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The Stacks project

7.50 Topologies and sheaves

Lemma 7.50.1. Let \mathcal{C} be a category endowed with a topology J. Let U be an object of \mathcal{C}. Let S be a sieve on U. The following are equivalent

  1. The sieve S is a covering sieve.

  2. The sheafification S^\# \to h_ U^\# of the map S \to h_ U is an isomorphism.

Proof. First we make a couple of general remarks. We will use that S^\# = LLS, and h_ U^\# = LLh_ U. In particular, by Lemma 7.49.1, we see that S^\# \to h_ U^\# is injective. Note that \text{id}_ U \in h_ U(U). Hence it gives rise to sections of Lh_ U and h_ U^\# = LLh_ U over U which we will also denote \text{id}_ U.

Suppose S is a covering sieve. It clearly suffices to find a morphism h_ U \to S^\# such that the composition h_ U \to h_ U^\# is the canonical map. To find such a map it suffices to find a section s \in S^\# (U) which restricts to \text{id}_ U. But since S is a covering sieve, the element \text{id}_ S \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, S) gives rise to a section of LS over U which restricts to \text{id}_ U in Lh_ U. Hence we win.

Suppose that S^\# \to h_ U^\# is an isomorphism. Let 1 \in S^\# (U) be the element corresponding to \text{id}_ U in h_ U^\# (U). Because S^\# = LLS there exists a covering sieve S' on U such that 1 comes from a

\varphi \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S', LS).

This in turn means that for every \alpha : V \to U, \alpha \in S'(V) there exists a covering sieve S_{V, \alpha } on V such that \varphi (\alpha ) corresponds to a morphism of presheaves S_{V, \alpha } \to S. In other words S_{V, \alpha } is contained in S \times _ U V. By the second axiom of a topology we see that S is a covering sieve. \square

Theorem 7.50.2. Let \mathcal{C} be a category. Let J, J' be topologies on \mathcal{C}. The following are equivalent

  1. J = J',

  2. sheaves for the topology J are the same as sheaves for the topology J'.

Proof. It is a tautology that if J = J' then the notions of sheaves are the same. Conversely, Lemma 7.50.1 characterizes covering sieves in terms of the sheafification functor. But the sheafification functor \textit{PSh}(\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J) is the left adjoint of the inclusion functor \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J) \to \textit{PSh}(\mathcal{C}). Hence if the subcategories \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J) and \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J') are the same, then the sheafification functors are the same and hence the collections of covering sieves are the same. \square

Lemma 7.50.3. Assumption and notation as in Theorem 7.50.2. Then J \subset J' if and only if every sheaf for the topology J' is a sheaf for the topology J.

Proof. One direction is clear. For the other direction suppose that \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J') \subset \mathop{\mathit{Sh}}\nolimits (\mathcal{C}, J). By formal nonsense this implies that if \mathcal{F} is a presheaf of sets, and \mathcal{F} \to \mathcal{F}^\# , resp. \mathcal{F} \to \mathcal{F}^{\# , \prime } is the sheafification wrt J, resp. J' then there is a canonical map \mathcal{F}^\# \to \mathcal{F}^{\# , \prime } such that \mathcal{F} \to \mathcal{F}^\# \to \mathcal{F}^{\# , \prime } equals the canonical map \mathcal{F} \to \mathcal{F}^{\# , \prime }. Of course, \mathcal{F}^\# \to \mathcal{F}^{\# , \prime } identifies the second sheaf as the sheafification of the first with respect to the topology J'. Apply this to the map S \to h_ U of Lemma 7.50.1. We get a commutative diagram

\xymatrix{ S \ar[r] \ar[d] & S^\# \ar[r] \ar[d] & S^{\# , \prime } \ar[d] \\ h_ U \ar[r] & h_ U^\# \ar[r] & h_ U^{\# , \prime } }

And clearly, if S is a covering sieve for the topology J then the middle vertical map is an isomorphism (by the lemma) and we conclude that the right vertical map is an isomorphism as it is the sheafification of the one in the middle wrt J'. By the lemma again we conclude that S is a covering sieve for J' as well. \square


Comments (3)

Comment #8092 by Andrea Panontin on

In the proof of theorem 00ZP you write "the sheafification functor is the right adjoint of the inclusion functor", it should be the left adjoint.

Comment #8093 by Andrea Panontin on

In Lemma 00ZO I think that should be .


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