : c → di with theproperty that for any other D-conef ′i : c ′ → di, ∃! arrow

f : c ′ → c s.t. di

c ′

f ′i==zzzzzzzz

f+3 c

fiaaCCCCCCCC

commutes for every object di ∈ D.

Topos Theoryand the

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MatthewGraham

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Why CategoryTheory?

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Example: Consider the arrowless diagram of two C objects aand b. A D-cone is an object c with two arrows f and g that

form a cfoo g // b . What is another name for a D-limit of

this diagram? Example: Let D be the diagram af //g// b . We

can write a D-cone in this case as the commutative

c h // af //g// b . What is another name for the D-limit of

this diagram?

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements

Co-cones and Co-limits

By duality a Co-cone fi : di → c of a diagram D consists ofan object c and arrows fi : di → c for each object di ∈ D. Aco-limit is defined analogously to limits. That is, a limitco-cone is a co-cone with the co-universal property that for anyother co-cone f ′i : di → c ′, ∃! arrow f : c → c ′ s.t. thefollowing commutes for every di ∈ D.

di g//

fi !!DDD

DDDD

Ddj

fjzzzz

zzzz

c

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Pop QUIZ !!!!!

• What is a cone of an empty diagram?

c

• What is the limit of the empty diagram?

c ′ +3 c

• What is another name for c?

The limit of an empty diagram is a terminal object.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements

Pop QUIZ !!!!!


c


c ′ +3 c



Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Pop QUIZ !!!!!


c


c ′ +3 c



Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Pop QUIZ !!!!!


c


c ′ +3 c



Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Pop QUIZ !!!!!


c


c ′ +3 c



Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Pop QUIZ !!!!!


c


c ′ +3 c



Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

• What is a cone (and limit) for the following diagram?

a b

a b

c

aaDDDDDDDD

<<zzzzzzzz

a a× boo // b

c ′

ccGGGGGGGGG

;;wwwwwwwww

KS

Dually:

• What is the co-limit of an empty diagram?An initial object.

• What diagram is the coproduct a limit of?

a b

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

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a b

a b

c

aaDDDDDDDD

<<zzzzzzzz

a a× boo // b

c ′

ccGGGGGGGGG

;;wwwwwwwww

KS

Dually:



a b

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements


a b

a b

c

aaDDDDDDDD

<<zzzzzzzz

a a× boo // b

c ′

ccGGGGGGGGG

;;wwwwwwwww

KS

Dually:



a b

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements


a b

a b

c

aaDDDDDDDD

<<zzzzzzzz

a a× boo // b

c ′

ccGGGGGGGGG

;;wwwwwwwww

KS

Dually:

• What is the co-limit of an empty diagram?

An initial object.


a b

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements


a b

a b

c

aaDDDDDDDD

<<zzzzzzzz

a a× boo // b

c ′

ccGGGGGGGGG

;;wwwwwwwww

KS

Dually:



a b

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

Definition

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a b

a b

c

aaDDDDDDDD

<<zzzzzzzz

a a× boo // b

c ′

ccGGGGGGGGG

;;wwwwwwwww

KS

Dually:



a b

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

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Acknowledgements


a b

a b

c

aaDDDDDDDD

<<zzzzzzzz

a a× boo // b

c ′

ccGGGGGGGGG

;;wwwwwwwww

KS

Dually:



a b

Topos Theoryand the

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MatthewGraham

Outline

Why CategoryTheory?

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Co-equalisers vs. Equiv. relations

The co-equaliser of a pair of parallel arrows f , g C-arrows is a

co-limit for the diagram af //g// b . Or equivalently, it can be

defined as a C-arrow q : b → c s.t.

1 q f = q g

2 If h : b → c satisfies h f = h g in C then ∃! arrowk : e → c s.t.

af //g// b

q //

h ""DDD

DDDD

D e

k

c

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Definition of k

Suppose that we have a k s.t. k fR = h. Then

k([a]) = k(fR(a)) = h(a)

So define k([a]) = h(a). This is well defined since if [a] = [b]then

fR(a) = fR(b) =⇒ h(a) = h(b).

This is a natural map (i.e. the only way to define this map) andhence unique.

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Pullback vs. Inverse Image

A Pullback of a diagram a f // c bgoo is a limit in C for

the diagram

b

g

a f // c

A cone and universal cone for this diagram is

df ′ //

g ′

h

???

????

b

g

a

f// c

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Outline

Why CategoryTheory?

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In Set, given a function f : A→ B, the inverse image of a setC ⊂ B is simply

f −1(C ) = x : x ∈ A, f (x) ∈ C .

Categorically this is represented by the pullback diagram below

f −1(C )f ∗ //

_

C _

A

f// B

where f ∗ is the restriction of f to f −1(C ).

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Pushouts

Are constructed by dualizing Pullbacks. They are the co-limitof this diagram.

a

b coo

OO

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Completeness

• A category C is complete if every diagram in C has a limitin C.

• Dually a category C is co-complete if every diagram in Chas a co-limit in C

• A category C is bi-complete if it is complete andco-complete.

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Exponentiation (all products exist)In Set theory: We are familiar with

BA =

f : A

f // B

.

This set is related to products by the evaluation function.

ev : BA × A→ B s.t. ev(< f , x >) = f (x)

ev has a universal property among functions of the form

C × Ag // B . ∀g of the above form ∃! g : C → BA s.t.

BA × Aev

##GGG

GGGG

GG

B

C × A

g×idA

KS

g

;;wwwwwwwww

The only choice of g : C → BA

making this commute isdefined:gc(a) = g(c , a) ∀a ∈ AThen g(a) = gc(a) ∀c ∈ C

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Why CategoryTheory?

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Exponentiation (all products exist)In Set theory: We are familiar with

BA =

f : A

f // B

.

This set is related to products by the evaluation function.

ev : BA × A→ B s.t. ev(< f , x >) = f (x)

ev has a universal property among functions of the form

C × Ag // B . ∀g of the above form ∃! g : C → BA s.t.

BA × Aev

##GGG

GGGG

GG

B

C × A

g×idA

KS

g

;;wwwwwwwww

The only choice of g : C → BA

making this commute isdefined:gc(a) = g(c , a) ∀a ∈ AThen g(a) = gc(a) ∀c ∈ C

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

C has exponentiation if a, b ∈ Cob =⇒1 a× b ∈ Cob

2 ∃ba ∈ Cob and an evaluation arrow ev : ba × a→ b

3 and ∀c ∈ Cob and ∀g : c × a =⇒ ∃!g : c → ba

making the diagram commute (i.e.∃!g s.t. ev (g × 1a) = g).

ba × aev

""EEE

EEEE

EE

b

c × a

g×ida

KS

g

<<xxxxxxxxx

This assignment of g to g establishes a isoC(c × a, b) ∼= C(c , ba).

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements





ba × aev

""EEE

EEEE

EE

b

c × a

g×ida

KS

g

<<xxxxxxxxx


Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements





ba × aev

""EEE

EEEE

EE

b

c × a

g×ida

KS

g

<<xxxxxxxxx


Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Subobjects vs. Subsets

• In category theory we don’t have access to the elementsdirectly since we only have objects and arrows.

A ⊂ B iff x ∈ A =⇒ x ∈ B

• How does one get at the notion of a subset withoutreferring to the elements contained within each set?

• The following logic begins to paint a neat picture:• If A ⊂ B then A

/ B is injective, hence monic.

• On the other hand any monic function C //f // B

determines a subset of B.• In fact it creates a bijection Im(f ) ∼= B.• Hence the domain (of a monic arrow), up to isomorphism,

is a subset of the codomain.• Should a subobject of an arrow g be any monic function

that has the same codomain as g?

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Subobjects vs. Subsets

• In category theory we don’t have access to the elementsdirectly since we only have objects and arrows.

A ⊂ B iff x ∈ A =⇒ x ∈ B

• How does one get at the notion of a subset withoutreferring to the elements contained within each set?

• The following logic begins to paint a neat picture:• If A ⊂ B then A

/ B is injective, hence monic.

• On the other hand any monic function C //f // B

determines a subset of B.• In fact it creates a bijection Im(f ) ∼= B.• Hence the domain (of a monic arrow), up to isomorphism,

is a subset of the codomain.• Should a subobject of an arrow g be any monic function

that has the same codomain as g?

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

• How do we make this analogy with subset stronger?

• We need to define subobject, the category equivalent ofsubset. But it might be the case that the subset of someset A is not even an object in our category!@!

• Which pushes us in the direction of defining the equivalentnotion of subset on the arrows.

• OK. Then we need to know (for g ∈ C→) how todetermine which f ∈ C→ satisfy f ⊇ g or f ⊆ g?

• And check that ⊆ makes (Sub(D),⊆) a poset.

Some interesting structure.• The relation of set inclusion is a partial ordering on the

power set P(D) of a set D.

• Hence (P(D),⊆) is a poset which is a category wherethere exists an arrow A→ B iff A ⊆ B.

• When A ⊆ B, ∃ a vertical arrow that makes the diagramcommute

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements










Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements










Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Subobject

• Suppose that we defined subobject of d ∈ Cob to be amonic arrow with codomain d .

• Then consider the following definition of ‘inclusion’relation between subobjects.

• Given a f // d , bg // d . f ⊆ g iff ∃h ∈ C→ s.t.

b g

???

????

?

a

h

OO

//f// d

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

• We need to check that this definition is:• Reflexive (f ⊆ f ) a

f

@@@

@@@@

@

a

1a

OO

//f// d

• Transitive (f ⊆ g , g ⊆ k =⇒ f ⊆ k)

c k

BBB

BBBB

B

b

l

OO

// g // d

a

hl

FF

h

OO

>> f

>>||||||||

• Antisymmetric (f ∼= g =⇒ f = g)

b g

???

????

?

l

a

h

VV

//f// d

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Actual definition of Subobject

• The antisymmetric diagram does not guarantee that f = gwhen it commutes only that f ∼= g . Easy check: it mightbe the case that a 6= b.

• Modify the definition to make equality hold:

A subobject of d is the equivalence class of all monic arrowswith codomain d .

• This definition makes (Sub(D),⊆) a poset.

• This definition corresponds to the usual subobjects(defined via elements) in the categories Rng ,Grp,Ab andR −Mod but apparently not in T op.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Actual definition of Subobject

• The antisymmetric diagram does not guarantee that f = gwhen it commutes only that f ∼= g . Easy check: it mightbe the case that a 6= b.

• Modify the definition to make equality hold:

A subobject of d is the equivalence class of all monic arrowswith codomain d .

• This definition makes (Sub(D),⊆) a poset.

• This definition corresponds to the usual subobjects(defined via elements) in the categories Rng ,Grp,Ab andR −Mod but apparently not in T op.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

• To each subset there is an associated characteristicfunction.

• Is the same true for every subobject?

• If so how do you define it categorially?

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Properties of Power Set

• P(D) ∼= 2D . There is a bijective correspondence betweenthe functions D → 2 and the subsets of D.

• You can get this by using the characteristic function χA ofa subset A.

χA =

1 x ∈ A0 x 6∈ A

• The characteristic function tells us if a particular elementx ∈ X is also an element in A.

• The characteristic functions ‘see’ the elements.

• Is there a universal property that would allow us to definea similar function for category theory?

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Properties of Power Set

• P(D) ∼= 2D . There is a bijective correspondence betweenthe functions D → 2 and the subsets of D.

• You can get this by using the characteristic function χA ofa subset A.

χA =

1 x ∈ A0 x 6∈ A

• The characteristic function tells us if a particular elementx ∈ X is also an element in A.

• The characteristic functions ‘see’ the elements.

• Is there a universal property that would allow us to definea similar function for category theory?

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

• It turns out that there is a universal property.

• Let Af = x : x ∈ D, f (x) = 1 = f −1(1).Then the set Af arises by the pullback square below.

Af

// D

f

1 // 2

• For a set A ⊂ D,

A

// D

χA

1 // 2

is a pullback square.

• The universal property is that χA is the only function thatcan make this a pullback square for the set A.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

• It turns out that there is a universal property.

• Let Af = x : x ∈ D, f (x) = 1 = f −1(1).Then the set Af arises by the pullback square below.

Af

// D

f

1 // 2

• For a set A ⊂ D,

A

// D

χA

1 // 2

is a pullback square.

• The universal property is that χA is the only function thatcan make this a pullback square for the set A.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Proof:

• Suppose there was some other function g that would makethe previous diagram a pullback. Then we get thefollowing diagram

Ag

k #@@

@@@@

@

@@@@

@@@ t

i

''OOOOOOOOOOOOOO

j

##

A

// D

g

1

true // 2

• For x ∈ A, g(x) = 1 =⇒ x ∈ Ag . Thus, A ⊆ Ag .

• Since the outer square commutes and i and j areinclusions then so is k. Hence, Ag ⊆ A.

• But then f is the characteristic function of A thereforef = χA.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Proof:

• Suppose there was some other function g that would makethe previous diagram a pullback. Then we get thefollowing diagram

Ag

k #@@

@@@@

@

@@@@

@@@ t

i

''OOOOOOOOOOOOOO

j

##

A

// D

g

1

true // 2

• For x ∈ A, g(x) = 1 =⇒ x ∈ Ag . Thus, A ⊆ Ag .

• Since the outer square commutes and i and j areinclusions then so is k. Hence, Ag ⊆ A.

• But then f is the characteristic function of A thereforef = χA.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Subobject Classifier

Given a category C with initial object 1. A subobject classifierfor C is Ω ∈ Cob together with a C-arrow true : 1→ Ω thatsatisfies the Ω-axiom below.

Ω-axiom: For each monic a // f // d there is one and only oneχf : d → Ω s.t. the following is a pullback square.

a // f //

d

χf

1

true // Ω

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Topos

An Elementary Topos is a category E s.t.

1 E is finitely complete.

2 E is finitely co-complete.

3 E has exponentiation.

4 E has a subobject classifier.

1 ⇐⇒ E has a terminal object and pullbacks.

2 ⇐⇒ E has an initial object and pushouts.

Also (1), (3), and (4) imply (2). So another way of defining atopos E is to say that E is cartesian closed category with asubobject classifier.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Topos









Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Topos









Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Topoi Examples

• Set - the prime example and the motivation for theconcept in the first place

• F inset is a topos with limits, exponentials and > : 1→ Ωexactly as in Set.

• Set2

• Set→ the category of functions.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Some true statements about Topoi

• ∼ 1963 Lawvere figured out new foundations forMathematics based on category theory. (i.e. what is sogreat about sets?)

• Even earlier than this the concept of tracking structure ofsystems via category theory became important in algebraicgeometry (via work by Grothendieck on the Weilconjectures).

• So Topos theory was invented from a merger of ideas fromgeometry and logic.

• Topos is a category with some extra properties that makeit look like Set.

• Many people study a space by studying the sheaves onthat space. Apparently the main utility of a topos arises insituations in math where topological intuition is veryeffective but the space does not allow a topology. It issometimes possible to formalize the intuition via a topos.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Some true statements about Topoi

• ∼ 1963 Lawvere figured out new foundations forMathematics based on category theory. (i.e. what is sogreat about sets?)

• Even earlier than this the concept of tracking structure ofsystems via category theory became important in algebraicgeometry (via work by Grothendieck on the Weilconjectures).

• So Topos theory was invented from a merger of ideas fromgeometry and logic.

• Topos is a category with some extra properties that makeit look like Set.

• Many people study a space by studying the sheaves onthat space. Apparently the main utility of a topos arises insituations in math where topological intuition is veryeffective but the space does not allow a topology. It issometimes possible to formalize the intuition via a topos.

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Some hand wavy connections withlogic

• The exponential Ωa is the topos analogue of 2A in Set.

• Some questions:• 2A ∼= P(A) is the same true for Ωa? Is it the “power set”

of a?

It turns out that it does behave similarly.• 2A logically represents all the mappings from A to the two

point set 0, 1, which can be thought of true and false.What does Ωa represent?

• What if Ω = 0, 12 , 1?

At a very basic level this means that if we regard a as‘containing’ statements of some sort then there are moreoptions than just true and false.(i.e. the law of excluded middle does not hold!!)

Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements




of a?



• What if Ω = 0, 12 , 1?


Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements




of a?



• What if Ω = 0, 12 , 1?


Topos Theoryand the

Connectionsbetween


MatthewGraham

Outline

Why CategoryTheory?

Definition

BasicExamples




Acknowledgements

Acknowledgements

I would like to thank:

• YOU !

• and the people that came to the last talk but found it tooboring to come to this one.

• and the fact that there exists a truck banana category,without which I don’t think that I could sleep at night.

Documents

Topos Theory and the Connections between Category and Set ...people.brandeis.edu/~nstambau/GSS/CategoryVsSetTheory.pdf · Topos Theory and the Connections between Category and Set