Simplicial Structures in Topology

I.2 Categories 29
3. For every pair of morphisms ( f ,g) with f ∈ C ′ (X ′ ,Y ′ ) and g ∈ C ′ (Y ′ ,Z ′ ),
the morphism obtained through composition in C ′ , namely, gf ∈ C ′ (X ′ ,Z ′ ),
coincides with the morphism gf ∈ C(X ′ ,Z ′ ) obtained through composition in C.
A subcategory C ′ of C is called full if, for every pair of objects (X ′ ,Y ′ ) of C ′ ,
the sets C ′ (X ′ ,Y ′ ) and C(X ′ ,Y ′ ) coincide. The category CTop is a subcategory of
Top × Top.
The set Top(X,Y ) of all maps from X to Y has a very important relation called
homotopy. LetI be the closed interval [0,1]. Twomapsf ,g: X → Y are homotopic
when there is a map
H : X × I → Y
such that H(−,0)= f and H(−,1)=g. Iff is homotopic to g, we write f ∼ g.
For instance, the maps f ,g: I → I given by f = 1I (that is to say, ∀x, f (x)=x)
and the constant map g from I to the point 0 ∈ I (in other words, ∀x, f (x) =0) are
homotopic; in fact, by constructing the map
H : I × I → I , (∀s,t ∈ I) H(s,t)=(1 − t)s,
it is clear that H(−,0)= f and H(−,1)=g. A homotopy may be indicated also by
the notation ft : X → Y ,wheret is the parameter t ∈ I and, if H(x,t) is the homotopy
function, then ft(x) := H(x,t). Consequently, f0 ∼ f1.
By Theorem (I.1.37), a homotopy H : X × I → Y from f to g determines a map
ˆH : I → M(X,Y ), thatistosay,apath in M(X,Y ) that links f to g. Conversely,if
X is compact Hausdorff, a path ˆH : I → M(X,Y ) determines a map H : X × I → Y ,
with H(−,0)= f and H(−,1)=g (Theorem (I.1.39)), in other words, a homotopy
from f to g.
(I.2.1) Lemma. The homotopy relation is an equivalence relation.
Proof. Clearly, any map f is homotopic to itself. Suppose now that f ∼ g; this
means that there is a map
consider the map
H : X × I → Y , H(−,0)= f , H(−,1)=g;
H ′ : X × I → Y , (∀x ∈ X)(∀t ∈ I) H ′ (x,t)=H(x,1 − t).
It is immediate to show that H ′ (−,0)=g and H ′ (−,1)= f and so, g ∼ f .
We finally prove that, if f ∼ g and g ∼ h,then f ∼ h. Consider the functions
H : X × I → Y , H(−,0)= f , H(−,1)=g,
G: X × I → Y , G(−,0)=g , G(−,1)=h.
Let us define a map K : X × I → Y by the conditions
�
1 H(x,2t) , 0 ≤ t ≤
(∀x ∈ x)(∀t ∈ I) K(x,t)=
2
G(x,2t − 1) , 1
2 ≤ t ≤ 1.
This map K has the required properties. �