The $C^k$ Sheaves

Question

In Sheaves in Geometry and Logic, the authors claim that $C^k$ are all sheaves because differentiability is local. How do I formally prove this?

Answer 1

Obviously, $C^k$ are functors ${\mathcal O(X)}^\mathrm{op} → \mathbf{Set}$.

Recall the definition of a sheaf: They are sheaves if for any open covering $U_α$ of some open subset $\bigcup_α U_α = U ⊂ X$ you have a glueing function on consistent systems $(s_α)$ of sections on $(U_α)$, i.e.:

• For all systems of sections $(s_α)$ in $C^k(U_α)$
• which are consistent on intersections, i.e. $∀α,β:~s_α|_{U_α∩U_β} = s_β|_{U_α∩U_β}$
• there is a unique glued-together section $s ∈ C^k(U)$
• which agrees with the given system, i.e. $∀α: s_α = s|_{U_α}$.

So, take an arbitrary open set $U ⊂ X$ and an arbitrary open covering $U = \bigcup_α U_α$, take an arbitrary system of functions $s_α \colon U_α → ℝ$ which are $C^k$ and agree on all intersections. Define a function $s\colon U → ℝ$ which is $C^k$ and agrees on all $U_α$ and prove such a function is unique.

To show that the defined function is $C^k$, you use that differentiability is a local property.

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To say that the diagram $$C^k(U) \overset{e}\rightarrow \prod_α C^k(U_α) \overset{p}{\underset{q}\rightrightarrows} \prod_{α,β} C^k(U_α ∩ U_β),$$ is an equalizer diagram in $\mathbf{Set}$, means exactly what I have written down before: Recall that

• $p((s_α)_α) = (s_α|_{U_α ∩ U_β})_{α,β}$
• $q((s_α)_α) = (s_β|_{U_α ∩ U_β})_{α,β}$, and
• $e(s) = (s_α)_α$.

The usual construction of the equalizer of the given diagram is the set $$E(p,q) = \{(s_α)_α ∈ \prod_α C^k(U_α);~ p((s_α)_α) = q((s_α)_α))\},$$ which by the above interpretation of $p$ and $q$ is exactly the set of “consistent systems of sections” (for any presheaf $\mathcal F$, the elements in $\mathcal F U$ are called “sections”), agreeing on the intersections.

To say that there’s a glueing function (as formulated above) is exactly to say that there’s a function $E(p,q) → C^k(U)$ commuting with $e$ (and the inclusion of $E(p,q) ⊂ \prod_α C^k(U_α)$), which is exactly to say that $C^k(U)$ is an equalizer as well: The glueing function gives a bijection, the inverse being given by the universal property of the equalizer.