Sheaf condition for subcoverings

Question

This is a refinement of my previous question. Let $X$ be a space and $\mathscr{F}$ a presheaf on $X$ with values in a complete category. Let $\mathscr{U} = \{ U_i \}$ be an open covering and suppose that $\mathscr{F}$ satisfies the sheaf condition with respect to some subcovering of $\mathscr{U}$. Does $\mathscr{F}$ automatically satisfy the sheaf condition with respect to $\mathscr{U}$? If $\mathscr{F}$ is a presheaf of sets then this is an easy calculation, but what if $\mathscr{F}$ takes values in e.g. spaces, where we cannot simply check that the relevant map is injective and surjective? Ideally there would be some purely category-theoretic proof.

Answer 1

This is an exercise in transferring proofs from $\mathbf{Set}$ to general categories via Yoneda. Here is the key lemma:

Let $\mathcal{C}$ be a category, let $\mathfrak{U}$ be a sieve on an object $C$ in $\mathcal{C}$, and let $\mathcal{A}$ be a locally small and complete category. Then a presheaf $F : \mathcal{C}^\mathrm{op} \to \mathcal{A}$ satisfies the sheaf condition for $\mathfrak{U}$ if and only if, for every $A$ in $\mathcal{A}$, the presheaf $\mathcal{A} (A, F) : \mathcal{C}^\mathrm{op} \to \mathbf{Set}$ satisfies the sheaf condition for $\mathfrak{U}$.

Indeed, the sheaf condition for $\mathfrak{U}$ is just the statement that some diagram of the form $$F (C) \to \prod_{f \in \mathfrak{U}} F (\operatorname{dom} f) \rightrightarrows \prod_{f \in \mathfrak{U}} \prod_{\operatorname{codom} g = \operatorname{dom} f} F (\operatorname{dom} g)$$ is an equaliser. The functor $\mathcal{A} (A, -) : \mathcal{A} \to \mathbf{Set}$ preserves limits, so $\mathcal{A} (A, F) : \mathcal{C}^\mathrm{op} \to \mathbf{Set}$ satisfies the sheaf condition for $\mathfrak{U}$ if $F$ does. Conversely, if $\mathcal{A} (A, F) : \mathcal{C}^\mathrm{op} \to \mathbf{Set}$ satisfies the sheaf condition for $\mathfrak{U}$ for all $A$, then we may deduce that the above diagram is indeed an equaliser diagram, because the functors $\mathcal{A} (A, -) : \mathcal{A} \to \mathbf{Set}$ are jointly conservative.

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There are more conceptual proofs, but they ultimately rely on proving it for presheaves of sets first. Suppose $\mathcal{C}$ is small and $\mathcal{A}$ is complete. Then, given a presheaf $F : \mathcal{C}^\mathrm{op} \to \mathcal{A}$, we can take the right Kan extension along the Yoneda embedding $\mathcal{C} \to [\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ and get a limit-preserving functor $F : [\mathcal{C}^\mathrm{op}, \mathbf{Set}]^\mathrm{op} \to \mathcal{A}$. Here is a fact:

$F : \mathcal{C}^\mathrm{op} \to \mathcal{A}$ is a sheaf if and only if $F : [\mathcal{C}^\mathrm{op}, \mathbf{Set}]^\mathrm{op} \to \mathcal{A}$ factors (up to isomorphism) through the associated sheaf functor $a : [\mathcal{C}^\mathrm{op}, \mathbf{Set}] \to \mathbf{Sh}(\mathcal{C}; \mathbf{Set})$.

It can be shown that the associated sheaf functor $a : [\mathcal{C}^\mathrm{op}, \mathbf{Set}] \to \mathbf{Sh}(\mathcal{C}; \mathbf{Set})$ is (up to equivalence) the universal functor that sends covering sieves $\mathfrak{U} \hookrightarrow \mathcal{C}(-, C)$ to isomorphisms. You can even check this just for a suitable generating families of covering sieves. The fact that the sheaf condition for a sieve implies the sheaf condition for any "larger" sieve then reduces to a handful of well-known facts:

• The associated sheaf functor $[\mathcal{C}^\mathrm{op}, \mathbf{Set}] \to \mathbf{Sh}(\mathcal{C}; \mathbf{Set})$ preserves finite limits (hence monomorphisms).
• If $\mathfrak{U}' \subseteq \mathfrak{U} \subseteq \mathcal{C}(-, C)$, then $a \mathfrak{U}' \subseteq a \mathfrak{U} \subseteq a \mathcal{C}(-, C)$; so if $a \mathfrak{U}' = a \mathcal{C}(-, C)$, then $a \mathfrak{U} = a \mathcal{C}(-, C)$.

In some sense, this is just like the squeeze theorem from calculus.

Incidentally, the principle here is sufficiently general to work in the homotopical setting. So, for instance, one can deduce that a presheaf satisfies the descent condition with respect to bounded hypercovers if and only if it satisifies the descent condition with respect to sieves, etc.