Post #1888487
2026-04-27 19:55 UTC
@jdw writes "Is there a notion of surjectivity for morphisms of locales".
Yes, and this t is an important notion.
In fact, my colleague Steve Vickers has argued that surjectivity behaves better for locales than for topological spaces, when you work constructively.
Here is one example:
https://sjvickers.github.io/papersfull.html#IntervalNote
But he also gave more examples in his 7WFTop talk in Venice recently.
Replies (1)
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@jdw@mathstodon.xyz 2026-04-27 20:09
@MartinEscardo @jonmsterling Thanks to both of you! In classical algebraic geometry, morphisms of schemes are called surjective if they are surjective on the underlying topological spaces. So for a map of spectra Spec(R) -> Spec(S) this means that prime ideals can be lifted along a ring homomorphism. A basic theorem is that morphisms induced by finite ring extensions are surjective (geometrically they correspond to some sort of finite coverings). I will try and see if localic surjectivity is a good replacement here.