Simplicial Structures in Topology

III.6 Homology of the Product of Two Polyhedra 141
be such that:
1. For every n ≥ 0, there is a set Mn of objects M i n ∈ C, i ∈ Jn (Jn is a set of
indexes).
2. For each M i n there is an element xi n ∈ F(Mi n )n such that, for every X ∈ C,theset
{F( f )n(x i n )| f ∈ C(Mi n ,X),i ∈ Jn}
is a basis for the free Abelian group F(X)n in the chain F(X).
Such a functor (if any exists!) is a free functor with models M = {Mn|n ≥ 0} and
universal elements {xi n|i ∈ Jn,n ≥ 0}.
We show the existence of free functors with models and universal elements by
means of an example.
Example. Starting from the category Csim of simplicial complexes, we consider
the functor
C : Csim −→ Clp
that takes each simplicial complex K to the augmented chain complex
C(K)={Cn(K;Z),∂n}.
For every n ≥ 0, we choose an n-simplex σn (fixed) and the corresponding oriented
simplicial complex σn; wethenset
Mn = {σn}
(the set Jn has therefore only one element); these are the sets of models. We now
look for the universal elements. For each n ≥ 0, let xn ∈ Cn(σn,Z) be the generator
represented by the oriented simplex σn (the only n-simplex of the simplicial
complex σn). For every model σn, the only simplicial functions f : σn →
K, which produce nontrivial homomorphisms are precisely the bijective simplicial
functions that take σn into the various oriented n-simplexes of K; hence, the
set {C( f )(xn)} contains all oriented n-simplexes of K and is therefore a basis of
Cn(K;Z).
Before we go over other examples, let us define the tensor product of two chain
complexes. Given C,C ′ ∈ C arbitrarily, we define the Abelian group
for every n ≥ 0, and the homomorphism
(C ⊗C ′ )n = �
Ci ⊗C
i+ j=n
′ j ,
d ⊗ n : (C ⊗C′ )n −→ (C ⊗C ′ )n−1
for every n ≥ 1 such that, for every xi ∈ Ci and y j ∈ C ′ j ,
d ⊗ i+ j (xi ⊗ y j)=∂i(x) ⊗ y j +(−1) i xi ⊗ ∂ ′ j (y j).