Good Math, Bad Math

As I said, I'm going to stick to a simple categorical definition of a sheaf of continuous functions on a topological space.

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:

• ∀ open set O ∈ T, the sheaf has a mapping F(O) ∈ Obj(Set) to an object in the category of sets.
• ∀ open sets N,O ∈ T where N ⊆ O, a morphism ρ_N,O : F(O) → F(N) in Set, called a restriction morphism, which must have the following properties:
  • ∀ open set M ∈ T, ρ_M,M = id_F(M). (The restriction morphism must be the identity morphism if N=O.)
  • ∀ open sets M, N, O ∈ T: ρ_M,N º ρ_N,O = ρ_M,O. *(The mapping to restriction morphisms must compose properly.)

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.

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 while I was researching this post. It's based on the idea of contravariant functors.

To remind you (or in case you're a new reader who didn't read my posts on category theory), I'll quickly review the definition of functor (or you can go back and look at my original post on functors):

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(id_o) = id_F(o); and ∀ f : x → y, g : y → z ∈ C, F(g º f) = F(g) º F(f).

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.

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.

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, 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 to C.

The neat thing about this is that the presheaf is supposed to include restricting morphisms. The contravariant functor from Top_S 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.

I thought that was pretty nifty :-)