Sheaf

up:: 027 MOC Category Theory

A Presheaf of sets over a Topological Space is said to be a sheaf if it also satisfies the following axioms: For any given open set and any open cover of , it follows that:

1. Gluing: For any family of elements , if , then there’ll be some which is the gluing of these elements, restricting to them accordingly ().
2. Separability: Given such that for all , then they must be equal . That is, the gluing of these ‘s is unique.

This is why The functor of continuous functions over a topological space is a (pre)sheaf: because, given continuous functions — locally continuous within — which coincide in , there will be a continuous function which is their (unique) gluing — that is, a globally continuous function in as a whole.

Categorical definition of a sheaf (over a topological space)

A presheaf is a sheaf if, for any and open covers of it , we have that the following Equalizer diagram commutes:

Given some element , we have that

In this way, it spells out outright what the condition is: Given some function ¹ — over the entire open set —, restricting it over each open set of one of its open covers yields “partial functions” which must coincide in their intersections; this “equalling” of the partial functions is what the equalizer is imposing/guaranteeing, through the equalling of (first index of intersecting open sets) and (second index).

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References

• III EBCT (Encontro Brasileiro em Teoria das Categorias) - 26/05 — ~02:24:00

Footnotes

1. It’s useful to think about the sheaf of continuous functions . ↩