Suppose we’ve got a topological space. So far, in our discussion of topology, we’ve tended to focus either very narrowly on local properties of **T** (as in manifolds, where locally, the space appears euclidean), or on global properties of **T**. We haven’t done much to *connect* those two views. How do we get from local properties to global properties?<\/p>\n
One of the tools for doing that is a sheaf (plural “sheaves”). A sheaf is a very general kind of structure that provides ways of mapping or relating local information about a topological space to global information about that space. There are many different kinds of sheaves; rather than being exhaustive, I’ll pretty much stick to a simple sheaf of functions on the topological space. Sheaves show up *all over* the place, in everything from abstract algebra to algebraic geometry to number theory to analysis to differential calculus – pretty much every major abstract area of mathematics uses sheaves.<\/p>\n
<\/p>\n
As I said, I’m going to stick to a simple categorical definition of a sheaf of continuous functions on a topological space.<\/p>\n
To figure out what a sheaf is, we’ll start with something weaker: a *presheaf*. A *presheaf* *F* of sets on a topological space **T** can be defined in terms of the category **Set** of sets as:<\/p>\n
The restriction morphisms are the really important thing in that definition. What we’re doing with the presheaf is providing a way of taking some statement about **T** as a whole, and providing a way of *restricting* it to one of the open sets of **T**. The properties of the restriction morphisms basically say (1) if you restrict a subset to itself, none of its properties change; and (2) if you restrict from the space down to an open set N, and then restrict the results of that down to a smaller open space M, you end up with the same property as if you restricted directly to M without going through N. So we’ve got a structure-preserving mapping that allows us to view properties of small parts of **T**.<\/p>\n
There’s also another way of describing a presheaf in terms of category theory which I hadn’t seen before, but which I discovered via [Wolfram’s Mathworld][mathworld] while I was researching this post. It’s based on the idea of *contravariant functors*.<\/p>\n
To remind you (or in case you’re a new reader who didn’t read my posts on [category theory][cat-archive]), I’ll quickly review the definition of functor (or you can go back and look at my original [post on functors][functors]): <\/p>\n
Suppose you’ve got two categories, C and D. A (covariant) functor *F* from C to D is a mapping that associates every object o in C with an object *F*(o) in D; and each morphism m : x → y in C with a morphism *F*(M) : F(x) → F(y) in D. The functor mapping *F* must preserve the composition structure of C, meaning that ∀ o ∈ Obj(C), F(ido<\/sub>) = idF(o)<\/sub>; and ∀ f : x → y, g : y → z ∈ C, F(g º f) = F(g) º F(f).\n<\/p>\n A *contravariant* functor is the same thing, except that the second property of the morphism mapping, which defines how the structure of composition is preserved is *reversed*: ∀ f : x → y, g : y → z ∈ C, F(gºf) = F(f) º F(g). To understand this, think of the normal way that composition works in categories (or in functions); a contravariant functor *reverses* direction of composition. One way of looking at that is to say that that the normal functor preserves structure by making all of the composition relations work normally – an arrow *m* from f to g in a category will compose with arrows from g to something else after applying the functor. So in some sense, when it maps a morphism, it’s preserving the relations with other morphisms that *start* where *m* ends. But it doesn’t say anything about what happens when the morphism is composed with something that *ends* where *m* *starts*. A contravariant functor does the opposite; it makes sure that through its transformation, the arrow composes properly with everything that *ends* where it *starts*, but doesn’t care about the other way.<\/p>\n To get what that means, you can think of it in terms of the category **Set** of sets and functions between them. A functor from **Set** to **Set** maps objects to objects, and functions to functions. A *covariant* functor says that for a set *A*, every function f : *A* → *B* will be mapped to a function *whose results* are acceptable as inputs to the mappings of things that that used to compose with f. It does *not* say anything about how the mapping of *f* will composite with things that used to provide valid *inputs* to f. The covariant functor can therefore be thought of as, in some sense, constraining the *output* of the function. The contravariant function does the opposite; it constrains the *input*.<\/p>\n Anyway – the point of that little diversion was to let us see an equivalent simpler definition of the presheaf, which works for presheafs of *all* of the different kinds of sheaves. Given a topological space **S**, we can define a category over the open sets of **S**, Top**S**<\/sub>, where the objects in the category are the open sets of **S**, and where for any two objects a,b ∈ **S**, there is an arrow from a to b if\/f a ⊆ b. Now, take a category, **C**, which which contains the values that you want the presheaf to operate over (which would be **Set** in the definition above); the presheaf is just a contravariant functor from Top**S**<\/sub> *to* **C**.<\/p>\n The neat thing about this is that the presheaf is supposed to include *restricting* morphisms. The *contravariant* functor from Top**S**<\/sub> to **C** guarantees that moving to subsets will *restrict* the values in *C*. That is, if you constrain the *input* to a function by only giving it a subset, *it will behave correctly*; it will *restrict* the output the same way that the *input* was restricted. So we’ve effectively found all of those restriction morphism embodied in the structure of the contravariant functor. [mathworld]: http:\/\/mathworld.wolfram.com\/Presheaf.html Suppose we’ve got a topological space. So far, in our discussion of topology, we’ve tended to focus either very narrowly on local properties of **T** (as in manifolds, where locally, the space appears euclidean), or on global properties of **T**. We haven’t done much to *connect* those two views. How do we get from local […]<\/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":[76,65],"tags":[],"class_list":["post-243","post","type-post","status-publish","format-standard","hentry","category-category-theory","category-topology"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-3V","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/243","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=243"}],"version-history":[{"count":2,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/243\/revisions"}],"predecessor-version":[{"id":2931,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/243\/revisions\/2931"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=243"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=243"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nI thought that was pretty nifty \ud83d\ude42<\/p>\n
\n[cat-archive]: http:\/\/scienceblogs.com\/goodmath\/goodmath\/category_theory\/
\n[functors]: http:\/\/scienceblogs.com\/goodmath\/2006\/06\/more_category_theory_getting_i.php<\/p>\n","protected":false},"excerpt":{"rendered":"