Access over 20 million homework documents through the notebank

Get on-demand Q&A homework help from verified tutors

Read 1000s of rich book guides covering popular titles

Anonymous

I have some questions in CATEGORY THEORY, are you familiar in this subject . you need to choose any 4 question out of the 5 ques posted , i am attaching book as well for help . how much ?

Category Theory Lecture Notes
Daniele Turi
Laboratory for Foundations of Computer Science
University of Edinburgh
September 1996 – December 2001
Prologue
These notes, developed over a period of six years, were written for an eighteen lectures
course in category theory. Although heavily based on Mac Lane’s Categories for the Working
Mathematician, the course was designed to be self-contained, drawing most of the examples
from category theory itself.
The course was intended for post-graduate students in theoretical computer science at the
Laboratory for Foundations of Computer Science, University of Edinburgh, but was attended
by a varied audience. Most sections are a reasonable account of the material presented
during the lectures, but some, most notably the sections on Lawvere theories, topoi and Kan
extensions, are little more than a collection of definitions and facts.
Contents
Introduction
1
1 Universal Problems
1
1.1
Natural Numbers in set theory and category theory . . . . . . . . . . . . . . .
1
1.2
Universals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2 Basic Notions
7
2.1
Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Initial and Final Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.4
Comma Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3 Universality
11
4 Natural Transformations and Functor Categories
14
5 Colimits
18
5.1
Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
5.2
Coequalisers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
5.3
Pushouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
5.4
Initial objects as universal arrows . . . . . . . . . . . . . . . . . . . . . . . . .
20
5.5
Generalised Coproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5.6
Finite Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
5.7
Colimits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
ii
6 Duality and Limits
25
6.1
Universal arrows from a functor to an object . . . . . . . . . . . . . . . . . .
25
6.2
Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
6.3
Equalisers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
6.4
Pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
6.5
Monos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
6.6
Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
7 Adjunctions
31
7.1
From Universal Arrows to Adjunctions . . . . . . . . . . . . . . . . . . . . . .
31
7.2
Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
7.3
From Adjunctions to Universal Arrows . . . . . . . . . . . . . . . . . . . . . .
35
7.4
Adjoints for Preorders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
8 Preservation of Limits and Colimits
39
9 Monads
41
9.1
Algebras of a Monad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
9.2
Comparison Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
9.3
Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
10 Lawvere Theories
45
11 Cartesian Closed Categories
48
11.1 Curry-Howard-Lawvere Isomorphism . . . . . . . . . . . . . . . . . . . . . . .
49
12 Variable Sets and Yoneda Lemma
50
13 Set theory without sets
53
13.1 Subobject Classifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
13.2 Topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
14 Kan Extensions
55
15 2-Categories
56
Further Reading
57
iii
Lecture I
Introduction
Aim
• Learn category theory?
There is a lot to learn and we do not have much time. The crucial notion is that of adjunction
and the course is geared towards getting there as quickly and as naturally as possible. Hard
in the beginning, but it pays-off. Above all we aim to:
• Learn to reason categorically!
When one learns a foreign language it is often advised to listen to the language first, learning
to understand the words before knowing how they are spelled. This is hard at first, but
it pays back. Similarly, in this course we shall try and refrain from relating categories to
other subjects and we shall try and work with examples and exercises from category theory
itself. Surprisingly, one can go a long way without mentioning the traditional examples
from mathematics and computer science: category theory is, by and large, a self-contained
discipline.
Before starting, let us recall what the words category and categorical mean in English:
Category: class or group of things in a complete system of grouping.
Categorical : (of a statement) unconditional, absolute.
(Definitions from the Oxford dictionary.)
1
Universal Problems
In this first lecture we introduce universal problems. Following [Mac86, §II.3], we show that
the recursion theorem is a categorical, compact way of expressing the Peano axioms for the
natural numbers. This leads to Lawvere’s notion of natural number object.
1.1
Natural Numbers in set theory and category theory
What are the natural numbers?
A1 Traditional, set-theoretic answer (Peano, one century ago):
The natural numbers form a set N such that:
1. ∃ zero ∈ N
2. ∀ n ∈ N, ∃ succ n ∈ N
3. ∀ n ∈ N, succ n 6= zero ∈ N
1
4. ∀ n, m ∈ N, succ n = succ m
5. ∀ A ⊆ N (zero ∈ A ∧ a ∈ A
+3 n = m (injectivity)
+3 succ a ∈ A)
+3 A = N
These axioms determine N uniquely up to isomorphism. (We shall prove this categorically in a moment.)
A2 Categorical answer (Lawvere, 60’s):
A natural number object
0∈N
s
/N
(in Set) consists of
– a set N
– with a distinguished element 0 ∈ N
– and an endofunction s : N
/N
which is universal in the sense that for every structure
e∈X
there exists a unique function f : N
g
/X
/ X such that
– f (0) = e
– f (s(n)) = g(f (n)) for all n in N
The two characterisations are equivalent:
Theorem 1.1 A1 ks
+3 A2
Proof.
(A1
+3 A2) ≡ Recursion Theorem (see proof in [Mac86, page 45].)
(A2
+3 A1)
(1) and (2) by definition, setting N = N , zero = 0 and succ n = s(n).
(3) by contradiction: assume s(n) = 0 and consider a structure
e∈X
g
/X
with:
• X = {e, d}
• g(e) = g(d) = d
Then, by A2, there exists f : N
f (0) = e
/ X such that
f (s(n)) = g(f (n))
But if s(n) = 0 then f (s(n)) = e 6= d = g(f (n)).
(4) Exercise 1.2.
(5) We first express A2 in a really categorical way. For this we need to establish
a language of diagrams.
2
Diagrams
Given functions f : X
/Y, g :Y
/ Z, h : X
/ Z we say that the diagram
f
/Y
AA
AA
g
h AA
XA
Z
commutes if and only if g(f (x)) = h(x) for all x in X. We write then:
g◦f =h
Of course, we can add identity functions wherever we want without affecting commutation.
Eg:
f
X
id X
/Y
g
X
/Z
h
Another example:
X@
f
f0
/Y
@@
@@
@
h @@
/Y0
g
Z
g0
/ Z0
k
commutes iff k(g(f (x))) = k(h(x)) = g 0 (f 0 (f (x))) for all x in X.
A trivial yet important remark is that every element x of a set can be regarded as a function
from a one-element (ie singleton) set {∗} to X. Moreover, this correspondence is a bijection.
From now on we then write 1 for the generic singleton set and
/X
x:1
as an alternative (very convenient!) notation for
x∈X
Finally, a dashed arrow
f
X _ _ _/ Y
indicates that there is a unique map f from X to Y .
A2 with diagrams We can now rephrase the two equations of A2 (and the uniqueness
condition) as follows:
1
id 1
0
1
e
/N
f
/X
3
s
g
/N
f
/X
Proof of Theorem 1.1 (continued)
First note that everything in sight in the diagram
0
1
id 1
1
id 1
0
/N
f
/A
1
0
s|A
i
/N
commutes, where
/N
f
/A
s
i
/N
s
/N
i:A
is the evident inclusion function associated to A ⊆ N , and
s|A : A
is the restriction of s : N
/A
/ N to A (which we assume it exists by A1.5).
Trivially, also
1
id 1
0
1
/N
0
s
id N
/N
/N
s
id N
/N
commutes, hence, by the uniqueness condition in A2, we have that
i ◦ f = id N
By the following lemma this implies that f is injective:
Lemma 1.2 (Right inverses are injective) Given two composable
functions f and g, if g ◦ f is the identity then f is injective.
We can then conclude that N ⊆ A, since the codomain A of the injective function
f is A ⊆ N .
Theorem 1.3 A2 (hence A1) determines N uniquely up to isomorphism.
Proof. We need to prove that if there exists another structure
1
00
/ N0
s0
/ N0
satisfying A2 then there exists an isomorphisms between N and N 0 . That is, we
/ N 0 and f 0 : N 0
/ N such that
would like to establish the existence of f : N
4
f 0 ◦ f = id N and f ◦ f 0 = id N 0 . We are going to find these two functions f and f 0
using the universal property of both structures given by A2:
1
id 1
0
1
id 1
00
1
0
/N
f
/ N0
f0
/N
/N
f
/ N0
f0
/N
s
s0
s
but also
1
id 1
0
1
/N
0
s
/N
id N
/N
s
id N
/N
hence
f 0 ◦ f = id N
Similarly,
f ◦ f 0 = id N 0
1.2
Universals
. . . there exists a unique function such that . . .
• Existence: define entities
• Uniqueness: prove properties
Theorem 1.4 A universal construction defines an entity uniquely up to isomorphism.
Proof. Very much like for Theorem 1.3.
Exercises
E 1.1 Prove that A2 implies the following (where X × N is the cartesian product of the sets
X and N ):
Primitive Recursion. For every set X with a distinguished element e ∈ X and
/ X there exists a unique function f : N
/ X such that
a function h : X × N
• f (0) = e
• f (s(n)) = h(f (n), n) for all n in N .
Hint: apply A2 to g : X × N
/ X × N , where g(x, n) = (h(x, n), n).
5
E 1.2 Use the above primitive recursion to prove that A2 implies the fourth Peano axiom
(injectivity). (Hint. You want to prove that the successor is injective. For this you can use
/N
Lemma 1.2 with respect to predecessor and successor. The predecessor function p : N
is definable by primitive recursion.)
6
Lecture II
2
Basic Notions
Category theory is the mathematical study of universal properties:
• it brings to light, makes explicit, and abstracts out the relevant structure, often hidden
by traditional approaches;
• it looks for the universal properties holding in the categories of structures one is working
with.
Category Theory vs Set Theory: primitive notions
Set Theory:
• membership and equality of those abstract collections called sets
– an object is determined by its content.
Category Theory:
• composition and equality of those abstract functions called arrows
– understand one object by placing it in a category and studying its relation with
other objects of the same category (using arrows), or related categories (using
functors, ie arrows between categories).
Informatics: we want to understand programs abstractly, independently from their implementation.
2.1
Categories
A category is a (partial) algebra of abstract functions:
• arrows with identities and a binary composition (partial) operation
– obeying generalised monoid laws.
More formally, a category C consists of:
• A collection ObjC of objects A, B, C, . . . , X, Y, . . ..
• For each pair of objects A and B, a collection C(A, B) of arrows f : A
to B;
– A is the domain and B is the codomain of f : A
7
/ B;
/ B from A
– the collection of all arrows f, g, h, k, . . . of C is denoted by ArrC ;
– arrows are also called maps or morphisms.
/ A.
• For each object A, an identity arrow id A : A
• For each pair of arrows
A
f
/B
g
a composite arrow
g◦f :A
/C
/C
These data have to satisfy the following generalised monoid laws.
1. Identity: if A
f
/ B, then
id B ◦ f = f = f ◦ id A
2. Associativity: if A
f
/B
g
/C
h
/ D, then
(h ◦ g) ◦ f = h ◦ (g ◦ f )
Examples
1. Categories freely generated by directed graphs.
2. Degenerate categories such as:
• Sets – ie categories where all arrows are identities.
• Monoids – ie categories with only one object.
• Preorders: categories with at most one arrow between every two objects.
3. 0, 1, ·
/ ·, ω.
4. Opposite categories Cop : obtained by reversing the arrows of given categories C, while
keeping the same objects.
5. Set: the category of (small) sets and functions; composition is the usual function composition (and so is in the remaining examples). Note that type matter: the identity on
the natural numbers is a different function from the inclusion of the natural numbers
into the integers.
6. Set∗ : pointed sets (ie sets with a selected base-point) and functions preserving the base
point.
7. N: finite ordinals and functions.
8. FinSet: finite sets and functions.
9. Preord: preorders and monotone functions.
10. Poset: partial orders and monotone functions.
8
11. Cpo: complete partial orders and continuous functions.
12. Mon: monoids and monoid homomorphisms.
13. Grp: groups and group homomorphisms.
14. SL: semi-lattices and join-preserving functions.
15. Top: topological spaces and continuous functions.
16. Met: metric spaces and non-expansive functions.
17. CMet: complete metric spaces and non-expansive functions.
2.2
Functors
A functor is a homomorphism of categories
F :C
/D
ie a morphism of categories preserving the structure, namely identities and composition.
/ D consists of a function X / F (X) from the objects of C to
Formally, a functor F : C
the objects of D and a function f / F (f ) from the arrows of C to the arrows of D such that:
F (id X ) = id F X
F (g ◦ f ) = F (g) ◦ F (f )
Thus functors preserve all commuting diagrams, hence, in particular, isomorphisms.
Functors between sets are functions, between preorders are monotone functions, between
monoids are monoid homomorphisms, between groups are isomorphisms (because a group
is a category with one object and where every map has an inverse, and functors preserve
isomorphisms).
For every category C, there is an evident identity functor
Id C : C
/C
/ D of functors F : B
Moreover, there is a composition G ◦ F : B
namely (G ◦ F )(X) = G(F (X)) and (G ◦ F )(f ) = G(F (f )).
/ C and G : C
/ D,
A category C is small if its collection ObjC of objects and its collection ArrC of arrows are sets; it is
locally small if the collection C(A, B) of arrows from A to B is a set for each pair of objects A and
B. N is small, while the category Finset is only locally small, although every finite set is isomorphic
to a finite ordinal. In the above examples, the categories 5, 6, and from 8 to 17 are not small, but only
locally small.
Cat is the corresponding category of all (small) categories and functors between them.
E 2.1 Check that Cat, the category of all small categories, is indeed a category, ie check
that the identity and the associative laws hold.
9
Lecture III
2.3
Initial and Final Objects
An object is initial in a category C if for every object X in C there exists a unique arrow in
C from it to X. (It need not exist!)
Initial objects are unique up to isomorphism.
Notation for ‘the’ initial object: 0.
An object is final (or terminal) in a category C if for every object X in C there exists a
unique arrow in C from X to it. (Again, a category may have no final object.) Note that:
A final in C iff A initial in Cop
therefore,
final objects are unique up to isomorphism.
Notation for ‘the’ final object: 1.
The initial object in a preorder is the bottom element, if it exists; the final object is the
top. In Set the initial object is the empty set, while the final object is the (unique up to
isomorphism) singleton set. In Cat the initial object is the empty category 0, with no objects
nor arrows, and the final object is the category 1 with only one object and one arrow (the
identity).
2.4
Comma Categories
Crucial for this course is Lawvere’s notion of a comma category. Given a category C, an
/ C, the comma category (A ↓ U ) is the category of
object A of C and a functor U : D
/ U Y i given by arrows of C of type A
/ UY ,
arrows from A to U , with objects hY, h : A
/
/
where Y can range over the objects of D. The homomorphisms f : hY, h : A
UY i
0
0
0
0
0
/ U Y i are given by arrows f : Y
/ Y of D such that U f ◦ h = h . I like to
hY , h : A
draw all this as follows.
U
Co
D
/ UY
DD
DD
Uf
h0 DD!
A DD
h
UY 0
Y
f
Y0
Please try and conform to this notation (with U going from right to left) as we shall use it
extensively throughout the course.
10
Exercises
E 2.2 Write down the proof that initial objects are unique up to isomorphism.
E 2.3 (We shall need this later on.) Just write down the more general notion of comma
category (T ↓ U ) involving two functors T and U with the same codomain
/Co
T
B
U
D
that we have seen in the lecture. Also, write the two ‘projection’ functors P : (T ↓ U )
/ D.
and Q : (T ↓ U )
3
/B
Universality
/ C consists of an initial
A universal arrow from an object A of C to a functor U : D
/ U FA of
object in the comma category (A ↓ U ), ie an object FA of D and arrow ηA : A
/ U Y of C there
C which are universal in the sense that for every Y of D and every h : A
]
]
/
exists a unique arrow h : FA
Y in D such that U h ◦ ηA = h. Diagrammatically:
Co
U
D
ηA
/ U FA
DD
DD
U h]
h DD!
A DD
UY
FA
h]
Y
It is at first hard to get accustomed to switching between the two categories C and D along
the functor U and in getting the right order in the quantifications involved, but this notion
is to category theory what the ∀∃δ formulation of continuity is to analysis. It is important
to realize that the object FA is by no means initial in D: there might be more than one
arrow from FA to Y , but h] is the unique arrow such that the triangle in the above diagram
commutes.
Exercise
E 3.1 (Important! ) Try and prove the following theorem.
Theorem 3.1 In the above situation, assume that, for every object A of C there
/ U FA from A to U . This defines a function F from
is a universal arrow ηA : A
def
the objects A of C to objects F A = FA of D. Then, by universality, this extends
/ D in the opposite direction of U .
to a functor F : C
11
Lecture IV
Proof of Theorem 3.1
The action of F on arrows is defined as follows:
Co
A
f
ηA
B
D
/ UFA
ηB
U
F A
UF f
/ UFB
def
F f = (ηB ◦f )]
FB
Using universality one can prove that F is a functor F : C
/ D, that is:
F g ◦ F f = F (g ◦ f )
F id A = id F A
2
Example 3.2 A monoid M = hM, e, mi consists of a carrier set M and an associative mul/ M with a unit e ∈ M :
tiplication operation m : M × M
m(x, m(y, z)) = m(m(x, y), z)
m(x, e) = x = m(e, x)
for all x, y, z in M . A monoid homomorphism is a function between carriers which respects
unit and multiplication. It is easy to see that monoids and their homomorphisms form a
category Mon.
There is a trivial yet powerful forgetful functor
Set o
Mon : U
which maps a monoid hM, e, mi to its carrier M and a homomorphism f : hM, e, mi
/ M 0 by forgetting that it is a homomorphism.
hM 0 , e0 , m0 i to itself f : M
/
An important property of this forgetful functor is that for every set A there is a universal
arrow from A to U , which gives rise, by the above theorem to a functor
F : Set
/ Mon
This F maps a set A to the set A∗ of finite words over A, with concatenation as multiplication
/ U F A = A∗ maps an
and with the empty word as unit. The universal arrow ηA : A
element a of A to the one letter word hai. For every monoid hM, e, mi and eve ...

Student has agreed that all tutoring, explanations, and answers provided by the tutor will be used to help in the learning process and in accordance with Studypool's honor code & terms of service.

This question has not been answered.

Create a free account to get help with this and any other question!

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors