用户: Scavenger/Recording Some Facts/A definition of Sites

Definition 0.1. Let $\mathcal{C}$ be a category, a sieve on $\mathcal{C}$ is a full subcategory ${\mathcal{C}}_0\subseteq \mathcal{C}$ having the property that if $C\to D$ is a morphism of $\mathcal{C}$ and $D\in {\mathcal{C}}_0$, then $C\in {\mathcal{C}}_0$.

Let $C\in \mathcal{C}$ be an object, a sieve on $C$ is a sieve on the category ${\mathcal{C}}_{/C}$. Given a morphism $f:D\to C$ of $\mathcal{C}$ and a sieve ${\mathcal{C}}_0\subseteq {\mathcal{C}}_{/C}$ on $C$, we let $f^{\ast }{\mathcal{C}}_0$ denote the sieve on $D$ such that $E\to D$ is an element of $f^{\ast }{\mathcal{C}}_0$ if and only if the composition $E\to D\to C$ is an element of ${\mathcal{C}}_0$.

A Grothendieck topology (or simply a topology) on $\mathcal{C}$ is the data of, for each element $C\in \mathcal{C}$, a collection of sieves on $C$, which we refer to as covering sieves, required to satisfy the following conditions:

(i) If $C\in \mathcal{C}$ is an object, then the sieve ${\mathcal{C}}_{/C}\subseteq {\mathcal{C}}_{/C}$ on $C$ is a covering sieve.

(ii) If $f:D\to C$ is a morphism of $\mathcal{C}$ and ${\mathcal{C}}_0\subseteq {\mathcal{C}}_{/C}$ is a covering sieve on $C$, then $f^{\ast }{\mathcal{C}}_0$ is a covering sieve on $D$.

(iii) Let $C$ be an object of $\mathcal{C}$, $C_0\subseteq {\mathcal{C}}_{/C}$ a covering sieve on $C$, and ${\mathcal{C}}_1\subseteq {\mathcal{C}}_{/C}$ an arbitrary sieve on $C$. Suppose that for each $f:D\to C$ belonging to the sieve ${\mathcal{C}}_0$, $f^{\ast }{\mathcal{C}}_1$ is a covering sieve on $D$, then ${\mathcal{C}}_1$ is a covering sieve on $C$.

A site is a category $\mathcal{C}$ together with a Grothendieck topology on $\mathcal{C}$.

Proposition 0.2 ([2] tag 00Z5). Let $\mathcal{C}$ be a category equipped with a topology. Let $C\in \mathcal{C}$ be an object, let ${\mathcal{C}}_0,{\mathcal{C}}_1\subseteq {\mathcal{C}}_{/C}$ be two covering sieves on $C$, then ${\mathcal{C}}_0\cap {\mathcal{C}}_1$ is again a covering sieve on $C$.

Let ${\mathcal{C}}_3\subseteq {\mathcal{C}}_4\subseteq {\mathcal{C}}_{/C}$ be sieves on $C$ such that ${\mathcal{C}}_3$ is a covering sieve, then ${\mathcal{C}}_4$ is a covering sieve.

Definition 0.3 ([1] Definition 1.3.1.1). Let $\mathcal{T}$ be a category equipped with a topology. Let $\mathcal{C}$ be a category. We say that a functor $F:{\mathcal{T}}^{\mathrm{op}}\to \mathcal{C}$ is a $\mathcal{C}$-valued sheaf on $\mathcal{T}$ if the following condition is satisfied: For every object $U\in \mathcal{T}$ and every covering sieve ${\mathcal{T}}_0\subseteq {\mathcal{T}}_{/U}$ on $U$, the composite map ${\mathcal{T}}_0^{▹}\to ({\mathcal{T}}_{/U}{)}^{▹}\to \mathcal{T}\stackrel{F^{\mathrm{op}}}{\to }{\mathcal{C}}^{\mathrm{op}}$ is a colimit diagram in ${\mathcal{C}}^{\mathrm{op}}$. We let ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{T})$ denote the full subcategory of $\mathrm{Fun}({\mathcal{T}}^{\mathrm{op}},\mathcal{C})$ spanned by the $\mathcal{C}$-valued sheaves on $\mathcal{T}$.

Example 0.4. Let $\mathcal{T}$ be a small category and $\mathcal{C}=\mathrm{Set}$, then this definition coincides with that of [2] tag 00Z8.

Proposition 0.5. Let $\mathcal{C}$ be a small category equipped with a topology. Then the category of sheaves of sets on $\mathcal{C}$ is a reflective subcategory of the category of presheaves of sets on $\mathcal{C}$.

Proof. See [2] tag 00ZM.

$\square$

Example 0.6. Let $\mathcal{C}$ be a category. Let $\mathrm{Cov}$ be a set, each of whose elements is a set $I$ together with an object $U$ of $\mathcal{C}$ and a family $\{U_i\to U{\}}_{i\in I}$ of morphisms of $\mathcal{C}$ with target $U$ indexed by $I$, which we refer to as coverings, required to satisfy the following conditions:

(We will not be majorly concerned with the index sets $I$, so we say that a collection of morphisms with fixed target $\{U_i\to U\}$ is a covering if there exists an index set $I$ and an element $(I,U,\{U_i\to U{\}}_{i\in I})$ of $\mathrm{Cov}$.)

(i) If $V\to U$ is an isomorphism in $\mathcal{C}$, then $\{V\to U\}$ is a covering.

(ii) If $\{U_i\to U{\}}_{i\in I}$ is a covering, and for each $i\in I$, $\{V_{ij}\to U_i{\}}_{j\in J_i}$ is a covering, then the collection of composite morphisms $\{V_{ij}\to U_i\to U\}$ is a covering.

(iii) If $\{U_i\to U\}$ is a covering and $V\to U$ is a morphism, then $U_i{\times }_{U}V$ exists for all $i$ and $\{U_i{\times }_{U}V\to V\}$ is a covering.

Let $U\in \mathcal{C}$ be an element. A sieve ${\mathcal{C}}_0\subseteq {\mathcal{C}}_{/U}$ on $U$ is called a covering sieve if there exists a covering $\{U_i\to U\}$ such that each $U_i\to U$ is an element of ${\mathcal{C}}_0$. Then these covering sieves form a topology on $\mathcal{C}$. Moreover, if $\mathcal{C}$ is small, then a presheaf ${\mathcal{C}}^{\mathrm{op}}\to \mathrm{Set}$ is a sheaf in this topology if and only if it is a sheaf with respect to the set of coverings $\mathrm{Cov}$.

References

[1]	Jacob Lurie, “Spectral Algebraic Geometry”, February 3, 2018 version, Avalible for download at https://www.math.ias.edu/ lurie/papers/SAG-rootfile.pdf
[2]	The Stacks project authors, “The Stacks project”, https://stacks.math.columbia.edu