4

My question is about whether it's more natural to see manifolds as ringed spaces or as locally ringed spaces. I think I have arguments for both points of view.

On the one hand, it's reasonable to expect functions that do not vanish at some point $p$ to be invertible in a neighbourhood of $p$. This is precisely saying that the stalks of the structure sheaf should be local rings.

On the other hand, limits and colimits of manifolds (when they exist), coincide with those taken in the category of ringed spaces, but not with those in the category of locally ringed spaces.

Another possible point is that I know that the category of manifolds is a naturally a full subcategory of $\mathsf{LRS}/\mathbb{R}$ but I'm not sure if it's also a full subcategory of $\mathsf{RS}/\mathbb{R}$.

Gabriel
  • 943
  • 3
    It's quite possible to be a full subcategory but the inclusion functor to not preserve (co)limits. – David Roberts Oct 21 '21 at 11:42
  • @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? – Gabriel Oct 21 '21 at 15:00
  • 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?). – Gabriel Oct 21 '21 at 15:01
  • In fact, you don't even need the sheaf structure. All (paracompact) manifolds are "affine" in some sense. See https://math.stackexchange.com/q/1764947 – Z. M Oct 21 '21 at 15:45
  • It might be neither... :-) – David Roberts Oct 21 '21 at 20:53
  • @DavidRoberts would you mind explaining more? – Gabriel Oct 22 '21 at 08:52
  • @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. – David Roberts Oct 22 '21 at 09:12
  • 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. – David Roberts Oct 22 '21 at 09:17
  • 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. – David Roberts Oct 22 '21 at 09:26
  • 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.) – Gabriel Oct 22 '21 at 13:53

0 Answers0