Download Intro to Categories

CATEGORIES AND FUNCTORS
L. MARIZZA A. BAILEY
1. Categories
Definition 1 (Category). A category, C consists of a class of objects, Obj[C], and
a set of morphisms, Mor(C), such that
(C1) for any two objects, A, B ∈ Obj(C) there exists a set M or(A, B). THIS
SET MAY BE EMPTY. This is contrary to what I had previously told you.
Between any two objects, it is not necessary that there be a morphism between them. Sorry.
(C2) for any objects A, B, C ∈ Obj(C) there exists a law of composition, ◦, such
that if f ∈ Mor(A, B) and g ∈ M or(B, C) then f ◦ g ∈ Mor(A, C).
(C3) If A, A0 , B, B 0 ∈ Obj(C) then Mor(A, B) ∩ Mor(A0 , B 0 ) = ∅ unless A = A0
and B = B 0 .
(C4) For any object A there exists and identity morphism idA ∈ Mor(A, A) such
that idA ◦ f = f for any f ∈ Mor(A, B).
(C5) The Law of composition is associative.
1.1. Examples of Categories.
• C = (Sets, Functions)
Obj[C] are sets and for any two A, B ∈ Obj[C], Mor(A, B) = {f : A → B |
f (x) 6= f (y) ⇒ x 6= y}.
• (Subsets of a set X, injective functions) Let X be a set.
Obj[C] = P(X) and for any two A, B ∈ Obj[C], Mor(A, B) = {f : A → B |
f (x) 6= f (y) ⇔ x 6= y}.
• (Groups, Group Homomorphisms) Obj[C] are groups and for any two
A, B ∈ Obj[C], Mor(A, B) = {f : A → B | f (xy) = f (x)f (y) ∧ f (1) = 1}.
f preserves the group operation.
• (Metric Spaces, Isometries) Obj[C] are metric spaces and for any two
A, B ∈ Obj[C], Mor(A, B) = {f : A → B | d(f (x), f (y)) = d(x, y)}.
f preserves distance.
Date: May 2, 2007.
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• (Metric Spaces, Continuous Functions) Obj[C] are metric spaces and for
any two A, B ∈ Obj[C], Mor(A, B) = {f : A → B | ∀ > 0∃δ > 0 3
d(x, y) < δ ⇒ d(f (x), f (y)) < }.
The pre-image of open sets are open.
• (Topological Spaces, Continuous Functions) Obj[C] are topological spaces
and for any two A, B ∈ Obj[C], Mor(A, B) = {f : A → B | Z ⊂
B is open f −1 (Z) ⊂ A is open }.
The pre-image of open sets are open.
• (Manifolds, Homeomorphism such that the maps commute) Obj[C] are
manifolds. Let A be a manifold with chart {Uα , φα .
Let B be a manifold with chart {Vα , ξα .
Mor(A, B) = {f : A → B | φα = ξα ◦ f ∀α}
f commutes with the chart maps.
• (Vector Spaces, Matrices or Linear Transformations) Obj[C] are (V, +, ∗),
two operations; vector addition and scalar multiplication (multiplication by
numbers).
(V, +) is a group, and ∗ has an identity 1 ∗ v = v, ∀v ∈ V .
Also ∗ is associative and distributes over vector addition, k ∗ (v + w) =
k ∗ v + k ∗ w and (v + w) ∗ k = v ∗ k + v ∗ w. For any two V, W ∈ Obj[C],
Mor(V, W ) = {f : V → W | f (v + w) = f (v) + f (w) ∧ f (k ∗ v) = k ∗ f (v)}.
f is a group homomorphism and respect scalar multiplication.
• (Covers of a topological space X, Continuous functions which commute
with cover) g is a morphism from Y to Z if for any X. If f1 : Y → X is a
cover, and f2 : Z → X is another cover,
g
Y −−−−→ Z


f1 y
f2 y
id
X −−−−→ X
• (Subspaces of a topological space X, Deformation Retractions)
g
X −−−−→ X


λ1 y
λ2 y
id
Z −−−−→ Y
In the diagram above, λ1 is a deformation retraction from X onto Z, and
λ2 is a deformation retraction from X onto Z. g : X → Y is a morphism if
λ2 ◦ g = λ1
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2. Universal Objects and Versal Objects
One of the most important, and useful, characteristics of categories, is the idea
of a universal object. Through these definitions, we can define the product, the
fiber product, and the co-product of two objects in any category.
Definition 2. Let C be a category.
U ∈ Obj[C] is a universally repelling object in C if for any A ∈ Obj[C] there exists
a unique morphism f : U → A.
It is also called an initial object
A universally repelling object, therefore, is an object that maps uniquely to every
other object in the category. The word unique here is very important. Below is an
example of a universally repelling object that is rather easy to see.
Example 1. Let C = (Groups, group homomorphisms)
We need a group which can map to every other group in existence.
So we need to ask ourselves, what is the common characteristic of all groups?
The identity.
They all must have an identity to be a group.
The universally repelling object in this category, then, is the trivial group, which
contains only the identity.
The unique morphism maps the identity to the identity.
Definition 3. Let C be category.
V ∈ Obj[C] is a versally repelling object in C if for any A ∈ Obj[C] there exists a
morphism, not necessarily unique, f : V → A.
The only difference here is the word ”unique”. There may be many maps from
the versally repelling objects to every other object in the category. The following
example will illustrate this.
Example 2. (Nonempty subsets of a set X, injective functions)
Let X be a set.
Obj[C] = P(X) and for any two A, B ∈ Obj[C], Mor(A, B) = {f : A → B | f (x) 6=
f (y) ⇔ x 6= y}.
A singleton set {x} is the versally repelling object because there will always be a
injective map from it to any subset of X, although the map can map to any element
of the set.
Definition 4. Let C be a category.
U ∈ Obj[C] is a universally attracting object in C if for any A ∈ Obj[C] there
exists a unique morphism f : A → U .
It is also called a terminal object.
A universally attracting object is an object to which every other object in the
category maps uniquely. The word unique here is still very important. Below is an
example of a universally attracting object that is rather easy to see.
Example 3.
(Nonempty subsets of a set X, embeddings functions) Let X be a set.
Obj[C] = P(X) and for any two A, B ∈ Obj[C], Mor(A, B) = {f : A → B | f (x) =
f (y) ⇔ x = y}.
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The whole set X is the universally attracting object because there will always be a
injective map into it from any subset, although the map is not uniquely determined
by the subset.
Definition 5. Let C be category.
V ∈ Obj[C] is a versally attracting object in C if for any A ∈ Obj[C] there exists
a morphism, not necessarily unique, f : V → A.
The only difference here is the word ”unique”. There may be many maps from
the versally repelling objects to every other object in the category. The following
example will illustrate this.
Example 4. Let C = (Topologies on a set X, continuous maps)
By this I mean that the objects consist of the same set with different topological
structures.
We need a topology whose open sets will always have an open preimage.
So we need to ask ourselves, which sets are always open under any topology?
The trivial topology, the topology whose open sets consist of the whole set and the
empty set.
If f : X → Y then f −1 (Y ) = X and f −1 (∅) = ∅.
The versally attracting object in this category, then, is X with the trivial topology.
What is the versal repelling object?
Note that the map is not unique because composition with a permutation of the
elements of X will still be continuous.
2.1. Product and Co-Product. Let C be a category. Its’ associated category C0
is the category whose objects are defined to be f : A → B morphisms f between
objects in A, B ∈ Obj[C] morphisms are pairs (g : A → A0 , h : B → B 0 ) between
two objects f : A → B and f 0 : A0 → B 0 such that the following diagram commutes.
g
A −−−−→ A0


fy
f 0y
h
B −−−−→ B 0
2.1.1. Product. In order to define the product of two objects A, B in a category C,
we need to create a new category D whose objects are objects Y ∈ Obj[C] with
morphisms p1 : Y → A and p2 : Y → B, and morphisms are commutative diagrams:
The product of A and B is the universally attracting object in this category.
In other words, it is the object A×B with maps πA : A×B → A and πB : A×B → B
such that for any other object Y ∈ Obj[C] with morphisms p1 : Y → A and
p2 : Y → B factor through πA and πB . uniquely. The diagram below should
illustrate this property.
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Example 5. The product of A, B in the category of sets is the cartesian product.
Any other object with maps onto A and B, would have to factor through A × B by
mapping the preimage of A to the copy of A and the preimage of B to the copy of
B.
The product in the category of groups is cartesian product with component wise
multiplication.
The product in the category of topological spaces is the cartesian product with the
topology induced by the projection maps.
In other words the coarsest topology ( that with the least number of open sets) which
makes the projection maps onto A and B continuous.
2.1.2. Coproduct. The coproduct, or the sum, is the universally repelling object in
the category with objects X with morphisms from f1 : A → X and f2 : B → X
such that for any other object Z with maps h1 : A → Z and h2 : B → Z there
exists a unique map g : X → Z such that the whole diagram commutes.
The following is a commutative diagram which illustrates my point of view.
Sometimes this is called the sum.
Example 6. The coproduct in the category of sets is the disjoint union.
The coproduct in the category of group is the amalgamated free product. What
does this mean?
The coproduct in the category of topological spaces is the
2.1.3. Free Group. Let X be any set. Let C be the category whose objects are
groups along with an injective function j : X → G. Then the universally repelling
object in this group is called the free group on X. All other groups with |X|
generators, would have to be aa quotient of the free group on |X| generators.
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3. Functors
A functor is a map between categories. It should send objects to objects and
morphisms to morphisms, with some extra conditions. We shall investigate what
these will be.
3.1. Covariant Functors. Let C and D be categories.
T : C → D is a covariant functor if and only if
(F1)
(F2)
(F3)
(F4)
If A ∈ Obj[C] then T (A) ∈ Obj[D].
If f ∈ Mor(A, B) then T (f ) ∈ Mor(T (A), T (B)).
If 1A is the identity on A, then T (1A ) = 1T (A)
T (f ◦ g) = T (f ) ◦ T (g)
A −−−−→
f

yT
B

yT
T (A) −−−−→ T (B)
g
Example 7. The Fundamental Group is a covariant functor from the category of
pointed topological spaces.
For each pointed topological space, one can assign a group of closed paths modulo
the equivalence relation defined by homotopy. In other words, if we let Λx0 be the
collection of all closed loops based at x0 , and ≡ defined by λ1 ≡ λ2 if λ1 is homotopic to λ2 . This forms a group under concatenation.
If λ : [0, 1] → X and γ : [0, 1] → X
(
λγ(t) =
λ( 12 t)
γ( 12 + 12 t)
t ∈ [0, 21 ]
t ∈ ( 21 , 1]
The inverse of λ(t) is λ(1−t) and associativity is easy to see, but painful to prove.
Morphisms in the category of topological spaces are continuous functions. So it
is necessary to see that they induce group homomorphisms.
Let f : X → Y be a continuous function. Then for each γinπ1 (X) we can assign
a closed path in Y by
f∗ : π1 (X) → π1 (Y )
and
f∗ (γ) = f ◦ γ
From this definition, it is to see f∗ (1X ) = 1f (X) . The constant path will definitely
act as the identity.
All that is left to see is that
f∗ (γλ) = f∗ (γ)f∗ (λ)
Example 8. The forgetful functor is also covariant. An example of a forgetful
functor would be the one from the category of groups and group homomorphisms
and sets and functions by forgetting the group structure on the set.
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Example 9. A covariant functor from the category of sets and functions, to the
category if free groups by assigning to each set, the free group generated by that set.
3.2. Contravariant Functors. A contravariant functor is defined almost exactly
the same way.
Let C and D be categories.
T : C → D is a covariant functor if and only if
(F1) If A ∈ Obj[C] then T (A) ∈ Obj[D].
(F2*) If f ∈ Mor(B, A) then T (f ) ∈ Mor(T (A), T (B)).
(F3) If 1A is the identity on A, then T (1A ) = 1T (A)
(F4) T (f ◦ g) = T (f ) ◦ T (g)
Note that the only difference is that the morphisms go in the opposite direction
after they pass through the functor.
Let C and D be categories.
T : C → D is a covariant functor if and only if
(F1) If A ∈ Obj[C] then T (A) ∈ Obj[D].
(F2) If f ∈ Mor(A, B) then T (f ) ∈ Mor(T (A), T (B)).
(F3) If 1A is the identity on A, then T (1A ) = 1T (A)
(F4) T (f ◦ g) = T (f ) ◦ T (g)
A −−−−→
f

yT
B

yT
T (A) ←−−−− T (B)
g
Example 10. A ring is a set with two operations, R(+, ∗).
It is a group under addition, and it has a multiplication.
The multiplication must satisfy
(R1) There exists a 1 ∈ R such that 1 ∗ r = r for all r ∈ R.
(R2) (r1 ∗ r2 ) ∗ r3 = r1 ∗ (r2 ∗ r3 )
(R3) r1 (r1 + r2 ) = r1 ∗ r2 + r1 ∗ r3 ( the same holds for left distribution.
The set of continuous real valued functions is a ring. You can add continuous real
valued functions and multiply them and they satisfy the above properties. They
do not, however, have a multiplicative inverse without losing continuity. Define
functor from the category topological spaces (compact and Hausdorff ) to Rings by
assigning to each topological space X, the ring of continuous functions on it.
This is a contravariant functor, because if we let f : X → Y be continuous.
Then for any p : Y → R continuous, we can assign f ∗ (p) : p ◦ f : X → R
which is also continuous. This is called the pull back function, and as you can see
f ∗ : Mor(X, Y ) → M or(F (Y ), F (X)).
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