Simplicial Structures in Topology

I.2 Categories 31
If conditions 2. and 3. are replaced by
2’. (∀ f ∈ C(A,B)) F( f ) ∈ C ′ (F(B),F(A))
3’. (∀ f ∈ C(A,B))(∀g ∈ C(B,C)) F(gf)=F( f )F(g)
we have a contravariant functor.
A very simple example of a (covariant) functor is the forgetful functor D: Top →
Set that merely “forgets” the topological space structure. Here is another example:
Let (X,x0) ∈ Top ∗ be a based space and let F : Top ∗ → Set∗ be the function that
takes each based space (Y,y0) into the set [X,Y]∗ of all homotopy classes of based
maps g: X → Y ; for each morphism f : Y → Z, wedefineF( f ): [X,Y]∗ → [X,Z]∗
as the function that takes any homotopy class [g] ∈ [X,Y ]∗ into [ fg].
We now give an example of contravariant functor. Given a based space (Y,y0),
we define F : Top ∗ → Set∗ with the condition F(X,x0)=[X,Y]∗ for every (X,x0);
here, for every f ∈ Top ∗((X,x0),(Z,z0)), the function F( f ) may only be defined
as F( f )([g]) = [gf] for every based map g: X → Z; notice that, if f ∈
Top ∗((X,x0),(Z,z0)), the arrow F( f ) has the opposite direction to that of f .
We now look into some less simple examples.
The suspension functor
Σ : Top ∗ → Top ∗
is defined on based spaces (X,x0) as the quotient
ΣX =
I × X
I ×{x0}∪∂I × X
where ∂I is the set {0,1}. We shall write either [t,x] or t ∧ x when indicating a
generic element of ΣX; then,
(∀ f ∈ Top ∗(X,Y))(∀t ∧ x ∈ ΣX) Σ( f )(t ∧ x)=t ∧ f (x).
The base point of ΣX is t ∧ x0 = 0 ∧ x = 1 ∧ x.
The behavior of the suspension functor is particularly interesting on spheres; in
fact, the suspension of an n-dimensional sphere is an (n + 1)-dimensional sphere.
This fact will be better understood after studying some maps, which will be useful
also later on. For every n ≥ 0, let S n be the unit n-sphere (it is the boundary ∂D n+1
of the unit (n + 1)-disk D n+1 ⊂ R n+1 ). Let us take the point e0 =(1,0,...,0) as the
base point for both S n and D n+1 . Let us now define the maps
cn : I × S n → D n+1 , (t,x) ↦→ (1 − t)e0 + tx,
i+ : D n+1 → S n+1 � �
, x ↦→ x, 1 −�x�2 �
,
i− : D n+1 → S n+1 � �
, x ↦→ x,− 1 −�x�2 �
and
˙kn+1 : I × S n → S n+1
�
˙kn+1(t,x)=
i+cn(2t,x) , 0 ≤ t ≤ 1
2
i−cn(2 − 2t,x) , 1
2 ≤ t ≤ 1.