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Yesterday's “answer” had to be deleted, because it was no good at all. Let's try to answer a little more fully.

An analogy is when phenomena and concepts that are different in origin develop and manifest themselves in a similar way. Homology is when phenomena and concepts that are similar in origin develop and manifest themselves in different ways.

Example: Petya and Zhenya are not related, but both work as taxi drivers. From the point of view of their profession, they are similar – we generally don't care which of them will give us a ride – but they are not homologous. But Zhora is Zhenya's twin brother, but works as a programmer. Zhora and Zhenya are homologous, since they have a similar origin, but are not analogous.

In the wild, the bat and the pterodactyl have developed an analogy – a flying membrane stretched over the fingers. It happened independently, and therefore-an analogy. And in whales, the legs have disappeared, but a small bone remains, which does not perform any role, but corresponds to a fully functional kneecap in their close relatives – ungulates. In biology, the latter phenomenon is called homology.

*Ponty mode on* Finally, as a mathematician, I cannot but say that for us homology is a covariant functor from the category of any spaces to the category of Abelian groups (or in general to some Abelian category), which in general satisfies the cut – out theorem-in the original all five Eulenberg-Steenrod axioms, but now many of them, such as the axiom of dimension and homotopy, can not always be correctly defined and should not be imposed in principle (otherwise there would be no – theories for C*-algebras and algebraic K-theory, which, however, are examples of the dual concept – cohomology). It is interesting, however, that the concept of homology came to mathematics from biology – that is, roughly speaking, they answer the same question: does this incomprehensible bone in a whale correspond to a kneecap?