Since the posts of sheaves have been more than a bit confusing, I’m going to take
\nthe time to go through a couple of examples of real sheaves that are used in
\nalgebraic topology and related fields. Todays example will be the most canonical one:
\na sheaf of continuous functions over a topological space. This can be done for *any* topological space, because a topological space *must* be continuous and gluable with
\nother topological spaces.<\/p>\n
Since the posts of sheaves have been more than a bit confusing, I’m going to take the time to go through a couple of examples of real sheaves that are used in algebraic topology and related fields. Todays example will be the most canonical one: a sheaf of continuous functions over a topological space. This […]<\/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-261","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-4d","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/261","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=261"}],"version-history":[{"count":1,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/261\/revisions"}],"predecessor-version":[{"id":3322,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/261\/revisions\/3322"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=261"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nLet’s quickly recall the definition of sheaves. A sheaf *F* is a mapping from open sets in
\na topological space **T** to objects in a category **C** that has a set of essential properties:
\n1. Every open set S in **T** is mapped to an object *F*(S) in **C**.
\n2. For all pairs of open sets in **T**, there is a *restriction morphism* that provides
\na mapping between objects that respects the intersection property of the open sets.
\n3. The category **C** has a terminal object t, and *F*(∅)=t.
\n4. For all sets U={ui<\/sub>} of open sets with compatible section in **T** there is exactly one unique section on which the restriction morphisms agree; more formally, there is exactly one section s∈F(U) such that ∀ i, ρui<\/sub>,U<\/sub>(S)=si<\/sub>.
\nYou can’t show examples of sheaves without starting with the most basic one: sheaves of continuous functions; that is, a sheaf mapping from topological functions to the category
\n**Top** of sets with continuous functions as morphisms. We can take any topological space **T**, and define a sheaf of continuous functions on it.
\nFirst we’ll define the presheaf. Let the presheaf *F* for **T** by doing the following:
\n* For each open set *O* in **T**, Let *F*(*O*) be the set of *continuous* functions f : *O* → ℜ.
\n* Given two open sets *O* and *P* such that *P* ⊂ *O*, let the restriction morphism
\nρ*P,O*<\/sub> take each function *f* and map it to the function generated by
\nrestricting the domain of *f* to the set *P*.
\nTo show that *F* is not just a presheaf but a full sheaf, we need to show that the
\nnormalization and gluing axioms hold. Normalization, as usual, is pretty trivial. There’s
\nonly one function f : ∅ → ℜ: the empty function. Bingo, normalization.
\nThe gluing axiom is harder. And what it comes down to is going to *look* circular. What we *want* is for the functions in the sheaf to behave in a particular way. So we’re going to
\n*define* the value of the functions included in the sheaf so that they work properly. As long as the functions for an open set *u* *exist* in the set of functions from *u* to ℜ, we
\ncan define the sheaf so that it only includes the functions that have the properties that we want. We’ll select exactly the functions that give us a sheaf of continuous functions
\nover the topology.
\nSo. We start by looking at a set of open sets: U = { ui<\/sub> }.
\nFor any function fi<\/sub> on an open set ui<\/sub>, there is a function
\nf*<\/sup> ∈ F(U) such that fi<\/sub> = f*<\/sup> restricted to
\nui<\/sub>. If we restrict the functions in the sheaf to all be restrictions of continuous
\nfunctions over **T**, then each of the *sections* over each open set *ui<\/sub> are
\nfunctions over **T** restricted to *ui<\/sub>*; and two sections are *compatible*
\nif they are restrictions of the *same* function over **T**. Then, basically by definition,
\nthe gluing axiom will hold – the sections will agree on overlaps, because they’ve been constructed that way.
\nSo what does this tell us? Basically that gluing topological spaces together will
\nalways result in a topological space – you can’t build a structure that won’t
\nbe a valid topological space by gluing topological spaces together.
\nBut it says more than that. What it really says is that the things you can do to
\nfunctions, you can do to topologies. So the fact that there is this sheaf of functions that can be constructed over any topological space means that you can do *algebra* on topological spaces. Since you can create continuous functions from a space **T** to ℜ by
\nadding together two continuous functions from **T** to ℜ, and you can create the additive inverse of a function from **T** to ℜ, that means that you’ve actually got a sheaf of groups. And do the same trick with multiplication, and you can see that this is also a sheaf of rings!<\/p>\n","protected":false},"excerpt":{"rendered":"