Definition 7.47.1. Let \mathcal{C} be a category. Let U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}). A sieve S on U is a subpresheaf S \subset h_ U.
7.47 Topologies
In this section we define what a topology on a category is as defined in [SGA4]. One can develop all of the machinery of sheaves and topoi in this language. A modern exposition of this material can be found in [KS]. However, the case of most interest for algebraic geometry is the topology defined by a site on its underlying category. Thus we strongly suggest the first time reader skip this section and all other sections of this chapter!
In other words, a sieve on U picks out for each object T \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) a subset S(T) of the set of all morphisms T \to U. In fact, the only condition on the collection of subsets S(T) \subset h_ U(T) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(T, U) is the following rule
7.47.1.1
\begin{equation} \label{sites-equation-property-sieve} \left. \begin{matrix} (\alpha : T \to U) \in S(T)
\\ g : T' \to T
\end{matrix} \right\} \Rightarrow (\alpha \circ g : T' \to U) \in S(T') \end{equation}
A good mental picture to keep in mind is to think of the map S \to h_ U as a “morphism from S to U”.
Lemma 7.47.2. Let \mathcal{C} be a category. Let U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}).
The collection of sieves on U is a set.
Inclusion defines a partial ordering on this set.
Unions and intersections of sieves are sieves.
Given a family of morphisms \{ U_ i \to U\} _{i\in I} of \mathcal{C} with target U there exists a unique smallest sieve S on U such that each U_ i \to U belongs to S(U_ i).
The sieve S = h_ U is the maximal sieve.
The empty subpresheaf is the minimal sieve.
Proof. By our definition of subpresheaf, the collection of all subpresheaves of a presheaf \mathcal{F} is a subset of \prod _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} \mathcal{P}(\mathcal{F}(U)). And this is a set. (Here \mathcal{P}(A) denotes the powerset of A.) Hence the collection of sieves on U is a set.
The partial ordering is defined by: S \leq S' if and only if S(T) \subset S'(T) for all T \to U. Notation: S \subset S'.
Given a collection of sieves S_ i, i \in I on U we can define \bigcup S_ i as the sieve with values (\bigcup S_ i)(T) = \bigcup S_ i(T) for all T \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}). We define the intersection \bigcap S_ i in the same way.
Given \{ U_ i \to U\} _{i\in I} as in the statement, consider the morphisms of presheaves h_{U_ i} \to h_ U. We simply define S as the union of the images (Definition 7.3.5) of these maps of presheaves.
The last two statements of the lemma are obvious. \square
Definition 7.47.3. Let \mathcal{C} be a category. Given a family of morphisms \{ f_ i : U_ i \to U\} _{i\in I} of \mathcal{C} with target U we say the sieve S on U described in Lemma 7.47.2 part (4) is the sieve on U generated by the morphisms f_ i.
Definition 7.47.4. Let \mathcal{C} be a category. Let f : V \to U be a morphism of \mathcal{C}. Let S \subset h_ U be a sieve. We define the pullback of S by f to be the sieve S \times _ U V of V defined by the rule
(\alpha : T \to V) \in (S \times _ U V)(T) \Leftrightarrow (f \circ \alpha : T \to U) \in S(T)
We leave it to the reader to see that this is indeed a sieve (hint: use Equation 7.47.1.1). We also sometimes call S \times _ U V the base change of S by f : V \to U.
Lemma 7.47.5. Let \mathcal{C} be a category. Let U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}). Let S be a sieve on U. If f : V \to U is in S, then S \times _ U V = h_ V is maximal.
Proof. Trivial from the definitions. \square
Definition 7.47.6. Let \mathcal{C} be a category. A topology on \mathcal{C} is given by a rule which assigns to every U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) a subset J(U) of the set of all sieves on U satisfying the following conditions
For every morphism f : V \to U in \mathcal{C}, and every element S \in J(U) the pullback S \times _ U V is an element of J(V).
If S and S' are sieves on U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}), if S \in J(U), and if for all f \in S(V) the pullback S' \times _ U V belongs to J(V), then S' belongs to J(U).
For every U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) the maximal sieve S = h_ U belongs to J(U).
In this case, the sieves belonging to J(U) are called the covering sieves.
Lemma 7.47.7. Let \mathcal{C} be a category. Let J be a topology on \mathcal{C}. Let U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}).
Finite intersections of elements of J(U) are in J(U).
If S \in J(U) and S' \supset S, then S' \in J(U).
Proof. Let S, S' \in J(U). Consider S'' = S \cap S'. For every V \to U in S(U) we have
S' \times _ U V = S'' \times _ U V
simply because V \to U already is in S. Hence by the second axiom of the definition we see that S'' \in J(U).
Let S \in J(U) and S' \supset S. For every V \to U in S(U) we have S' \times _ U V = h_ V by Lemma 7.47.5. Thus S' \times _ U V \in J(V) by the third axiom. Hence S' \in J(U) by the second axiom. \square
Definition 7.47.8. Let \mathcal{C} be a category. Let J, J' be two topologies on \mathcal{C}. We say that J is finer or stronger than J' if and only if for every object U of \mathcal{C} we have J'(U) \subset J(U). In this case we also say that J' is coarser or weaker than J.
In other words, any covering sieve of J' is a covering sieve of J. There exists a finest topology on \mathcal{C}, namely that topology where any sieve is a covering sieve. This is called the discrete topology of \mathcal{C}. There also exists a coarsest topology. Namely, the topology where J(U) = \{ h_ U\} for all objects U. This is called the chaotic or indiscrete topology.
Lemma 7.47.9. Let \mathcal{C} be a category. Let \{ J_ i\} _{i\in I} be a set of topologies.
The rule J(U) = \bigcap J_ i(U) defines a topology on \mathcal{C}.
There is a coarsest topology finer than all of the topologies J_ i.
Proof. The first part is direct from the definitions. The second follows by taking the intersection of all topologies finer than all of the J_ i. \square
At this point we can define without any motivation what a sheaf is.
Definition 7.47.10. Let \mathcal{C} be a category endowed with a topology J. Let \mathcal{F} be a presheaf of sets on \mathcal{C}. We say that \mathcal{F} is a sheaf on \mathcal{C} if for every U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) and for every covering sieve S of U the canonical map
\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S, \mathcal{F})
is bijective.
Recall that the left hand side of the displayed formula equals \mathcal{F}(U). In other words, \mathcal{F} is a sheaf if and only if a section of \mathcal{F} over U is the same thing as a compatible collection of sections s_{T, \alpha } \in \mathcal{F}(T) parametrized by (\alpha : T \to U) \in S(T), and this for every covering sieve S on U.
Lemma 7.47.11. Let \mathcal{C} be a category. Let \{ \mathcal{F}_ i \} _{i\in I} be a collection of presheaves of sets on \mathcal{C}. For each U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) denote J(U) the set of sieves S with the following property: For every morphism V \to U, the maps
\mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ V, \mathcal{F}_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S \times _ U V, \mathcal{F}_ i)
are bijective for all i \in I. Then J defines a topology on \mathcal{C}. This topology is the finest topology in which all of the \mathcal{F}_ i are sheaves.
Proof. If we show that J is a topology, then the last statement of the lemma immediately follows. The first and third axioms of a topology are immediately verified. Thus, assume that we have an object U, and sieves S, S' of U such that S \in J(U), and for all V \to U in S(V) we have S' \times _ U V \in J(V). We have to show that S' \in J(U). In other words, we have to show that for any f : W \to U, the maps
\mathcal{F}_ i(W) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ W, \mathcal{F}_ i) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S' \times _ U W, \mathcal{F}_ i)
are bijective for all i \in I. Pick an element i \in I and pick an element \varphi \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S' \times _ U W, \mathcal{F}_ i). We will construct a section s \in \mathcal{F}_ i(W) mapping to \varphi .
Suppose \alpha : V \to W is an element of S \times _ U W. According to the definition of pullbacks we see that the composition f \circ \alpha : V \to W \to U is in S. Hence S' \times _ U V is in J(W) by assumption on the pair of sieves S, S'. Now we have a commutative diagram of presheaves
\xymatrix{ S' \times _ U V \ar[r] \ar[d] & h_ V \ar[d] \\ S' \times _ U W \ar[r] & h_ W }
The restriction of \varphi to S' \times _ U V corresponds to an element s_{V, \alpha } \in \mathcal{F}_ i(V). This we see from the definition of J, and because S' \times _ U V is in J(W). We leave it to the reader to check that the rule (V, \alpha ) \mapsto s_{V, \alpha } defines an element \psi \in \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(S \times _ U W, \mathcal{F}_ i). Since S \in J(U) we see immediately from the definition of J that \psi corresponds to an element s of \mathcal{F}_ i(W).
We leave it to the reader to verify that the construction \varphi \mapsto s is inverse to the natural map displayed above. \square
Definition 7.47.12. Let \mathcal{C} be a category. The finest topology on \mathcal{C} such that all representable presheaves are sheaves, see Lemma 7.47.11, is called the canonical topology of \mathcal{C}.
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