Previously: Coverages and Sites

The definition of a sheaf is rather complex and involves several layers of abstraction. To help us navigate this maze we can use some useful intuitions. One such intuition is to view objects in our category as some kind of sets (in particular, open sets, when we talk about topology), and arrows as set inclusions. An arrow from v to u means that v is a subset of u.

A cover of u is a family of arrows \{ p_i \colon u_i \to u \}. A coverage assigns a collection of covers to every object, satisfying the sub-coverage conditions described in the previous post. A category with coverage is called a site.

The next layer of abstraction deals with presheaves, which are set-valued contravariant functors. Interestingly, there is a way to interpret a presheaf as an extension of the original category. I learned this trick from Paolo Perrone.

We may represent a presheaf P using virtual hom-sets. First we add one virtual object, let’s call it \bullet , to our category. The set P u is then interpreted as the set of arrows from u to \bullet.

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Moreover, we can represent the action of P on arrows as simple composition. Take an arrow f \colon v \to u. The presheaf lifts it to a function between sets: P f \colon P u \to P v (contravariance means that the arrow is reversed). For any h \in P u we can define the composition h \circ f to be (P f) h.

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Incidentally, if the functor P is representable, it means that we can replace the virtual object \bullet with an actual object in our category.

Notice that, even though the category of open sets with inclusions is a poset (hom-sets are either singletons or empty, and all diagrams automatically commute), the added virtual hom-sets usually contain lots of arrows. In topology these hom-sets are supposed to represent sets of continuous functions over open sets.

We can interpret the virtual object \bullet as representing an imaginary open set that “includes” all the objects u for which P u is non-empty, but we have to imagine that it’s possible to include an object in more than one way, to account for multiple arrows. In fact, in what follows we won’t be assuming that the underlying category is a poset, so virtual hom-sets are nothing special.

To express the idea of intersections of open sets, we use commuting diagrams. For every pair of objects u_i and u_j that are in the cover of u,  an object v is in their intersection if  the following diagram commutes:

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Note that in a poset all diagrams commute, but here we’re generalizing this condition to an arbitrary category. We could say that v is in the intersection of u_i and u_j seen as covers of u.

Equipped with this new insight, we can now express the sheaf condition. We assume that there is a coverage defined in our category. We are adding one more virtual object \bullet for the presheaf P, with bunches of virtual arrows pointing to it.

For every cover \{ p_i \colon u_i \to u \} we try to select a family of virtual arrows, s_i \colon u_i \to \bullet. It’s as if the objects u_i, besides covering u, also covered the virtual object \bullet.

We call the family \{s_i \} a matching family, if this new covering respects the existing intersections. If v is in the intersection of u_i and u_j (as covers of u, see the previous diagram), then we want the following diagram to also commute:
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In other words, the \{u_i\}‘s intersect as covers of \bullet.

A presheaf P is a sheaf if, for every covering family p_i and every matching family s_i there exists a unique s \colon u \to \bullet that factorizes those s_i‘s:
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Translating it back to the language of topology: There is a unique global function s defined over u whose restrictions are s_i‘s.

The advantage of this approach is that it’s easy to imagine the sheafification of an arbitrary presheaf by freely adding virtual arrows (the s‘s and their compositions with p_i‘s in the above diagram) to all intersection diagrams.

Next: Covering Sieves