3 ms·
TL;DR To objects/sets are equal if and only if there is a canonical isomorphism/bijection between them. What that is is not necessarily clear, but there is an
by k0k0r0 4y ago
TL;DR
To objects/sets are equal if and only if there is a canonical isomorphism/bijection between them. What that is is not necessarily clear, but there is an exception regarding abelian groups accotiated to a field, where the canonical isomorphism is clear up to a minus sign by the main theorem of global class theory[0].
[0] https://en.m.wikipedia.org/wiki/Class_field_theory https://en.m.wikipedia.org/wiki/Class_field_theory
- ur-whale 4y agoWhat is the definition of 'canonical' then? I've often found the label 'canonical' to be rather arbitrary.
- k0k0r0 4y agoExactly that the point. It's not clear. Except for whenever the main theorem of global class theory applies. Then it's clear up to a minus sign.
- Koshkin 4y ago'Canonical' means constructible from the definition (the axioms) of the structure alone. A canonical isomorphism is not necessarily unique.
- k0k0r0 4y agoThat's maybe true in some fields of mathematics. In others that's misleading, and the usual understanding of canonical as unprecisely defined "obvious choice" is definitely more helpful.