Homology theory

Definition
Let $$C$$ be the category of compact polyhedral pairs. A homology theory on $$C$$ is defined as follows.

Data

 * For every nonnegative integer $$n$$, a functor $$H_n: C \to Ab$$ where $$Ab$$ denotes the category of Abelian groups
 * For every positive integer, a natural transformation $$\partial_n: H_n \to H_{n-1} \circ R$$ where $$R$$ is the functor that sends $$(X,A)$$ to $$(A,\emptyset)$$.

Axioms

 * Homotopy axiom: If $$f_0,f_1:(X,A) \to (Y,B)$$ are homotopic, then $$H_n(f_0) = H_n(f_1)$$
 * Exactness axiom: For every pair $$(X,A)$$ with inclusions $$(A,\emptyset) \to (X,\emptyset) \to (X,A)$$, there is a long exact sequence:

$$\ldots \to H_n(A,\emptyset) \to H_n(X,\emptyset) \to H_n(X,A) \to H_{n-1}(A,\emptyset) \to \ldots $$
 * Excision axiom: For every open subset $$U$$ whose closure lies in the interior of $$A$$, the map of homotopy groups induced by the inclusion $$(X - U, A - U) \to (X,A)$$ is an isomorphism
 * Dimension axiom: If $$X$$ is a one-point space, then $$H_n(X,\emptyset)$$ is trivial for all $$n > 0$$. One calls $$H_0(X,\emptyset)$$ the coefficient group of the homology theory.

For a homology theory, the homology of a topological space $$X$$ is defined as the homology of the pair $$(X,\emptyset)$$.