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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 $\mathsf{RS}/\mathbb{R}$.

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    $\begingroup$ It's quite possible to be a full subcategory but the inclusion functor to not preserve (co)limits. $\endgroup$
    – David Roberts ♦
    Commented Oct 21, 2021 at 11:42
  • $\begingroup$ @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? $\endgroup$
    – Gabriel
    Commented Oct 21, 2021 at 15:00
  • $\begingroup$ 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?). $\endgroup$
    – Gabriel
    Commented Oct 21, 2021 at 15:01
  • $\begingroup$ In fact, you don't even need the sheaf structure. All (paracompact) manifolds are "affine" in some sense. See math.stackexchange.com/q/1764947 $\endgroup$
    – Z. M
    Commented Oct 21, 2021 at 15:45
  • $\begingroup$ It might be neither... :-) $\endgroup$
    – David Roberts ♦
    Commented Oct 21, 2021 at 20:53

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