A formal analogy between flat morphisms and open maps

By applying the functional approach to general topology proposed by Clementino, Giuli, and Tholen, we may make a formal analogy between flat morphisms of affine schemes and open maps of topological spaces. First, the set-up:

  • We are given a category 𝒞 and a class ℱ of morphisms in 𝒞; for convenience, we assume 𝒞 has pullbacks.
  • A proper morphism is a morphism f : XY such that, for any pullback f′ : X′ → Y′ of f : XY along any morphism Y′ → Y, f′ : X′ → Y′ is a member of ℱ.
  • A closed embedding is a monomorphism in 𝒞 that is also a proper morphism.
  • A dominant morphism is a morphism that is left orthogonal to every closed embedding.
  • A open morphism is a morphism f : XY such that pullback along f preserves dominant morphisms in the following sense: for any pullback f′ : X′ → Y′ of f : XY along any morphism Y′ → Y, the pullback of any dominant morphism Y″ → Y′ along f′ : X′ → Y′ is a dominant morphism X″ → X′.

If we take 𝒞 to be the category of topological spaces and ℱ to be the class of closed maps (i.e. continuous maps such that the image of every closed subset is again a closed subset), then these definitions agree with the classical ones. On the other hand, we could take 𝒞 to be the category of affine schemes and ℱ to be the class of closed immersions, in which case the dominant morphisms correspond to injective homomorphisms of commutative rings.

Unfortunately, the analogy ends there: although open immersions of schemes are flat monomorphisms, they are also required to be locally of finite presentation, which is not automatic. There are also some subtleties when trying to apply these definitions to the category of all schemes: although a dominant morphism in the sense above is a morphism whose scheme-theoretic image is the whole codomain, the connection between open morphisms in the sense above and flat morphisms of schemes is much less obvious.

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