It would be nice to allow forgetful functors for manifolds. In particular:

• differentiable manifolds -> topological manifolds
• (Pseudo-)Riemannian manifolds -> differentiable manifolds (not a category yet)
• topological manifolds -> sets (?)

and probably even more. The last point, if getting to work somehow, might also be useful and/or should be seen in view of #30832.

At the moment, we have the following behavior (which could be seen as a bug):

sage: from sage.categories.functor import ForgetfulFunctor
sage: from sage.categories.manifolds import Manifolds
sage: F = ForgetfulFunctor(Manifolds(RR).Differentiable(), Manifolds(RR).Topological())
sage: M = Manifold(2, 'M')
sage: F(M)
2-dimensional differentiable manifold M

Depends on #31247

CC: @tscrim @egourgoulhon @mkoeppe

Component: manifolds

Issue created by migration from https://trac.sagemath.org/ticket/31241