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Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

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A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?

A spectrum $E$ defines a homology theory via $E_*(X)=\pi_*(E\wedge X)$. … So the homology theories given by spectra are a bit better behaved than a homology theory as is usually defined. …