Simplicial Structures in Topology

30 I Fundamental Concepts
The quotient set obtained from Top(X,Y ) and the relation ∼ is denoted by [X,Y ];
its elements are equivalence classes for homotopy of maps, or homotopy classes of
maps from X to Y.
The category HTop,thehomotopy category associated with Top, has topological
spaces for objects and its morphisms are homotopy classes of maps: the set of
morphisms is, therefore,
HTop(X,Y)=[X,Y].
When dealing with based maps f ,g ∈ Top ∗((X,x0),(Y,y0)), we say that f ∼ g if
there is a map H : X × I → Y such that
(∀t ∈ I) H(x0,t)=y0 , H(−,0)= f , and H(−,1)=g.
In this case, we use the notation [X,Y]∗ = Top ∗(X,Y)/∼.
Relative homotopy in CTop is a useful concept: two maps of pairs f ,g: (X,A) →
(Y,B) are homotopic relative to X ′ ⊂ X if f |X ′ = g|X ′ and there exists a map
such that
H : (X × I,A × I) −→ (Y,B)
(∀x ∈ X) H(x,0)= f (x) H(x,1)=g(x)
(∀x ∈ X ′ , ∀t ∈ I) H(x,t)= f (x)=g(x).
This being the case, we denote the relative homotopy from f to g with
f ∼ relX ′ g
and we say that f is homotopic to g rel X ′ . If X ′ = /0, we have a free homotopy
in CTop. In the category Top ∗ the based homotopy coincides with the homotopy
relative to the base point.
We say that two spaces X,Y ∈ Top are of the same homotopy type (or simply,
type) iftherearemapsf : X → Y and g: Y → X such that gf ∼ 1X and fg∼ 1Y ;
the map f is called a homotopy equivalence.
The “functions” between categories are called functors, which take objects to
objects and morphisms to morphisms. Specifically, a covariant functor or simply
functor
F : C → C ′
is a relation between these two categories such that
1. (∀X ∈ C) F(X) ∈ C ′
2. (∀ f ∈ C(A,B)) F( f ) ∈ C ′ (F(A),F(B))
3. (∀ f ∈ C(A,B))(∀g ∈ C(B,C)) F(gf)=F(g)F( f )
4. (∀A ∈ C) F(1A)=1FA