I’ve mostly been taking it easy this week, since readership is way down during the holidays, and I’m stuck at home with my kids, who don’t generally give me a lot of time for sitting
\nand reading math books. But I think I’ve finally got time to get back to the stuff
\nI originally messed up about sheaves.
\nI’ll start by talking about the intuition behind the idea of sheaves. The basic idea of
\na sheave is to provide a way of taking some local property of a topological space, and
\ndemonstrating that it holds everywhere. The classic example of this is manifolds, where the *local* property of being locally almost euclidean around a point is expanded to being almost euclidean around *all* points.<\/p>\n
I’ve mostly been taking it easy this week, since readership is way down during the holidays, and I’m stuck at home with my kids, who don’t generally give me a lot of time for sitting and reading math books. But I think I’ve finally got time to get back to the stuff I originally messed […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[65],"tags":[],"class_list":["post-257","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-49","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/257","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=257"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/257\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=257"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=257"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nTo be able to do that step from the local to the universal requires a step of abstraction; it needs to exploit the particular structural properties of the relationships between points in the space. The easiest way to talk about the structural properties is by choosing a mathematical structure that’s appropriate for the property you want to study, and creating
\na particular kind of correspondence between the open sets of the topological space (that is, the topological structure) and objects of the type that you want to use for reasoning.
\nTo make that work, you basically need to do two different things: one is to generate the mapping between the open sets of the space in a way that preserves the important properties of the topological structure of the space; the second is to show that joining the mapped objects
\npreserves structure – that it, given objects corresponding to overlapping open sets, you can show that the topological properties are preserved when those objects are joined.
\nIf you do the first step – showing a mapping between open sets and some other mathematical
\nobjects in a structure-preserving manner – what you get is a *pre-sheaf*. If you take a
\npre-sheaf, and show that it has the properties needed to making joining objects corresponding
\nto sets work, then you’ve got a sheaf.
\nSo what’s a pre-sheaf? As I said above, a pre-sheaf *F* over a a topological space **T** is a mapping from open sets in **T** to some kind of objects where the mapping preserves the topological structure. Since the topological structure is defined by subset relations, that means that the mapping has to preserve the basic properties of the subset relations. Formally, you can described a pre-sheaf *F* as a mapping to a category *C* with a special set of morphisms, Ρ such that:
\n1. Every open set of **T** is mapped by *F* to an object in **C**.
\n2. For every pair of open sets x, y in **T** such that x ⊆ y, Ρ contains a ρx,y<\/sub> such that:
\n1. For all points *x* in **T**, ρx,x<\/sub> = the identity morphism for *F*(x). *(The mapping preserves the identity {*x*} ⊆ {*x*}.)*
\n2. For all triples of open sets M, N, and O in **T**, ρM,N<\/sub>ºρN,O<\/sub> = ρM,O<\/sub>. *(The mapping preserves the transitivity of the subset relation: M ⊆ N and N ⊆ O ⇒ M ⊆ O.)*
\nOr alternatively, the presheaf can be seen as a *contravariant* functor from the topological category of **T** to the category **C**. The contravariant functor guarantees that the mappings will behave the way that the restriction morphism describe above.
\nNow, suppose we’ve got a presheaf *F* mapping to the category *C*; that’s called a *C*-valued presheaf. If we have an open set from **T** called *O*, then *F(O)* is a set which represents the part of the structure of **T** enclosed by the open set *O*. We call *F(O)* the *sections* of *F* over *O*. Assuming that the objects in *C* are sets (which is not *always* the case, but will be the case in pretty much every category we use to describe anything topological), then each element of *F(O)* is a *section* of *O*.
\nWhat we want to be able to do when we go from pre-sheaves to sheaves is to show that the way that the pre-sheaf represents the structure of **T** allows us to reason about properties of subsets like *O*, and then expand that reasoning to the entire topological space by showing that all subsets of *O* possess the desired property, *and* (this and is the really important part!) that when two subsets overlap, it’s *not* just the case that both subsets have the desired property, but that both subsets *agree* on how that property is represented in the sheaf on the sections where they overlap.
\nSo how do we go about saying that?
\nSuppose we have a presheaf *F* over the topological space **T**. First, we need to show that the category *C* used by *F* has a necessary property for small-to-large reasoning to work. In category theoretic terms, *C* must have a terminal object *t*, and *F(∅)=t*. (A terminal object is sort of like a kind of universal lower bound; if the objects in a category are sets, the terminal object is the empty set.) This property is called the *normalization axiom*.
\nBy far the more interesting part is the *gluing axiom* – that’s the part that lets us define
\nwhat it means for two subsets to agree on an overlapping section.
\nWhat does is mean for two subsets to *agree* on an overlapping section? We’ll start with a
\nslightly weaker notion, of *compatibility* between sections, and then from there, we’ll extend
\nit to *agreement* between sections. Suppose we have open sets *D* and *E*, and that
\n*D*∩*E*&neq;∅. Then if we do the presheaf mapping on *D* and *E*, we get a set of
\nsections over *D* and *E* in *F*. Two sections *sd<\/sub>* and *se<\/sub>* are
\n*compatible* if the restriction morphisms respect the rule *ρD∩E,D<\/sub>(sd<\/sub>) = ρD∩E,E<\/sub>(se<\/sub>)*. The open subsets *D* and *E* are compatible if\/f for all pairs of sections *d∈F(D)* and *e∈F(E)*, *d* and *e* are compatible. (This is why the normalization axiom was important: if the sections do *not* overlap, there must be a unique value to represent that fact: the terminal object of the category.)
\nNow, finally, we can get to agreement, and what the gluing axiom requires for a presheaf to be a sheaf. Given a topological space *T* and a presheaf *F*, *F* is a sheaf if and only if for all sets *U* = {*ui<\/sub>*} of open sets of **T** with compatible sections *s={si<\/sub>}*, there is exactly *one* unique section **s**∈F(U) such that ρui<\/sub>,U<\/sub>(**s**)=si<\/sub>. The unique section **s** for
\nthe set of open subsets *U* is called the *gluing* of the set *U*.
\nThat’s a very fancy way of saying that whenever a group of open sets have overlapping sections, that the members agree that there is exactly *one* mapping between their respective sections, and that the structural properties in that section are consistent.
\nIt’s worth reiterating here: the gluing axiom says *nothing* about coordinates, distances, or directions. All that it says is that the mapping of the open subsets of the topological space preserves their subset structure. So whatever properties the subset structure of the space has must also be represented in the sheaf mapping; and when sections with interesting properties
\nare glued together, the gluing respects those properties.
\nWhen we work with manifolds, we often do it using atlases, which are mappings from subspaces
\nof some euclidean space ℜN<\/sup> to sections of a manifold, and we glue together
\nsections of the those mappings. That works because of the gluing axiom, but the *coordinate
\nsystems* that we get from doing that aren’t necessary for gluing; in fact, it’s the opposite: the metric structure of the sections is a local property, and the gluing axiom allows us to combine metrizable subspaces in a way that produces a single consistent metric for the entire manifold.<\/p>\n","protected":false},"excerpt":{"rendered":"