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Not so long ago I asked to clarify what is "homomorphism" and had few nice answers. Now I am kind of reiterating what I've learned already about category theory, and accidentally found following:

People also write $\mathsf{A}(A, B)$ as $\mathsf{Hom}_{\mathsf{A}}(A, B)$. The notation "Hom" stands for homomorphism...

But when I get back to the definition of the $\mathsf{A}(A, B)$, it does not require morphism between objects $A$ and $B$ to be structure-preserving maps; more than that, it does not require morphism to be maps at all, those are just "arrows" and thus any attempt to define "structure-preservation" (without additional details of what kind of objects are, etc.) is senseless.

So I feel confusion, whether the answers I got from the question referenced above are incomplete, or where else am I missing the point? Maybe "homomorphism" has also some informal, historical meaning?

Zazaeil
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1 Answers1

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The notation $\operatorname{Hom}(A,B)$ predates the invention of category theory. In particular, for instance, it was used to denote the group of homomorphisms between two abelian groups $A$ and $B$. So, when categories came along, and people wanted to talk about the set of morphisms between two objects in a general category, it was natural to extend this familiar notation to other settings.

Eric Wofsey
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