[TYPES] origin of the term "hom-set"?

[Uday S Reddy]

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".

Cheers,
Uday