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MatthewGraham
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Topos Theory and the Connections betweenCategory and Set Theory
Matthew Graham
March 13, 2008
Topos Theoryand the
Connectionsbetween
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MatthewGraham
Outline
Why CategoryTheory?
Definition
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ToposDefinition andExamples
Motivationand History ofTopos theory
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
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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.
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• 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!
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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?
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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
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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
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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.
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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).
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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
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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
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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).
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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.
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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.
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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.
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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.
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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′.
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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).
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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 .
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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
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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”
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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)〉.
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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.
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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
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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.
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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
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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.
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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 .
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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
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Why are the statements ofcategory theory so complicated?
They don’t have to be.
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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.
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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?
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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
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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.
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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.
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Why CategoryTheory?
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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
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MatthewGraham
Outline
Why CategoryTheory?
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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
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MatthewGraham
Outline
Why CategoryTheory?
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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
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MatthewGraham
Outline
Why CategoryTheory?
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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.
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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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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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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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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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).
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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).
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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).
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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?
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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?
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• 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
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• 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
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Outline
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• 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
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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
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• 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
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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.
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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.
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• To each subset there is an associated characteristicfunction.
• Is the same true for every subobject?
• If so how do you define it categorially?
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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?
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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?
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• 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.
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• 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.
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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!!)
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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
Category andSet Theory
MatthewGraham
Outline
Why CategoryTheory?
Definition
BasicExamples
Relations toSet Theory
ToposDefinition andExamples
Motivationand History ofTopos theory
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
Category andSet Theory
MatthewGraham
Outline
Why CategoryTheory?
Definition
BasicExamples
Relations toSet Theory
ToposDefinition andExamples
Motivationand History ofTopos theory
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.
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