Lemma 17.19.1. Let X be a topological space. Let \mathcal{B} be a basis for the topology on X. Let \mathcal{F} be a sheaf of sets on X. There exists a set I and for each i \in I an element U_ i \in \mathcal{B} and a finite set S_ i such that there exists a surjection \coprod _{i \in I} j_{U_ i!}\underline{S_ i} \to \mathcal{F}.
17.19 Constructible sheaves of sets
Let X be a topological space. Given a set S recall that \underline{S} or \underline{S}_ X denotes the constant sheaf with value S, see Sheaves, Definition 6.7.4. Let U \subset X be an open of a topological space X. We will denote j_ U the inclusion morphism and we will denote j_{U!} : \mathop{\mathit{Sh}}\nolimits (U) \to \mathop{\mathit{Sh}}\nolimits (X) the extension by the empty set described in Sheaves, Section 6.31.
Proof. Let S be a singleton set. We will prove the result with S_ i = S. For every x \in X and element s \in \mathcal{F}_ x we can choose a U(x, s) \in \mathcal{B} and s(x, s) \in \mathcal{F}(U(x, s)) which maps to s in \mathcal{F}_ x. By Sheaves, Lemma 6.31.4 the section s(x, s) corresponds to a map of sheaves j_{U(x, s)!}\underline{S} \to \mathcal{F}. Then
\coprod \nolimits _{(x, s)} j_{U(x, s)!}\underline{S} \to \mathcal{F}
is surjective on stalks and hence surjective. \square
Lemma 17.19.2. Let X be a topological space. Let \mathcal{B} be a basis for the topology of X and assume that each U \in \mathcal{B} is quasi-compact. Then every sheaf of sets on X is a filtered colimit of sheaves of the form
17.19.2.1
\begin{equation} \label{modules-equation-towards-constructible-sets} \text{Coequalizer}\left( \xymatrix{ \coprod \nolimits _{b = 1, \ldots , m} j_{V_ b!}\underline{S_ b} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{a = 1, \ldots , n} j_{U_ a!}\underline{S_ a} } \right) \end{equation}
with U_ a and V_ b in \mathcal{B} and S_ a and S_ b finite sets.
Proof. By Lemma 17.19.1 every sheaf of sets \mathcal{F} is the target of a surjection whose source \mathcal{F}_0 is a coproduct of sheaves the form j_{U!}\underline{S} with U \in \mathcal{B} and S finite. Applying this to \mathcal{F}_0 \times _\mathcal {F} \mathcal{F}_0 we find that \mathcal{F} is a coequalizer of a pair of maps
\xymatrix{ \coprod \nolimits _{b \in B} j_{V_ b!}\underline{S_ b} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{a \in A} j_{U_ a!}\underline{S_ a} }
for some index sets A, B and V_ b and U_ a in \mathcal{B} and S_ a and S_ b finite. For every finite subset B' \subset B there is a finite subset A' \subset A such that the coproduct over b \in B' maps into the coproduct over a \in A' via both maps. Namely, we can view the right hand side as a filtered colimit with injective transition maps. Hence taking sections over the quasi-compact opens V_ b, b \in B' commutes with this coproduct, see Sheaves, Lemma 6.29.1. Thus our sheaf is the colimit of the cokernels of these maps between finite coproducts. \square
Lemma 17.19.3. Let X be a spectral topological space. Let \mathcal{B} be the set of quasi-compact open subsets of X. Let \mathcal{F} be a sheaf of sets as in Equation (17.19.2.1). Then there exists a continuous spectral map f : X \to Y to a finite sober topological space Y and a sheaf of sets \mathcal{G} on Y with finite stalks such that f^{-1}\mathcal{G} \cong \mathcal{F}.
Proof. We can write X = \mathop{\mathrm{lim}}\nolimits X_ i as a directed limit of finite sober spaces, see Topology, Lemma 5.23.14. Of course the transition maps X_{i'} \to X_ i are spectral and hence by Topology, Lemma 5.24.5 the maps p_ i : X \to X_ i are spectral. For some i we can find opens U_{a, i} and V_{b, i} of X_ i whose inverse images are U_ a and V_ b, see Topology, Lemma 5.24.6. The two maps
\beta , \gamma : \coprod \nolimits _{b \in B} j_{V_ b!}\underline{S_ b} \longrightarrow \coprod \nolimits _{a \in A} j_{U_ a!}\underline{S_ a}
whose coequalizer is \mathcal{F} correspond by adjunction to two families
\beta _ b, \gamma _ b : S_ b \longrightarrow \Gamma (V_ b, \coprod \nolimits _{a \in A} j_{U_ a!}\underline{S_ a}), \quad b \in B
of maps of sets. Observe that p_ i^{-1}(j_{U_{a, i}!}\underline{S_ a}) = j_{U_ a!}\underline{S_ a} and (X_{i'} \to X_ i)^{-1}(j_{U_{a, i}!}\underline{S_ a}) = j_{U_{a, i'}!}\underline{S_ a}. It follows from Sheaves, Lemma 6.29.3 (and using that S_ b and B are finite sets) that after increasing i we find maps
\beta _{b, i}, \gamma _{b, i} : S_ b \longrightarrow \Gamma (V_{b, i}, \coprod \nolimits _{a \in A} j_{U_{a, i}!}\underline{S_ a}) , \quad b \in B
which give rise to the maps \beta _ b and \gamma _ b after pulling back by p_ i. These maps correspond in turn to maps of sheaves
\beta _ i, \gamma _ i : \coprod \nolimits _{b \in B} j_{V_{b, i}!}\underline{S_ b} \longrightarrow \coprod \nolimits _{a \in A} j_{U_{a, i}!}\underline{S_ a}
on X_ i. Then we can take Y = X_ i and
\mathcal{G} = \text{Coequalizer}\left( \xymatrix{ \coprod \nolimits _{b = 1, \ldots , m} j_{V_{b, i}!}\underline{S_ b} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{a = 1, \ldots , n} j_{U_{a, i}!}\underline{S_ a} } \right)
We omit some details. \square
Lemma 17.19.4. Let X be a spectral topological space. Let \mathcal{B} be the set of quasi-compact open subsets of X. Let \mathcal{F} be a sheaf of sets as in Equation (17.19.2.1). Then there exist finitely many constructible closed subsets Z_1, \ldots , Z_ n \subset X and finite sets S_ i such that \mathcal{F} is isomorphic to a subsheaf of \prod (Z_ i \to X)_*\underline{S_ i}.
Proof. By Lemma 17.19.3 we reduce to the case of a finite sober topological space and a sheaf with finite stalks. In this case \mathcal{F} \subset \prod _{x \in X} i_{x, *}\mathcal{F}_ x where i_ x : \{ x\} \to X is the embedding. We omit the proof that i_{x, *}\mathcal{F}_ x is a constant sheaf on \overline{\{ x\} }. \square
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