Simplicial Structures in Topology

144 III Homology of Polyhedra
We may now concern ourselves with the Acyclic Models Theorem:
(III.6.2) Theorem. Let C be any category and let two (covariant) functors
F,G: C −→ Clp;
be given; suppose that F has models M = {Mn|n ≥ 0} that are acyclic for the
functor G. Then, for every homomorphism ¯f : Z → Z, there exists a natural transformation
: F −→ G
η ¯f
such that, for each object X ∈ C, η ¯f (X) is an extension of ¯f . Moreover, if τ is
another natural transformation from F to G with the same property held by η,there
exists a natural homotopy of chain complexes E such that
d G E + Ed F = η − τ.
Before we begin the proof of the Acyclic Models Theorem, we observe that the
assertion for each X ∈ C the morphism
η ¯f (X): F(X) −→ G(X)
is an extension of ¯f means that there is a commutative diagram
···F(X)n
η ¯f (X)n
��
···G(X)n
d F n ��
F(X)n−1
η ¯f (X)n−1
d ��
G n ��
G(X)n−1
d F n−1 ��
···
d G n−1 ��
···
d F 1 ��
F(X)0
η ¯f (X)0
d ��
G 1 ��
G(X)0
ε ��
Z
Note that the boundary homomorphisms depend on X; and from now on, for the
sake of a simpler notation, we shall write ηn instead of η ¯f (X)n.
We advise the reader to go back to Sect. II.3 to review the definition of homotopy
among chain complexes and for properly interpreting the last part of the statement
on each given object X ∈ C.
Proof. We begin by building η0. For each M i 0 ∈ M0, there is an element x i 0 ∈
F(M i 0 )0 such that, for every X ∈ C,theset
{F( f )0(x i 0 )| f ∈ C(Mi 0 ,X),i ∈ J0}
is a basis for F(X)0. Sinceε ′ : G(Mi 0 )0 → Z is surjective, for each element ¯f ε(xi 0 ) ∈
Z there exists yi 0 ∈ G(Mi 0 )0 such that
we define
ε ′ (y i 0 )= ¯f ε(x i 0 );
η0(F( f )0(x i 0 )) := G( f )0(y i 0 )
on the generators F( f )0(x i 0 ) and linearly extend η0 to the entire free group F(X)0.
ε ′
¯f
��
��
Z