Refine category of RealSet

[mkoeppe]

we should put it in a suitable subcategory of TopologicalSpaces.

Depends on #31880

CC: @mjungmath @egourgoulhon @tscrim

Component: categories

Author: Matthias Koeppe

Branch/Commit: 5b0f85d

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/31877

[mjungmath]

comment:1

Possibly a good idea to consider: #30832 comment:6.

[mkoeppe]

Branch: u/mkoeppe/refine_category_of_realset

[mkoeppe]

New commits:

4b09685

RealSet: Put it in a suitable subcategory of TopologicalSpaces()

[mkoeppe]

Commit: 4b09685

[mkoeppe]

Author: Matthias Koeppe

[mjungmath]

comment:4

I personally don't like adding RealSet to TopologicalSpaces. Imo the difference between RealSet and OpenInterval should be exactly that: seeing them as sets or manifolds (i.e. top. spaces).

Then a forgetful functor would also make more sense.

[mkoeppe]

comment:5

A RealSet instance is a union of finitely many (possibly degenerate) intervals, so it is always a topological subspace of the real line, but not always a manifold. It would belong to the category of manifolds with corners of mixed dimension, but I think it's a long way until we have implemented that.

[tscrim]

comment:6

I think this is good. As Matthias said, they are naturally topological spaces from the subspace topology. The only question I have is why is ambient() a @staticmethod? I feel like it should be a normal method because the ambient space is associated with the instance rather than as a constructor function (cf. open_closed).

@mjungmath I don't understand what you mean by forgetful functor. These are parents, and have nothing to do with functors. For example, there is a forgetful functor from Rings -> Sets, and applying that functor to ZZ still obtains ZZ. It still has all of the behavior of an object in Sets, but there is no reason to actually forget about the ring structure.

[tscrim]

Reviewer: Travis Scrimshaw

[mkoeppe]

comment:8

Replying to @tscrim:

The only question I have is why is ambient() a @staticmethod? I feel like it should be a normal method

OK, I'll make this change.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

46eed0e

RealSet.ambient: Change to a normal method

[sagetrac-git]

Changed commit from 4b09685 to 46eed0e

[mkoeppe]

comment:10

Replying to @tscrim:

I don't understand what you mean by forgetful functor. These are parents, and have nothing to do with functors. For example, there is a forgetful functor from Rings -> Sets, and applying that functor to ZZ still obtains ZZ. It still has all of the behavior of an object in Sets, but there is no reason to actually forget about the ring structure.

I think the idea would be that after applying the functor to the parent, asking for example for the homset would give a different answer.

[tscrim]

comment:11

Replying to @mkoeppe:

Replying to @tscrim:

I don't understand what you mean by forgetful functor. These are parents, and have nothing to do with functors. For example, there is a forgetful functor from Rings -> Sets, and applying that functor to ZZ still obtains ZZ. It still has all of the behavior of an object in Sets, but there is no reason to actually forget about the ring structure.

I think the idea would be that after applying the functor to the parent, asking for example for the homset would give a different answer.

IMO, it is better to pass that category information to the homset construction (which you can do) if you want to consider morphisms in a weaker category. This way you have less proliferation of "equal" parents and dealing with elements that maybe shouldn't have the additional structure. In this example, should you be able to add two integers that are in the ZZ as a Sets object? Plus, the category you want for the morphisms is a property of the morphism, not the parent.

[tscrim]

comment:12

Thanks for the change.