3 ms·

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)

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.

dboreham 11mo ago
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.
samasblack 11mo ago
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-an-identification-of-form-and-function-the-case-study-of-polynomials/ https://terrytao.wordpress.com/2023/08/25/yonedas-lemma-as-a...
deleted 11mo ago
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bmacho 11mo ago
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) https://en.wikipedia.org/wiki/Variety_(universal_algebra)

dboreham 11mo ago

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.

samasblack 11mo ago

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-an-identification-of-form-and-function-the-case-study-of-polynomials/ https://terrytao.wordpress.com/2023/08/25/yonedas-lemma-as-a...

deleted 11mo ago

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bmacho 11mo ago

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) https://en.wikipedia.org/wiki/Variety_(universal_algebra)