Definition of sheaves in wikipedia

Question

In wikipedia, sheaves were first defined in the case of concrete categories (with usual identity and gluing axioms), then in the general case. (writing it as an "exact" sequence)

Do these two definitions agree? I find the definition for concrete categories case very strange, for if we consider a topological space of two points a,b with discrete topology, and let us consider a sheaf of topological spaces on it that assigns A to a, B to b. According to the "concrete-category-case" definition, we need the global sections to look like $A \times B$ such that the projection maps are continuous and nothing else. But if we look at the "equalizer" definition, we would require the global sections to carry the product topology as well.

So is wikipedia wrong? Or am I misunderstanding something? Thanks!

Edit: There is another not-quite-related question. In wikipedia, for the "equalizer" definition they require the category, where the sheaf's taking values in, to have products. Is this really necessary? In EGA Chapter 0 p.23 for example, the product is just "splitted", and we consider the large family of maps all together. It seems that these two approaches are just the same. Or am I wrong?

Answer 1

I think you're quite right; Wikipedia's "concrete definition" is only correct for concrete categories whose underlying-set functor is (not just faithful but) conservative, i.e. such that any morphism which is a bijection on underlying sets is an isomorphism in the category. The page does say that the concrete definition "applies to the most common examples such as sheaves of sets, abelian groups and rings," all of which have this property, but it ought to be fixed to make clear in exactly what situations this definition applies.

Secondly, I observe that the "normalisation" condition in the Wikipedia concrete definition is also odd. Since the empty set is covered by the empty family, the "local identity" and "gluing" conditions already imply that the underlying set of $F(\emptyset)$ is terminal. Saying that in addition, $F(\emptyset)$ itself is terminal is an additional condition, which is in fact a special case of the second, more generally applicable, definition.

Thirdly, I think you're also right that for the correct general definition, the category doesn't need to have any limits a priori; you can just assert that $F(U)$ is the limit of the appropriate diagram of the $F(U_i)$ and $F(U_i\cap U_j)$.

Finally, let me go out on a limb and say that it seems to me that defining "sheaves with values in an arbitrary category" is often a misguided thing to do. More often, it seems like rather than "a sheaf with values in the category of X," the important notion is "an internal X in the category of sheaves of sets." For familiar cases such as groups, abelian groups, rings, small categories—in fact, for any finite limit theory—the two are the same, which may be what leads to the confusion. But the good notion of "sheaf of local rings," for instance, is not a sheaf with values in the category of local rings, but rather a sheaf of rings whose stalks are local (at least, when there are enough points), and that's the same as an internal local ring in the category of sheaves of sets. The situation is similar, I think, for "sheaves of topological spaces" (or locales). I'd be happy for people to point out where I'm wrong about this, though.

Answer 2

I agree, the "definition" for a concrete category is wrong, and your example with a two-point discrete space shows that it's wrong.

Now, if only one could edit Wikipedia...

(I make the following conjecture. Category theory beginners are often more keen on so-called concrete categories than is entirely healthy. Some such person may have written that passage.)

Answer 3

Short answer: You can't take sheaves of topological spaces. This is because a "sheaf of abelian groups" is in fact an abelian group object in the category of sheaves. This equivalence of concepts does not hold for topological spaces because the forgetful functor adjunction of Top and Set is not monadic. If you read Mac Lane's book Sheaves in Geometry and Logic, they explain precisely what this means and why it's important.

Long answer (I am not fully competent to answer this part): taking sheaves of topological spaces can be done, but the enrichment must be in the category of compactly generated (weak hausdorff) spaces or else we won't have some things that we want like being cartesian closed.