Simplicial Structures in Topology

III.1 The Category of Polyhedra 101
This line of reasoning allows us to define a function
m+n
ψ : |K|×|L|−→|K × L| , ψ(p,q)=
∑
i=0
(ci − ci−1)zi
which is the inverse of φ. In fact, if we take p ∈|K| and q ∈|L| as before,
|pr1|ψ(p,q)=
m
∑
i=0
γixi .
Let zr < zr+1 < ... < zr+t be the elements zi of ψ(p,q) with xs for first coordinate,
as described by the following table:
vertices coefficients in ψ(p,q)
zr−1 =(xs−1,yα) cr−1 − cr−2
zr =(xs,yα) cr − cr−1
... ...
... ...
... ...
zr+t =(xs,yα+t) cr+t − cr+t−1
zr+t+1 =(xs+1,yα+t) cr+t+1 − cr+t .
We note that the coefficient γs in |pr1|ψ(p,q) is equal to cr+t − cr−1; moreover,
by the definitions of zr−1 and zr, we may conclude that cr−1 = as−1 (similarly,
cr+t = as). It follows that
and so, |pr1|ψ(p,q)=p. The proof of
γs = cr+t − cr−1 = as − as−1 = αs
|pr2|ψ(p,q)=q
is completely similar; these two results show that φψ = 1 |K|×|L|.
Let us now take an element
u =
s
∑
r=0
ζrzr ∈|K × L|
where ∑ s r=0 ζr = 1andz0 < z1 < ... < zs. Let x0,...,xm (respectively, y0,...,yn)
be the distinct vertices that appear as first (respectively, second) coordinates of
z0,...,zs;then
{x0,...,xm}∈Φ , {y0,...,yn}∈Ψ
and besides, z0 =(x0,y0) and zs =(xm,yn). By the definitions given here, we are
able to write the equality
|pr1|(u)=
m
∑
i=0
αixi ,