Download Functors and natural transformations A covariant functor F : C → D is

Functors and natural transformations
A covariant functor F : C → D is a set map
F : Ob(C) → Ob(D) together with a set map
F : M or(A, B) → M or(F (A), F (B)) for each
morphism f : A → B in C such that
1. F (idA) = idF (A).
2. F respects composition of morphisms, i.e.
F (f ◦ g) = F (f ) ◦ F (g)
if g : T → A and f : A → B are morphisms
in C.
Thus covariant functors preserve the direction
of arrows: f : A → B goes to F (f ) : F (A) →
F (B).
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A contravariant functor H : C → D reverses the
direction of arrows. Thus one has a set map
H : Ob(C) → Ob(D) together with a set map
F : M or(A, B) → M or(F (B), F (A)) for each
morphism f : A → B in C such that
1. F (idA) = idF (A).
2. F respects composition of morphisms, i.e.
F (f ◦ g) = F (g) ◦ F (f )
if g : T → A and f : A → B are morphisms
in C, where F (g) : F (A) → F (T ) and F (f ) :
F (B) → F (A).
Example 1: Forgetful functors come from forgetting extra structure required for an object
of D to give an object of C.
For example, if C is the category of groups and
D is the category of sets, we have a forgetful
functor F : C → D defined by letting F (A) be
the underlying set of a group A. Given a group
homomorphism f : A → B we just let F (f ) be
the underlying set map.
Similarly, one has a forgetful functor from the
category of vector spaces over a given field L
and the category of abelian groups.
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Example 2: The category T of pointed topological spaces consists of pairs (X, x) in which
X is a topological space and x ∈ X. A morphism f : (X, x) → (Y, y) is a continuous map
f : X → Y such that f (x) = y. The coprod`
uct (X, x) (X 0, x0) is the so-called connected
sum, which is the quotient of the disjoint union
` 0
X X by the equivalence relation generated by
setting x∼x0.
Is there a product in the category T ?
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There is a covariant functor F : T → Groups
defined by
F (X, x) → π1(X, x)
If f : (X, x) → (Y, y) is a morphism in T , then
F (f ) : π1(X, x) → π1(Y, Y ) results from taking
images under f of loops based at x.
The Van-Kampen theorem says F respects coproducts:
F ((X, x)
a
0
0
(X , x )) = π1(X, x)
T
a
π1(X 0, x0).
Groups
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Example 3: Representation functors
Suppose C is a category and X is an object of C.
One gets covariant and contravariant functors
FA : C → Sets and F A : C → Sets defined by
FA(B) = M or(A, B)
and
F A(B) = M or(B, A)
If f : B → B 0 is a morphism in C,
FA(f ) : M or(A, B) → M or(A, B 0 )
is just the composition h → f ◦ h. Similarly,
F A(f ) : M or(B 0, A) → M or(B, A) is the composition
h→h◦f
If the morphisms M or(A, B) have additional
structure, we can do better than using Sets
as the range. For example, if C is the category
of abelian groups, we can replace Sets by C.
Problem: What is FZ : Groups → Groups?
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Natural transformations, isomorphisms
and equivalences
Suppose F and J are covariant functors from
C to D.
A natural transformation (or morphism of functors ) from F to J is a map which assigns to
each object A of C a morphism
ηA ∈ M orD (F (A), J(A))
such that the following is true. For each pair
of objects A, B of C and each f ∈ M orC (A, B),
there is a commutative diagram in D
F (A)
ηA
−→
F (f )↓
F (B)
J(A)
↓J(f )
ηB
−→
.
(1)
J(B)
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If each ηA is an isomorphism, say η is an isomoprhism of functors.
One says that C and D are isomorphic categories if there are functors F : C → D and
H : C → D such that the compositions H ◦ F
and F ◦ H are the identity functors.
One says that C and D are equivalent categories if there are functors F : C → D and
H : C → D such that the compositions H ◦ F
and F ◦ H are naturally isomorphic to the identity functors.
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Intuition for Isomorphisms of categories:
Isomorphic categories C and D are the same
up to relabelling specified by F : C → D and
H : D → C.
In order for H ◦ F = IdC and F ◦ H = IdD , the
functors F and H have to induce a bijection
on objects.
Further, for objects A and B of C, the map
F : M or(A, B) → M or(F (A), F (B))
is a bijection.
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Intuition for Equivalences of Categories:
Suppose F : C → D and H : D → A define
an equivalence of categories. Then F (A) is
a “model” of F (A) inside D, in the following
sense.
The composition
H ◦F :C →C
is naturally isomorphic to IdC via some
η : H ◦ F → IdC .
Thus
η : H(F (A)) → IdC (A) = A
is an isomorphism in C (but not necessarily the
identity map).
Example: One of the homework problems is
to show that if G is a group, then the category
of C of all finite G-sets is equivalent by not
isomorphic to the category D of all finite Gsets contained in Z.
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Applications of representation functors
Various problems in group theory and geometry can be phrased by asking whether a covariant functor
T : C → Sets
is isomorphic to a the representation functor
FA : C → Sets
defined by FA(∗) = M orC (A, ∗) for some object
A of C. Such an isomorphism comes about
from compatible isomorphisms (= set bijections)
ηB : T (B) → FA(B) = M orC (A, B)
for all objects B of C.
In this way, one can reconstruct T from the
arrows in C from A to the other objects of C.
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Example: Suppose T : Groups → Sets sends a
group B to the set of elements of B of order
1 or 2. This is a functor, because a group
homomorphism B → B 0 sends such elements
to elements of order 1 or 2.
Let’s check that T is represented by A = Z/2.
We need a bijection
ηB : T (B) → FZ/2(B) = HomGroups (Z/2, B)
for all B. One can let η(b) be the homomorphism which sends the non-trivial element of
Z/2 to b.
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Example: Classifying spaces.
A topological group G is a group with a topology such that the group law and the map giving inverses are continuous maps.
For example, one can always take the discrete
topology on G, in which every set is open.
The real topology on GL(n, R) makes this a
topological group.
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A principal G-bundle is a continuous surjection
of topological spaces f : E → B with these
properties:
1. G acts continuously on E and trivially on
B
2. f respects this action. So
f (gα) = gf (α) = f (α)
for G ∈ G and α ∈ E.
3. Each b ∈ B has a neighborhood Ub over
which f is trivial, in the following sense.
f −1(Ub) is isomorphic to G × Ub as a topological G-space.
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For example suppose E is the unit two sphere
S 2 in R3, and B = RP 2 is the real projective
plane. So points of B are equivalence classes
{x, −x} = [x] of x ∈ S 2 under the antipodal
map.
Define f : S 2 → RP 2 by
x → [x] = {x, −x}
Then f is a principal G = Z/2 bundle which
is not trivial (i.e. not G-isomorphic to RP 2 ×
Z/2.)
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We now fix G. We have a contravariant functor
FG : Topological spaces = T → Sets
defined by
FG(B) = isomophism classes of topological
G − bundles overB
For B 0 → B, one constructs F (B) → F (B 0 ) by
pulling back G-bundles on B to G-bundles on
B 0.
Exercise: Given a G-bundle E → B and a continuous map B 0 → B, construct the pullback
G-bundle E 0 → B 0 having the appropriate universal property.
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For W a topological space, let [B, W ] be the
set of homotopy classes of maps f : B → W .
Theorem For each topological group G, there
is a space BG called the classifying space of
G and an isomorphism of functors between FG
and the functor B → [B, BG].
The homotopy class of the identity map BG →
BG then gives a “universal” G-bundle EG →
BG. For all topological G-bundles E → B,
there is a continuous funtion t : B → BG unique
up to homotopy for which E is G isomorphic
to the pullback of EG via t.
For example, if G = Z/2, then BG = RP ∞ is
the union of over k the quotients RP k of the ksphere S k by the antipode map x → −x, where
one puts these spheres in bigger and bigger
real Euclidean spaces.
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