What are the benefits of viewing a sheaf from the "espace étalé" persepctive? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T04:49:26Z http://mathoverflow.net/feeds/question/96264 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96264/what-are-the-benefits-of-viewing-a-sheaf-from-the-espace-etale-persepctive What are the benefits of viewing a sheaf from the "espace étalé" persepctive? Simon Rose 2012-05-07T21:01:24Z 2012-05-08T00:40:01Z <p>I learned the definition of a sheaf from Hartshorne---that is, as a (co-)functor from the categor of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. While fairly abstract at the outset, this seems to be (to me) an intuitive view; in particular, all of the manipulations and constructions with sheaves fit nicely into this schema.</p> <p>I know that the older view of a sheaf on $X$ was to consider it as a triple $$(E, X, \pi)$$ where $\pi : E \to X$ is a local homeomorphism, and so that the "sheaf of sections" of this map $\pi$ is the sheaf in the functorial sense described above. </p> <p>This view makes much less sense to me, but I have to wonder if that is simply due to my having learned it second. However, it also makes me wonder if I am missing something, and so my question is as follows.</p> <blockquote> <p>What are some (edit:) <b>specific</b> benifits of viewing a sheaf in this sense? What is gained by considering a sheaf as the espace étalé over $X$?</p> </blockquote> http://mathoverflow.net/questions/96264/what-are-the-benefits-of-viewing-a-sheaf-from-the-espace-etale-persepctive/96266#96266 Answer by Tom Leinster for What are the benefits of viewing a sheaf from the "espace étalé" persepctive? Tom Leinster 2012-05-07T21:26:54Z 2012-05-07T23:24:15Z <p>Let me expand on Yosemite Sam's comment. Pullbacks are indeed easier to define if you view a sheaf as a local homeomorphism. On the other hand, pushforwards are easier to define if you view a sheaf as a set-valued functor. </p> <p>Suppose we have a continuous map $f: X \to Y$ of topological spaces. </p> <p>Given a sheaf $F$ on $Y$, viewed as a local homeomorphism $\pi: F \to Y$, we can simply pull $\pi$ back along $f$ to obtain a map into $X$; it is easily shown to be a local homeomorphism too. This is the pullback sheaf $f^* F$.</p> <p>On the other hand, given a sheaf $E$ on $X$, viewed as a functor $\mathrm{Open}(X)^{op} \to \mathbf{Set}$ (where $\mathrm{Open}(X)$ is the poset of open subsets of $X$), we can simply compose $E$ with the functor $\mathrm{Open}(Y) \to \mathrm{Open}(X)$ that takes inverse images along $f$. This gives a set-valued functor on $\mathrm{Open}(Y)$; it is easily shown to be a sheaf too. This is the pushforward sheaf $f_* F$.</p> <p>So, there are advantages to proving the equivalence between the two definitions early on. </p> http://mathoverflow.net/questions/96264/what-are-the-benefits-of-viewing-a-sheaf-from-the-espace-etale-persepctive/96270#96270 Answer by David Carchedi for What are the benefits of viewing a sheaf from the "espace étalé" persepctive? David Carchedi 2012-05-07T22:26:28Z 2012-05-07T22:26:28Z <p>One advantage is that it gives you a geometric representation for slice topoi of sheaves over a space:</p> <p>Given a topos $E$, $E$ is equivalent to the full-subcategory of $Top/E$, the category of topoi over $E$ consisting of etale morphisms of topoi. The equivalence sends an element $e \in E$ to the morphism $E/e \to E$ where $E/e$ is the slice topos.</p> <p>Topoi are generalizations of spaces, and to view a space as a topos, we send a space $X$ to its topos of sheaves $Sh(X)$. Etale geometric morphisms $Sh(X) \to Sh(Y)$ are in bijection with local homeomorphisms $X \to Y$ (when Y is sober.). So, this means that if $F \in Sh(X)$, the local homeomorphism $E(F) \to X$ which corresponds to $F,$ viewed as a map of topoi is nothing but the etale geometric morphism $Sh(X)/F \to Sh(X)$. In particular, this implies that $Sh(X)/F \cong Sh(E(F)).$</p> http://mathoverflow.net/questions/96264/what-are-the-benefits-of-viewing-a-sheaf-from-the-espace-etale-persepctive/96280#96280 Answer by Benjamin Steinberg for What are the benefits of viewing a sheaf from the "espace étalé" persepctive? Benjamin Steinberg 2012-05-08T00:12:41Z 2012-05-08T00:27:58Z <p>To me the obvious answer involves sheafification of a presheaf. If you look at the construction of the associated sheaf to a presheaf in, say, Hartshorne it goes through the etale space construction without specifically telling you and to me it makes the construction somewhat unmotivated.</p> <p>Namely, taking if $P$ is a presheaf on $X$, then taking the stalk $P_x$ at each point of $X$ gives you an $X$-indexed set, or as Tom would say above, a set over $X$. One can then define a topology on $\biguplus_{x\in X}P_x$ so that the natural projection $\biguplus_{x\in X}P_x\to X$ is a local homeomorphism in the obvious way namely if $U$ is a neighborhood in $X$ and $s\in P(U)$, then $(s,U) = \lbrace germ_x(s)\mid x\in U\rbrace$ is a basic neighborhood. This topology immediately makes $(s,U)$ homeomorphic to $U$ and makes $s$ a section over $U$ via $x\mapsto germ_x(s)$ for $x\in U$. The sheaf of sections of $p$ is the associated sheaf of $P$. If find this construction completely unmotivated without going through etale spaces.</p> <p><b>Added.</b> Another good reason is it is convenient for defining actions of a topological groupoid on a sheaf. If $G=(G_0,G_1)$ is a groupoid, a $G$-sheaf is an etale space $p:X\to G_0$ over $G_0$ together with an action map $G_1\times_{d,p} X\to X$ satisfying obvious axioms. This is more difficult to phrase in the sheaf as a functor language. A theorem of Joyal and Tierny says that every Grothendieck topos is equivalent to the topos of sheaves on a localic groupoid.</p> <p><b> Additional additions</b> From the etale space point of view it is clear that covering spaces are indeed elements of the topos $Sh(X)$ of sheaves on $X$ and that the fundamental group of $Sh(X)$ (in the sense of Barr and Diaconescu) is the usual fundamental group of $X$ if $X$ is locally simply connected. </p> <p>Of course it is not hard to see that covering spaces correspond to locally constant sheaves but I don't think this is the way people think about covering spaces.</p>