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Stream: theory: mathematics

Topic: What is the "correct definition" of a stalk?

Jack Jia (Jan 09 2025 at 15:34):

Sorry if the title is misleading, I really meant to ask whether the computation of the stalk should happen at the set level or in an arbitrary category.

Say I am only working with presheaves on topological spaces: $F: \text{Top}^{\text{op}}(X) \rightarrow \mathcal{D}$, where $\mathcal{D}$ is cocomplete. In many books, when people talk about sheaves, they simply do the colimit on the level of sets and state that this colimit also carries the structure of $\mathcal{D}$.

I know that if $\mathcal{D}= \mathsf{Ab}, \mathsf{Vect}_k$ then it does not matter if we take said approach or compute the colimit in these categories directly, since the filtered colimits agree. But what about the category of commutative rings $\mathsf{cRing}$? Do they agree?

I also found something on the stacks project:
https://stacks.math.columbia.edu/tag/007G

Regarding the conditions in the lemma: I think all presheaves on topological spaces are faithful since all hom-sets have size at most 1, but I don't know if all presheaves (or even sheaves) should commute with directed colimits.

So my ultimate question is: Should the stalks be defined on the set level and one would reasonably suspect that the colimit carry other structures, or should the stalks be defined in the target category one choose to work with?

Todd Trimble (Jan 09 2025 at 16:00):

The principled approach would be to take the filtered colimit of the $F(U)$ (ranging over open $U$ containing the given point) in the category $\mathcal{D}$.

But if filtered colimits in $\mathcal{D}$ are preserved and reflected by a "forgetful" functor to $\mathsf{Set}$, then as you say, it's more or less harmless practice to operate at the level of sets, and leave the $\mathcal{D}$-structure to look after itself.

This is the case whenever $\mathcal{D}$ is the category of models for a finitary algebraic theory, including for example the category of commutative rings. I imagine this observation handles most cases that come up in practice.