Comment by perihelions
3 months ago
Tangential question that's been nagging me; in spaces of functions defined pointwise, is there a natural way to lift algebraic structures (group/ring/whatever) from the base set up to the function space? It's generally true across many separate objects in math, but I don't know a clear explanation of why. (Category theory, I assume).
Many of the propositions in the author's Appendix A are of this form.
I.e., if you look at how addition on function spaces is defined pointwise, (f+g)(x) = f(x)+g(x) -- that's different meanings of (+) on either side -- that looks exactly like the defining relation of a group homomorphism, except that the symbols are backwards.
Let T^S = {f| f:S->T} be all functions from a source set S to a target structure (set + operations + relations + axioms) T.
You can lift all the operations and relations from T to T^S, and you'll get a structure with the same type signature.
Universal equations involving operations remain true when lifted. Therefore if T is a variety[0], T^S is a variety of the same type.
So for example if S a set with 2 elements, then T^S is TxT + lifted properties. If T is an Abelian group then TxT is also an Abelian group. If T is a ring, TxT is also a ring. If T is a field, TxT is not a field since (0,1) has no inverse.
What about the relations, what types of identities remain true when lifted form T to T^S?
[0] : https://en.wikipedia.org/wiki/Variety_(universal_algebra)
You are observing the evaluation map ev_p: C(M, R) -> R, ev_p(f) = f(p), is a ring homomorphism. In that spirit, you might find this Terry Tao article on the Yondea lemma interesting: https://terrytao.wordpress.com/2023/08/25/yonedas-lemma-as-a...
ianam but I'm not sure there needs to be a "why" -- these things are defined to have certain properties, and so they have those properties. That's the purpose for which they were defined.