Timeline for Are manifolds "naturally" ringed or locally ringed spaces?
Current License: CC BY-SA 4.0
11 events
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| Oct 22, 2021 at 13:53 | comment | added | Gabriel | I need to verify carefully, but I'm pretty sure that the inclusion functor to ringed spaces preserves limits and colimits. (That's what I've said in the post.) So, from this standpoint, it's perhaps better to see manifolds as ringed spaces. (But there's also the other point of view discussed in the post.) | |
| Oct 22, 2021 at 9:26 | comment | added | David Roberts♦ | Considering limits, the limit of manifolds, if it exists, is preserved by the underlying topological space functor. The forgetful functor from ringed spaces to topological spaces preserves limits, so this case is less clear. | |
| Oct 22, 2021 at 9:17 | comment | added | David Roberts♦ | And, as it happens, the inclusion of locally ringed spaces into ringed spaces preserves colimits, as it is a left adjoint. So if colimits of manifolds considered as locally ringed spaces gives something different to when computed in the category of manifolds, then doing it in ringed spaces isn't going to help. | |
| Oct 22, 2021 at 9:12 | comment | added | David Roberts♦ | @Gabriel I mean that if you are unhappy with the inclusion of manifolds as locally ringed spaces not preserving (co)limits, then it might be the case that the inclusion functor to ringed spaces, even if it is fully faithful, might also not preserve (co)limits. | |
| Oct 22, 2021 at 8:52 | comment | added | Gabriel | @DavidRoberts would you mind explaining more? | |
| Oct 21, 2021 at 20:53 | comment | added | David Roberts♦ | It might be neither... :-) | |
| Oct 21, 2021 at 15:45 | comment | added | Z. M | In fact, you don't even need the sheaf structure. All (paracompact) manifolds are "affine" in some sense. See math.stackexchange.com/q/1764947 | |
| Oct 21, 2021 at 15:01 | comment | added | Gabriel | Being really explicit, I asked a technical question (is $\mathsf{Man}$ a full subcategory of $\mathsf{LRS}/\mathbb{R}$?) and a heuristic question (should we think about manifolds as ringed or locally ringed spaces?). | |
| Oct 21, 2021 at 15:00 | comment | added | Gabriel | @DavidRoberts Sure! That happens in this case. But if we're going to see manifolds as (locally) ringed spaces, it is desirable to be able to compute (co)limits in the larger category (where they always exist). Isn't it? | |
| Oct 21, 2021 at 11:42 | comment | added | David Roberts♦ | It's quite possible to be a full subcategory but the inclusion functor to not preserve (co)limits. | |
| Oct 21, 2021 at 9:02 | history | asked | Gabriel | CC BY-SA 4.0 |