Previously: Sheaves as Virtual Objects.

In order to define a sheaf, we have to start with coverage. A coverage defines, for every object u, a family of covers that satisfy the sub-coverage conditions. Granted, we can express coverage using objects and arrows, but it would be much nicer if we could use the language of functors and natural transformations.

Let’s start with the idea that, categorically, a cover of u is a bunch of arrows converging on u. Each arrow p_i \colon u_i \to u is a member of the hom-set \mathcal C (u_i, u). Now consider the fact that \mathcal C (-, u) is a presheaf, \mathcal C^{op} \to \mathbf{Set}, and ask the question: Is a cover a “subfunctor” of \mathcal C (-, u)?

A subfunctor of a presheaf P is defined as a functor S such that, for each object v, S v is a subset of P vand, for each arrow f \colon v \to w, the function S f \colon S w \to S v is a restriction of P f.

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In general, a cover does not correspond to a subfunctor of the hom-functor. Let’s see why, and how we can fix it.

Let’s try to define S, such that S u_i is non-empty for any object u_i that’s in the cover of u, and empty otherwise. As a presheaf, we could represent it as a virtual object with arrows coming from all \{ u_i \}‘s.

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Now consider an object v that is not in the cover, but it has an arrow f \colon v \to u_k connecting it to some element u_k of the cover. Functoriality requires the (virtual) composition s_k \circ f to exist.Untitled Artwork

Thus v must be included in the cover–if we want S to be a functor.

In particular, if we are looking at a category of open sets with inclusions, this condition means that all (open) sub-sets of the covering sets must also be included in the cover. Such a “downward closed” family of sets is called a sieve.

Imagine sets in the cover as holes in a sieve. Smaller sets that can “pass through” these holes must also be parts of the sieve.

If you start with a cover, you can always extend it to a covering sieve by adding more arrows. It’s as if you started with a few black holes, and everything that could fall into them, would fall.

We have previously defined sheaves in terms of coverings. In the next installment we’ll see that they can equally well be defined using covering sieves.

Next Sieves and Sheaves.