A Note On Sheafification

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A presheaf {\mathcal{F}} on a topological space {X} with values in a category {\mathcal{D}} (usually with values in GrpCommRing, or Set), can be associated to a sheaf {\widetilde{\mathcal{F}}} on {X} in a natural way. In fact, this association is left adjoint to the inclusion functor from the category of sheaves on {X} to the category of presheaves on {X}.

The standard construction of {\widetilde{\mathcal{F}}} is fairly straightforward: consider the topology on the disjoint union of stalks {\text{Tot}(\mathcal{F}) = \bigcup_{x \in X} \mathcal{F}_x} generated by the collection of sets

\displaystyle \begin{array}{rcl} \mathfrak{V}(b, U) &=& \{ (b_x, x) : x \in U\}, \end{array}

where {U \subseteq X} is open, {b\in \mathcal{F}(U)} and {b_x} is the image of {b} in {\mathcal{F}_x}. Let {p : \text{Tot}(\mathcal{F}) \rightarrow X} be the natural projection. For every {U \subseteq X} define {\widetilde{\mathcal{F}}(U)} to be the set of continuous sections {s : U \rightarrow \text{Tot}(\mathcal{F})} of {p}, i.e. {p\circ s = \text{id}_U}. It follows that {\widetilde{\mathcal{F}}} is a sheaf and there is a natural morphism {i : \mathcal{F} \rightarrow \widetilde{\mathcal{F}}}, such that {\mathcal{F}} is a sheaf if, and only if, {i} is an isomorphism, (this is a fairly easy exercise).

This shows that every sheaf arises as the sheaf of sections of some continuous map onto {X}. This also justifies the terminology: an element {a \in \mathcal{F}(U)} is called a section. At first glance, it is hard to see how this sheaf relates to {\mathcal{F}}. Indeed, the canonical map {i} fails to be either surjective or injective, in general. This says that we cannot “complete” {\mathcal{F}} by simply adding more elements, or by glueing together elements; we must do both. Thus, we can can’t view {\mathcal{F}} as a subpresheaf of {\widetilde{\mathcal{F}}} or as a cover. However, there is another natural way to think of {\widetilde{\mathcal{F}}}.

Let {\mathcal{G}} be a sheaf on {X} with values in some category {\mathcal{D}}, as above. One of the advantages of having a sheaf on {X} is that a section {s \in \mathcal{G}(U)} is uniquely determined by the collection of elements {\{s_x \in \mathcal{G}_x : x \in U\}}. This can fail for presheaves. Thus, we need {i(U): \mathcal{F}(U) \rightarrow \widetilde{\mathcal{F}}(U)} to glue together elements {a, b \in \mathcal{F}(U)} such that {a_x = b_x} for all {x \in U}. This is evident in the commutative diagram:

\displaystyle \begin{array}{rcl} \begin{array}{ccc} \mathcal{F}(U) & \longrightarrow & \widetilde{\mathcal{F}}(U) \\ \downarrow & \; & \downarrow \\ \mathcal{F}_x & \longrightarrow & \widetilde{\mathcal{F}}_x \end{array} \end{array}

This says that {i(U)(a)} and {i(U)(b)} have the same image in {\widetilde{\mathcal{F}}_x} for all {x \in U}. Thus, because {\widetilde{\mathcal{F}}} is a sheaf, {i(U)(a) = i(U)(b)}.

Aside from glueing sections together that “should” be equal, we also need to add more elements to our presheaf {\mathcal{F}}. Suppose we have an open cover of an open subset {U \subseteq X}: {U = \bigcup_\alpha U_\alpha}. If we can find a collection {\{b_\alpha\}, b_\alpha \in U_\alpha}, such that {b_\alpha|_{U_\alpha \cap U_\beta} = b_{\beta}|_{U_\alpha \cap U_\beta}}, then we should be able to find a “global” section {b} on {U} such that {b|_{U_\alpha} = b_\alpha}. Because our presheaf may be too “small” and may not contain this element, we need to ensure that our sheafification {\widetilde{\mathcal{F}}} does. The set {\{b_\alpha\}} is compatible in the obvious sense, so we can glue them together to form a section {\hat{b} \in \widetilde{\mathcal{F}}(U), \hat{b} : x \mapsto (b_{\alpha})_x}. This map is continuous because {\hat{b}^{-1}(\mathfrak{V}(a, W)) = \{x \in W \cap U : \hat{b}(x) = a_x\}}. is open. Indeed, if {a_x = b(x) = (b_\alpha)_x} for some {x}, then there is an open subsets {W' \subseteq W \cap U} such that {x\in W'} and {a|_{W'} = (b_\alpha)|_{W'}} in {\mathcal{F}(W')}, so {W' \subseteq \hat{b}^{-1}(\mathfrak{V}(a, W))}. It is also clear that {p \circ \hat{b} = \text{id}_U}.

To sum up our construction, we note the following: to construct {\widetilde{\mathcal{F}}} we find the most “efficient” sheaf such that

  1. sections that have the same germ at every point are equal
  2. compatible sets of sections have a common extension.

This gives the following alternative description of {\widetilde{\mathcal{F}}}:

\displaystyle \begin{array}{rcl} \widetilde{\mathcal{F}}(U) &=& \{ \{(b_{\alpha}, W_\alpha)\}_{\alpha} : U = \bigcup_{\alpha} W_\alpha, b_\alpha \in \mathcal{F}(W_{\alpha}), (b_\alpha)_x = (b_\beta)_x \text{ for all } x \in W_\alpha \cap W_\beta\} /\sim \end{array}

where {\{(b_\alpha, W_\alpha)\}_\alpha \sim \{(a_\beta, V_{\beta})\}_\beta}, if for all {\alpha, \beta} and {x \in W_\alpha \cap V_{\beta}}, {(b_\alpha)_x = (a_\beta)_x}. Note that condition {2} and the assumption that this extension is unique is just the sheaf axiom and, hence, it implies {1}.

We claim that this definition of {\widetilde{\mathcal{F}}} is equivalent to the definition given above. Indeed, we can view each collection {\{(b_\alpha, W_\alpha)\}_\alpha} as a section {\hat{b}} in the natural way: {\hat{b}(x) = (b_\alpha)_x} if {x \in W_\alpha}. This is independent of the representative of {\{(b_\alpha, W_\alpha)\}_\alpha} by {\sim}. Note that {\hat{b}(x)} is continuous, as we showed earlier. Further, any section {s : U \rightarrow \text{Tot}(\mathcal{F})} arises in this way. Indeed, let {x \in U}, then {s(x) \in \mathfrak{V}(b^x, W_x)} for a neighborhood {W_x} of {x} and an element {b^x \in \mathcal{F}(W_x)}. Thus, because {s} is continuous there exists {W_x' \subseteq s^{-1}( \mathfrak{V}(b^x, W_x)) = \{ y \in W \cap U : s(y) = (b^x)_y\}}. Consider the collection {b = \{(b^x, W_x')\}_{x\in U}}. Because {(b^x)_y = s(y) = (b^z)_y} for all {y \in W_x' \cap W_z'}, it follows that {b} corresponds to {s}. This shows that the correspondence is surjective. We claim that this correspondence is also injective. Indeed, if {\{(b_\alpha, W_\alpha)\}_\alpha} and {\{(a_\beta, V_\beta)\}_\beta} correspond to the same section {s}, then {(b_\alpha)_x = s(x) = (a_\beta)_x} for all {x \in W_\alpha \cap V_\beta}. Thus, {\{(b_\alpha, W_\alpha)\}_\alpha = \{(a_\beta, V_\beta)\}_\beta} Finally we note that this correspondence is actually a morphism of presheaves: Restricting a collection {\{(b_\alpha, W_\alpha)\}_\alpha \in \widetilde{\mathcal{F}}(U)} to a subset {U'\subseteq U}, yields the element {\{(b_\alpha|_{U'}, W_\alpha\cap U')\}_{\alpha}} and this corresponds to the restricted section {\hat{b}|_{U'}}.

This new definition has some utility. It helps to illuminate the following claims:

Claim 1 Let {i : \mathcal{F} \rightarrow \widetilde{\mathcal{F}}} be the natural map: {i(U)(b) = \{(b, U)\}}. Then {\mathcal{F}} is a sheaf if, and only if, {i} is an isomorphism.

Proof: If {\mathcal{F}} is a sheaf then each collection {\{(b_{\alpha}, W_\alpha)\}_{\alpha} \in \widetilde{\mathcal{F}}(U)} can be written uniquely as {\{(b, U)\}} for some {b \in \mathcal{F}(U)}. Conversely, suppose {i} is an isomorphism, let {U \subseteq X} be an open subset, let {U = \bigcup W_\alpha}, and let {\{b_\alpha : b_\alpha \in \mathcal{F}(W_\alpha)\}} be a collection such that {b_{\alpha}|_{W_\alpha \cap W_\beta} = b_{\beta}|_{W_\alpha \cap W_\beta}}. Then {(b_{\alpha})_x = (b_\beta)_x} for all {x \in W_\alpha \cap W_\beta}, so {\{(b_\alpha, W_\alpha)\}_\alpha \in \widetilde{\mathcal{F}}(U)}. Thus, there exists a unique element {b \in \mathcal{F}(U)} such that {i(U)(b)= \{(b_\alpha, W_\alpha)\}_\alpha}. Therefore, for each {\alpha}, {b_x = (b_\alpha)_x} for all {x \in W_\alpha}. In particular, for each fixed {\alpha}, {i(W_\alpha)(b|_{W_\alpha}) = \{(b|_{W_\alpha}, W_\alpha)\} = \{(b_\alpha, W_\alpha)\} = i(W_\alpha)(b_\alpha)}, whence, {b|_{W_\alpha} = b_\alpha}, as desired. \Box

Now, suppose that {\mathcal{G}} is a sheaf on {X}. Let {j : \mathcal{F} \rightarrow \mathcal{G}} be a morphism of presheaves.

Claim 2 There exists a unique map {\widetilde{j} : \widetilde{\mathcal{F}} \rightarrow \mathcal{G}}, such that {\widetilde{j} \circ i = j}.

Proof: Let {U \subseteq X} be an open subset and let {\{(b_\alpha, W_\alpha)\}_\alpha \in \widetilde{\mathcal{F}}(U)}. Observe that the commutative diagrams

\displaystyle \begin{array}{rcl} \begin{array}{ccc} \mathcal{F}(W_\alpha) & \stackrel{j(W_\alpha)}{\longrightarrow} & \mathcal{G}(W_\alpha) \\ \downarrow & \; & \downarrow \\ \mathcal{F}(W_\alpha\cap W_\beta) & \stackrel{j(W_\alpha\cap W_\beta)}{\longrightarrow} & \mathcal{G}(W_\alpha\cap W_\beta) \\ \downarrow & \; & \downarrow \\ \mathcal{F}_x & \longrightarrow & \mathcal{G}_x \end{array} \end{array}

ensure that {j(W_\alpha)(b_\alpha)_x = j(W_\beta)(b_\beta)_x} for all {x \in W_\alpha \cap W_\beta}. Thus, {j(W_\alpha)(b_\alpha)|_{W_\alpha \cap W_\beta} = j(W_\beta)(b_\beta)|_{W_\alpha \cap W_\beta}} and so by the sheaf axiom for {\mathcal{G}}, there is a unique extension {\widetilde{j}(U)(\{(b_\alpha, W_\alpha)\}_\alpha) \in \mathcal{G}(U)}. For convenience, we can view {\mathcal{G}} as {\widetilde{\mathcal{G}}}, with {\widetilde{j}(U)(\{(b_\alpha, W_\alpha)\}_\alpha) =\{( j(U)(b_\alpha), W_\alpha)\}_\alpha}. Now, if {U' \subseteq U} is an open set, then

\displaystyle \begin{array}{rcl} \widetilde{j}(U')( \{(b_\alpha, W_\alpha)\}_\alpha|_{U'}) &=& \widetilde{j}(U')(\{(b_\alpha|_{U'}, W_\alpha\cap U')\}_\alpha) \\ &=& \{(j(U')(b_\alpha|_{U'}), W_\alpha\cap U')\}_\alpha \\ &=& \{(j(U)(b_\alpha)|_{U'}, W_\alpha\cap U')\}_\alpha \\ &=& \{(j(U)(b_\alpha), W_\alpha )\}_\alpha|_{U'} \\ &=& \widetilde{j}(U) (\{(b_\alpha, W_\alpha)\})|_{U'}. \end{array}

Thus, {\widetilde{j}} is a morphism {\widetilde{j} : \widetilde{\mathcal{F}} \rightarrow \mathcal{G}}. From the definition of {\widetilde{j}}, it is clear that {\widetilde{j} \circ i = j} and {\widetilde{j}} is unique. \Box


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