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Is there somebody who can explain/show me why a sheaf is something that can permit us to move from the local to the global?

An explanation for the layman would be fine. Usually I tend to abhor them, but, being self-thaught and without any guidance, I think that kind of explanation could be a useful starting point to get what to look at during a first reading on the topic (they make something more salient).

Thanks a lot.

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  • $\begingroup$ For the moderators: I am not sure this has been already asked: I looked around, but I did not really find anything. Also, concerning the tags, I added the soft-question one. I hope it is fine. $\endgroup$ – Kolmin Mar 25 '16 at 0:54
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    $\begingroup$ Do you already know the definition of a sheaf? What is your background? $\endgroup$ – Lorenzo Mar 25 '16 at 1:11
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    $\begingroup$ So you are asking specifically about the use of sheaves in algebraic topology? (As opposed to algebraic geometery, or complex geometry, etc.) $\endgroup$ – Lorenzo Mar 25 '16 at 1:21
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    $\begingroup$ A sheaf by definition permits you to move from local to global. $\endgroup$ – Future Mar 25 '16 at 1:56
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    $\begingroup$ Do you want to just confine yourself to sheaves on topological spaces? Or do you want to enter into discussions on Grothendieck coverages? Anyway, for me, the fact that the spectrum functor in algebraic geometry needs the structure of a locally ringed space to be fully faithful was reason enough. This is related to the fact that there is no universal local ring construction in sets, but if we allow rings to be objects of sheaves (over any topological space) there is a universal local ring construction. I think of categories of sheaves as other places to do "mathematics" other than Set. $\endgroup$ – Rachmaninoff Mar 25 '16 at 13:01
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In my understanding, one of the main points of introducing sheaves is to introduce sheaf cohomology. Often this will allow you to produce relationships between vector spaces that you might not have been able to see otherwise, and it can force very useful isomorphisms. If you have seen the long-exact sequence in homology associated to a short exact sequence, know that similar things happen here.

Often (of course not always) one is interested in the global sections of a sheaf, but various tricks can be used to say something about the global sections by means of understanding the higher cohomology groups. For example, a famous result in algebraic geometry uses downward induction from the higher cohomology groups to prove finite generation of global sections of certain sheaves. This result can be proven more directly, but the proof via cohomology is very clean.

Computing cohomology is fairly local in nature (for the initiated: I just mean that when you are computing cohomology via a Cech cover, you are making a bunch of local computations and organizing them into a complex. You do eventually make local computations everywhere, so its not like you are stealing information out of nowhere!) but results that it can force about global sections are global in nature.

I guess something else you may care to hear: On a locally contractible space, computing sheaf cohomology with the constant sheaf computes the usual singular cohomology. So it encompasses a theory that you will probably agree is very powerful.

The point I am trying to make is this: Sheaves can be constructed to keep track of a lot of different things. Sheaf cohomology lets you exploit hidden relationships between those things, often by computing things locally sort off. It is incredibly useful.

(Also, another point of sheaves is to define schemes and abstract algebraic varieties. The main philosophy there is that "the functions on a space define its geometry." Sheaves are a way of keeping track of "the functions on a space," which are especially useful in situations where there are not many global functions - like in algebraic geometry or complex analysis. The only holomorphic functions on a compact Riemann surface are the constant functions. But there are many holomorphic functions defined on open sets that surface. A sheaf keeps track of this data.)

(Another thing: You may wonder why all of these cohomology theories that can be developed on a real smooth manifold agree. (De Rham, Singular, etc.) The argument I have seen (in Warner's Introduction to Differentiable Manifolds) uses sheaves heavily.)

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  • $\begingroup$ Thanks a lot. This gives me a lot of hints! $\endgroup$ – Kolmin Mar 25 '16 at 17:06
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    $\begingroup$ @Kolmin You are welcome! BTW: The beginning of Ravi Vakil's Foundations of Algebraic Geometry gives a very readable introduction to the basic techniques of sheaf theory. There are a lot of technical details in the beginning, but it becomes more intuitive after time. $\endgroup$ – Lorenzo Mar 25 '16 at 17:22
  • $\begingroup$ Thanks a lot for the advice! Actually, looking around on the site I already found some posts mentioning it, and I took a look (rather quick) at it... and here it really comes the question. As I tried to explain when you asked me about my background, this question (with related potential answers) was something I needed in order to get what to look at in a book such as Vakil's, being self-thaught. It's a bit like talking to a friend (or a professor!) about something, and getting from him what is important, what you should care about out of the huge amount of info you usually find in a book. :) $\endgroup$ – Kolmin Mar 26 '16 at 1:05
  • $\begingroup$ @Kolmin Absolutely, I know what you mean. $\endgroup$ – Lorenzo Mar 26 '16 at 2:07
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For sake of simplicity, let $X$ be a topological space and let $\mathcal{C}$ be the sheaf of continuous functions on $X$ with values in $\mathbb{R}$ (or $\mathbb{C}$, it is indifferent) with the natural topology; for clarity: \begin{equation} \forall U\subseteq X\,\text{open},\,\mathcal{C}(U)=\{f:U\to\mathbb{R}\mid f\,\text{is continuous}\}. \end{equation} The stalk of $\mathcal{C}$ in $x\in X$, by definition, is: \begin{equation} \mathcal{C}_x=\lim_{\overrightarrow{x\in U,\,U\subseteq X\,\text{open}}}\mathcal{C}(U); \end{equation} let $f_x\in\mathcal{C}_x$, by construction, $f_x$ codifies the continuity in $x$ of one function $f$ defines at least in $x$.

What mean I? We know that a function $f$ is continuous on $U$ (open subset of $X$) if and only if $f$ is continuous on any point of $U$; therefore if we know which are the continuous functions on all points $x$ in $U$, we can construct the continuous functions on $U$ and only these!

Formally: $\mathcal{C}(U)$ is (obviously) the set of continuous functions on $U$, considering the function \begin{equation} \varphi_U:f\in\mathcal{C}(U)\to(f_x)\in\prod_{x\in U}\mathcal{C}_x; \end{equation} by sheaf axioms, we can prove that $\varphi_U$ is an injective map, that is: a continuous function $f$ on $U$ is uniquely determined by its germs at the points of $U$.

In general $\varphi_U$ is not surjective!

Example: Let $X=\mathbb{R}$ with the natural topology, considering the function: \begin{equation} f(x)=\begin{cases} 0\iff x\in\mathbb{Q}\\ |x|\iff x\notin\mathbb{Q} \end{cases} \end{equation} is continuous only at $0$!, or in complicated way: $f\notin\mathcal{C}(U)$ for any open neighbourhood $U$ of $0$.

Let $U$ be an open neighbourhood of $0$ and let $\varphi_U\left(|\cdot|_U\right)=(g_x)_{x\in U}$ and $\varphi_U(0_U)=(h_x)_{x\in U}$; considered \begin{equation} (f_x)\in\prod_{x\in U}\mathcal{C}_x\mid\begin{cases} f_x=g_x\iff x\in U\setminus\mathbb{Q}\\ f_x=h_x\iff x\in U\cap\mathbb{Q} \end{cases}, \end{equation} if there exists $\overline{f}\in\mathcal{C}(U)$ such that $\varphi_U\left(\overline{f}\right)=(f_x)_{x\in U}$ then $\overline{f}$ should be the zero function: indeed for any $x\in U\cap\mathbb{Q}$ there exists an open neighbourhood $V_x$ of $x$ such that: \begin{equation} \overline{f}_{|V_x}=0_{V_x}, \end{equation} because: $U\cap\mathbb{Q}$ is a dense subset of $U$, $U$ is a separated space, $\overline{f}$ is continuous and zero on $U\cap\mathbb{Q}$, then $\overline{f}$ is the zero function on $U$. Because $\varphi_U$ is injective: \begin{equation} (f_x)_{x\in U}=\varphi_U\left(\overline{f}\right)=\varphi_U(0_U)=(g_x)_{x\in U}\neq(f_x)_{x\in U}\surd \end{equation} that is a contraddition; then $\varphi_U$ is not surjective.

From all this: the stalks $\mathcal{C}_x$ of $\mathcal{C}$ codify the date of continuous functions at $x$ (point-wise information), and the $\mathcal{C}(U)$'s codify the date of continuous functions on $U$ (local and\or global information); in particular, the function $\varphi_U$ shows as we can pass from the point-wise date to local\global date.

The same reasoning and ideas hold for the sheaf of smooth maps and of differential forms on a manifold, the sheaf of holomorphic maps on a complex manifold, the sheaf of regular maps on an algebraic variety or on a scheme or on a (locally) ringed spaces.

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  • $\begingroup$ I am not sure I follow your example. I mean, of course $\varphi_U$ is not injective, it almost never is. But what is the purpose of your function $f(x)$ ? It is not an element of $\prod_{x\in U}\mathcal{C}_x$, it is not even an element of $\mathcal{C}_0$ despite $f$ being continuous at 0. Do you mean to take the family $(f_x)_{x\in U}$ where $f_x$ is the 0 function if $x\in\mathbb{Q}$ and the absolute value function if $x\not\in\mathbb{Q}$ ? That would work, but it is a bit unclear... $\endgroup$ – Roland Mar 25 '16 at 12:51
  • $\begingroup$ First of all: sorry me, but I have no time. $\varphi_U$ is injective ever; see stacks.math.columbia.edu/tag/0079. I'll clarify the example later. However, thank you for your response. $\endgroup$ – Armando j18eos Mar 25 '16 at 13:04
  • $\begingroup$ @Roland Is it all clear now? $\endgroup$ – Armando j18eos Mar 25 '16 at 22:20
  • $\begingroup$ sorry for my late respond. Yes it is clearer but overly complicated. It is really easy to see that $\varphi$ is not onto and it has nothing to do with $f$ not being continuous (after all the family $(f_x)$ with $f_x$ being the constant function with value $x$ is not in the image). But at least the answer is clearer $\endgroup$ – Roland Mar 26 '16 at 22:41
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Moving from the local to global is exactly the idea behind homology, which measures the failure of a set of local solutions (e.g,. forms satisfying the differential equations given by de Rham cohomology) to patch together and give a global solution.

Given a left-exact functor $F$, we can extend a short exact sequence \begin{align*} 0 \to L \to M \to N \to 0 \end{align*} to a long exact sequence \begin{align*} 0 \to F(L) \to F(M) \to F(N) \to R^1 F(L) \to R^1 F(M) \to R^1 F(N) \to R^2 F(L) \to \dots \end{align*} for some functor $R^* F$. By abstract nonsense, the $R^*F$ are well-defined. (Specifically, the technical requirement we need here is that the underlying category has enough injectives, which holds for the most familiar abelian category of modules over a ring). For particular choices of $F$, the $R^*F$ are just the ordinary cohomology theories. More specifically,taking $F$ to be the space of global sections $F = \Gamma(\cdot, {\scr F})$ for various sheaves ${\scr F}$ gives the de Rham, singular, etc. cohomology theories.

For the category of CW-complexes, this approach may seem like overkill. (I'd argue that it isn't, but this post is long enough already). For more pathological spaces, like varieties, that's the best we can do; the Zariski topology isn't even Hasudorff, so the usual approaches of algebraic topology over nicer spaces won't even work. In the CW-complex setting, continuity is more or less the only criterion for 'nice' functions that we can or need to impose. (That's not quite true; it's very useful to consider the sheaf of sections of an arbitrary bundle. Forms on a smooth manifold $X$, for example, are smooth sections of some $\bigwedge^p T^*X$). In the algebraic geometry setting, that's no longer the case; we have many sheaves available, and one of the interesting things to study is how cohomologies of related sheaves are themselves related. For example, the Riemann-Roch theorem on a Riemann surface of genus $g$ can be written as \begin{align*} \dim H_0(X, L) = \dim H_0(X, L^{-1}\otimes K) = \operatorname{deg}(L) - g + 1 \end{align*} for a (holomorphic) bundle $L \to X$, where $K$ is the canonical (i.e,. determinant) bundle on $X$.

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