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Stream: learning: questions

Topic: Descent, Cohomology, and Local-Global

John Onstead (Jul 31 2024 at 06:34):

I want to start a new thread to continue a quest of mine to understand local-global correspondences in the most general form. My goal is to eventually develop a suitable definition for a "local-global structure" on an object in a general category. The previous posts were this one exploring the world of resolutions, and this one which was a side quest from that to understand the Dold-Kan correspondence. This stemmed from this post talking about effective descent.

With that out of the way, I want to begin by talking about what a "local object" is. The n-lab defines a local object as something that is defined with respect to some class of morphisms in a category, so it is not even an absolute notion. An object c is then local with respect to this morphism class if every morphism in this class is sent to an isomorphisms in Set by Hom(-, c). What I want to know is a few things. First, is there any way to "unpack" this definition so that I can see more clearly what it is trying to "do"? Second, what is the motivation of this definition- is there some motivating concrete example? (I will get to sheaves and presheaves soon, so here I'm more interested in if there's examples of local objects that have absolutely nothing to do with sheaves). Thirdly and most importantly, why is this object "local"? What, if anything, does this definition have to do with locality?

John Baez (Jul 31 2024 at 09:58):

For what it's worth, this concept is a generalization of [[localization]] in ring theory, where we take a ring and get a new ring by decreeing that a certain set of elements are invertible - i.e., by throwing in formal inverses of these elements. In algebraic geometry this is a way of chopping out part of an affine scheme and thus working "locally" in the rest of this affine scheme - that's the origin of the term "local". But can do localization not just for rings, for any monoid. And then you can do it for it categories. And eventually it becomes what you're talking about.

John Onstead (Aug 01 2024 at 00:14):

Thanks for the background, I'll start reading up on these concepts. I'll be back when I have any questions or to clarify my understanding!

John Onstead (Aug 24 2024 at 06:27):

Hi, I've finally gone through everything and learned all about about ring localizations, localizations of categories, reflective localizations, etc. Apologies it's taken so long, but it's been difficult to learn them! I can also see how ring, monoid, and category localizations connect, since monoids and rings can be seen as one object categories and preadditive categories respectively. But localizing a ring only has the interpretation of "locality" when you take the geometric perspective as you mention (or the one I commonly see, which is in terms of the spectrum of a ring where localizing at a prime ideal is like "zooming into" a point). If this is the case, then I'm still left wondering how I am supposed to interpret a "local object" in a category.

I've done some thinking about this, but the most I've come up with is just based on the definition of a local object. The hom functor, if I understand it, sends a morphism f from a to b to a function Hom(b, c) -> Hom(a, c) that represents composition with f. If this is a bijection for some morphism f, then it means any morphism from a to c can be expressed as a composition of some morphism from b to c composed with f. If c is W-local, then this property holds for all morphisms f in W. I still don't understand what exactly this "means" or is telling us. Maybe a concrete example of a W-local object would help?

John Baez (Aug 24 2024 at 09:25):

It's good to hear you've learned all that stuff about localization!

I'm much more familiar with the localization of a category, which has many important applications, than [[local objects]], so I would tend to think of the latter as a spinoff of studying the former.

I think people don't always take an explicitly geometrical perspective on localization of categories (or local objects). But it should be possible in examples. How about checking out what it amounts to if we start with a category of sheaves on some space $X$, and then localize it by inverting morphisms that become isomorphisms when restricted to some open $U \subseteq X$. Maybe we can figure out which objects are local in this case.

Todd Trimble (Aug 24 2024 at 11:57):

Along lines similar to what John suggests, one source of examples is sheaves as local objects in a category of presheaves. This is relatively easy to explain in the classical case of (set-valued) sheaves over a topological space, but it readily generalizes to the fancier case of sheaves with respect to a Grothendieck topology.

Recall that a presheaf on a topological space $X$ is a functor $\mathcal{O}(X)^{op} \to \mathsf{Set}$ where $\mathcal{O}(X)$ is the category of open subsets and inclusions between them. A morphism between presheaves is a natural transformation between such functors. For the class $S$ of presheaf morphisms you want to localize with respect to, take inclusions of the form

$C \hookrightarrow \hom(-, U)$

where $C$ is the presheaf obtained as a coequalizer attached to a covering $\{U_i\}$ of a given open set $U$:

$\sum_{i, j} \hom(-, U_i \cap U_j) \rightrightarrows \sum_i \hom(-, U_i) \to C.$

In case it's unclear what those two parallel arrows are, they are defined so that their restrictions to a typical summand $\hom(-, U_i \cap U_j)$ are induced by the inclusions $U_i \cap U_j \hookrightarrow U_i$, $U_i \cap U_j \hookrightarrow U_j$:

$\hom(-, U_i \cap U_j) \hookrightarrow \hom(-, U_i) \hookrightarrow \sum_i \hom(-, U_i),$

$\hom(-, U_i \cap U_j) \hookrightarrow \hom(-, U_j) \hookrightarrow \sum_i \hom(-, U_i).$

What's going on here is that given a presheaf $F$, a natural transformation $\sigma: C \to F$ is precisely tantamount to a compatible family of sections $s_i \in F(U_i)$. The $s_i$ arise from the composition

$\sum_i \hom(-, U_i) \to C \overset{\sigma}{\to} F$

whose restriction to a typical summand $\hom(-, U_i)$ is a transformation $\hom(-, U_i) \to F$; by the Yoneda lemma, this corresponds to an element $s_i \in F(U_i)$. The fact that the map $\sum_i \hom(-, U_i) \to C \overset{\sigma}{\to} F$ must coequalize the two parallel arrows gives you precisely the compatibility condition.

Thus, the sheaf condition -- for all coverings $\{U_i\}$ of open sets $U$, the set $F(U)$ is the equalizer

$F(U) \to \prod_i F(U_i) \rightrightarrows \prod_{i, j} F(U_i \cap U_j)$

-- is neatly encapsulated by realizing that a priori, this equalizer is really the set of transformations $[\mathcal{O}(X)^{op}, \mathsf{Set}](C, F)$, and $F(U) \cong [\mathcal{O}(X)^{op}, \mathsf{Set}](\hom(-, U), F)$ (again by Yoneda), so the sheaf condition is saying that homming any $S$-morphism $C \to \hom(-, U)$ into $F$ produces an isomorphism

$[\mathcal{O}(X)^{op}, \mathsf{Set}](\hom(-, U), F) \to [\mathcal{O}(X)^{op}, \mathsf{Set}](C, F)$.

(In the more general case of a Grothendieck topology $j: \Omega \to \Omega$, the sheaf condition similarly says that sheaves are local objects with respect to "$j$-dense inclusions". It takes a bit of mental energy to work through this.)

John Onstead (Aug 24 2024 at 22:06):

Thanks for the help, I'll try to think through all this. Hopefully it won't take as long as with learning about ring localization!

John Onstead (Aug 31 2024 at 23:26):

John Baez said:

How about checking out what it amounts to if we start with a category of sheaves on some space X, and then localize it by inverting morphisms that become isomorphisms when restricted to some open U⊆X. Maybe we can figure out which objects are local in this case.

I'm not sure how I would work through this with symbols and all that. But if you invert morphisms that would become isomorphisms when restricted to some open set, then it seems you are doing this very restriction. So my guess is that the local objects in this case are the sheaves on the open U? If so then it does make geometric sense to call this "localization", since you are going from sheaves on the whole space X down to sheaves on a part of X given by U. Let me know if I'm on the right track here! Also, if this is a reflective localization, this would imply that every category of sheaves on an open set is a reflective subcategory of the overall category of sheaves. Is this true?

Todd Trimble said:

one source of examples is sheaves as local objects in a category of presheaves.

This is actually why I'm trying to learn more about local objects in the first place, so I can have a better intuitive understanding of sheaves. On the surface it makes sense that sheaves are local objects in a presheaf category, since sheafification exhibits the category of sheaves as a reflective subcategory of the category of presheaves -> all reflective subcategories are examples of reflective localizations -> therefore sheafification must be a reflective localization. But from the following explanation it seems that it is no trivial task to identify the class S of presheaf morphisms one must localize with respect to. From what it looks like, and I could be getting this wrong, the statement of sheaves in terms of being local objects really is just a convoluted way of restating the usual definition of a sheaf in terms of the equalizer condition. So maybe realizing sheaves as local objects really doesn't add any additional intuition to what is going on?

John Onstead (Aug 31 2024 at 23:34):

That said, going over the "descent" page, it seems one can more naturally express the statement of sheaves in terms of being local objects in terms of a descent condition. I think this is where I might find my intuitive explanation for sheaves and their definition. So I think I will start heading in that direction.

Now, I will be going on a vacation for the next week so I won't be able to view or respond to messages here. But please leave resources and any information you think is relevant to descent and its relation to sheaves and I will be sure to look at it when I get back!

(Also, I've recently been getting into learning more about fiber bundles since for some reason fiber bundles are much easier for me to intuitively understand than sheaves, and the two are related because any sheaf can be given as a fiber bundle as per the discussion of Baez's topos theory blog. More particularly, I realize that many fundamental concepts in mathematical physics can be given in terms of fiber bundles and I'm wondering if fiber bundles might be "the" common language for discussing mathematical physics. So at some point I might start a discussion of fiber bundles, maybe after completing this train of thought!)

John Baez (Sep 01 2024 at 21:51):

John Onstead said:

John Baez said:

How about checking out what it amounts to if we start with a category of sheaves on some space X, and then localize it by inverting morphisms that become isomorphisms when restricted to some open U⊆X. Maybe we can figure out which objects are local in this case.

I'm not sure how I would work through this with symbols and all that. But if you invert morphisms that would become isomorphisms when restricted to some open set, then it seems you are doing this very restriction. So my guess is that the local objects in this case are the sheaves on the open U? If so then it does make geometric sense to call this "localization", since you are going from sheaves on the whole space X down to sheaves on a part of X given by U. Let me know if I'm on the right track here!

That would be my guess too. It should be straightforward to check this, but since I'm feeling too lazy to do so, maybe some expert could weigh in and tell us if this is correct.

Also, if this is a reflective localization, this would imply that every category of sheaves on an open set is a reflective subcategory of the overall category of sheaves. Is this true?

Maybe we're guessing that the localization discussed above is naturally isomorphic to the inverse image functor

$i^{-1} : \mathsf{Sh}(X) \to \mathsf{Sh}(U)$

where

$i: U \to X$

is the inclusion. The inverse image is left adjoint to the direct image

$i_* : \mathsf{Sh}(U) \to \mathsf{Sh}(X)$

So if you want $\mathsf{Sh}(U)$ to be a reflective subcategory of $\mathsf{Sh}(X)$, you just need the direct image functor $i_*$ to be fully faithful. Right? I tend to get these things backwards, so don't trust me.

It feels fully faithful to me!

Peva Blanchard (Sep 01 2024 at 22:02):

What a coincidence, I've just watched this lecture (Laurent Lafforgue, on topos theory).

He does spend some time discussing inclusion of topos. It is defined as a geometric morphism $(f^*, f_*) : \mathcal{E}_U \to \mathcal{E}_X$ such that the direct image functor $f_*$ is fully faithful.

He justifies this definition by mentioning the case of sheaf toposes of topological spaces: an inclusion of topological space $X \subseteq Y$ yields a geometric morphism $\text{Sh}(X) \to \text{Sh}(Y)$ whose direct image functor is fully faithful.