[TYPES] origin of the term "hom-set"?
Uday S Reddy
u.s.reddy at cs.bham.ac.uk
Wed Nov 5 13:52:15 EST 2014
Derek Dreyer writes:
> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]
> Hi. Does anyone know the origin of the term "hom-set"? It came up
> yesterday in class, and after a little googling, I have not turned up
> the answer. I always assumed it stood for "homomorphism set" (?) but
> even that much I have not been able to verify, and it doesn't explain
> why we are talking about "homomorphisms" as opposed to "morphisms" (or
> "sets" rather than "collections", since the hom-set is not always a
> set). I presume there is some historical reason for this?
Originally, "morphism" was a shortened form of "homomorphism". However,
some mathematicians these days restrict "homomorphism" to mean a
structure-preserving map (as in algebra) whereas morphisms don't have any
such requirement. With that terminology, the two are distinct concepts.
As for why it should be called a "hom-set", recall that Mac Lane works in a
set theory with universes. So, what we call a collection or class is a
"set" for him and what we call a set is a "small set".
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