Simplicial Structures in Topology

Chapter IV
Cohomology
IV.1 Cohomology with Coefficients in G
In Sect. II.5, we have seen that the homology groups Hn(K;Q) of an oriented
simplicial complex K, with rational coefficients, have the structure of vector spaces
and may therefore be dualized. The possibility of dualizing such vector spaces
led mathematicians to ask whether it was also possible to “dualize” the homology
groups with coefficients in a different Abelian group G. Let us remember that
we used the tensor product to change the coefficients of the homology groups; to
“dualize” homology or change the coefficients of the dualized theory, we use the
functor
Hom(−,G): Ab → Ab
where G is a fixed Abelian group. More precisely, given an Abelian group A, we
define Hom(A,G) to be the Abelian group of all homomorphisms from A to G with
the addition
Hom(A,G) × Hom(A,G)) → Hom(A,G)
defined, for each pair (φ,ψ) and for each a ∈ A,by
(φ + ψ)(a)=φ(a)+ψ(a) .
On morphisms, Hom(−,G) acts as follows: given f : A → A ′ ,
�f = Hom( f ,G): Hom(A ′ ,G) → Hom(A,G) , φ ↦→ φ f .
The homomorphism �f = Hom( f ,G) is called adjoint of f . Note that the functor
is contravariant.
Hom(−,G): Ab → Ab
D.L. Ferrario and R.A. Piccinini, Simplicial Structures in Topology, 151
CMS Books in Mathematics, DOI 10.1007/978-1-4419-7236-1 IV,
© Springer Science+Business Media, LLC 2011