Continuing from where we left off yesterday…
\nYesterday, I managed to describe what a *presheaf* was. Today, I’m going to continue on that line, and get to what a full sheaf is.
\nA sheaf is a presheaf with two additional properties. The more interesting of those two properties is something called the *gluing axiom*. Remember when I was talking about manifolds, and described how you could describe manifolds by [*gluing*][glue] other manifolds together? The gluing axiom is the formal underpinnings of that gluing operation: it’s the one that justifies *why* gluing manifolds together works.
\n[glue]: http:\/\/scienceblogs.com\/goodmath\/2006\/11\/better_glue_for_manifolds.php<\/p>\n
Continuing from where we left off yesterday… Yesterday, I managed to describe what a *presheaf* was. Today, I’m going to continue on that line, and get to what a full sheaf is. A sheaf is a presheaf with two additional properties. The more interesting of those two properties is something called the *gluing axiom*. Remember […]<\/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-244","post","type-post","status-publish","format-standard","hentry","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-3W","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/244","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=244"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/244\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=244"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=244"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nBefore the gluing axiom, there’s another necessary property for a sheaf, called the *normalization axiom*. It’s *much* easier to describe in terms of the second, category-theoretic definition of presheaf. If we have a presheaf *F* based on a category **C**, then for *F* to be a sheaf, the gluing axiom says that **C** must have a [terminal object][terminal] t, and *F*(∅) = t.
\nThe gluing axiom for a sheaf is the really neat one. What it says is that we can glue
\ntwo subsets together if they agree on the overlap. To make that formal:
\nSuppose we have a topological space **T**, and a presheaf *F*.
\nLet { Ti<\/sub> }i=1..n<\/sub> be a collection
\nof open subsets of **T**; and since we’ll want to look at the union of the open subsets in that set, let *U* be the union ∪i=1..n<\/sub> Ti<\/sub>.
\nFor each open set Ti<\/sub>, we can use *F* to define a *section* si<\/sub>. Given those sections, we can say that two sections si<\/sub> and sj<\/sub> are *compatible* if\/f ρTi<\/sub>∩Tj<\/sub>,Ti<\/sub><\/sub>(si<\/sub>) = ρTi<\/sub>∩Tj<\/sub>,Tj<\/sub><\/sub>(sj<\/sub>). (That is just a fancy way of saying that their restriction functions *agree* on the overlap.) We can extend the definition of *compatibility* to the entire set *U* by saying *U* is a compatible group of open subsets if all pairs of sets within it are compatible.
\nNow, finally, we can get to what the gluing axiom for a sheaf *F* says!
\nGiven a topological space **T**, and a *sheaf* *F*, for all sets *U* of open subsets of **T** U={ui<\/sub>}i∈I<\/sub> with *compatible sections* { si<\/sub> }i ∈ I<\/sub>, there there exists exactly one section s ∈ F(U) such that &rhoui<\/sub>,U<\/sub> = si<\/sub>.
\nIn simple terms, it says that if you use the restriction maps of compatible sections in a sheaf, they’ll agree on *exactly one* mapping for the overlapped subsection. So the overlap is defined exactly once, and there can be no disagreement about the correct way of looking at it.
\nIt’s a lot of work to get to this point, but it’s worth it. What we’ve done is go from the
\ninformal but intuitive idea that *gluing* topological spaces together works, to a careful and precise definition of exactly what it *means* for it to work. And hopefully, if I’ve done an adequate job of explaining this, you can see why this says that gluing two manifolds results in a manifold.
\nYou see, for something to be a manifold, what that means is that for all points in the space, there is a *local* property that the space around the point looks euclidean. But it’s not enough for that *local* property to be true in just some places, it must be true *everywhere*.
\nBy defining a sheaf over the topological space, we can easily show that that *local* property holds true for the entire space. And when we glue two manifolds together, we’re forming a new topological space; the glue defines a mapping which we can use to define a sheaf for the entire new space; and if the glue is done right, the resulting sheaf *must* also show that the local property of being almost euclidean is true for the entire *new* space.
\n[terminal]: http:\/\/scienceblogs.com\/goodmath\/2006\/06\/category_theories_some_definit.php<\/p>\n","protected":false},"excerpt":{"rendered":"