====== Grothendieck topos  ======
===== 1. Introduction  =====
This note proceeds the previous note on the topoi of sheaves on topological spaces. The plan at the moment is not to introduce various results of Grothendieck topoi but to covey the readers on how this concept of Grothendieck topology and topos generalizes the ordinary notion of sheaves on topological spaces. The purpose is to provide a stepping stone toward our next topic which is about the Lawvere–Tierney topology.

===== 2. Grothendieck topology  =====

So far we have used the structure of open sets $\mathcal{O}(X)$ to model the concept of locality. For each open $U \in \mathcal{O}(X)$, we attach a set of data locally to $U$, which means basically to define a functor $\mathcal{O}(X)^{op} \to \mathbf{Set}$ where $\mathcal{O}(X)$ is seen as a category. Hence, a straightforward generalization of the usual notion of topology in our point of view is to replace $\mathcal{O}(X)$ with an arbitrary category $\mathcal{C}$ with some structure.

For an object $C$, we say $S$ a **sieve** on $C$ when $S$ is a family of morphisms to $C$ such that 
\[
(f : X \to C)  \in S \Rightarrow f \circ g \in S
\]
for any $g : Y \to X$. (Note that it is equivalent to a subsubject of $\mathbf{y}_C$ in $\mathbf{Sets}^{\mathcal{C}^{op}}$.) Given a morphism $h : D \to C$, define a mapping from the sieves on $C$ to the sieves on $D$ by
\[
h^* : S \mapsto \{f \mid h\circ f \in S\}.
\]
Intuitively, seeing $C$ as an open set, a sieve on $C$ is a collection of opens in $C$.
Hence, a topology determines amongst the sieves, which are open coverings:
A mapping $J$ is a //**Grothendieck topology**// if it maps each object $C$ to a collection of //**sieves**// on $C$ such that 
  1. the maximal sieve $\mathbf{y}_C$ is in $J(C)$, 
  2. for each $S \in J(C)$ and $h : D \to C$, $h^*(S) \in J(D)$, and 
  3. for any sieve $R$ on $C$, if there is $S\in J(C)$ such that $h^*(R) \in J(D)$ for each $h\in S$, then $R\in J(C)$
A pair $(\mathcal{C}, J)$ of a small category and its Grothendieck topology is called a **site**. As the analogy between $\mathcal{O}(X)$ and $(\mathcal{C}, J)$ suggests, we call a sieve $S$ on $C$ a $J$**-covering** of $C$ when $S \in J(C)$. 

A presheaf on a small category $\mathcal{C}$ is a contravariant functor $P : \mathcal{C}^{op} \to \mathbf{Set}$ and we write $\mathbf{PSh}(\mathcal{C})$ for the functor category. For a presheaf $P$ and a Grothendieck topology $J$, for a cover $S \in J(C)$, a **matching family** assigns to each $(f : D \to C) \in S$ an element $x_f \in P(D)$ such that
\[
P(g)(x_f) = x_{f \circ g} \quad \text{for each }g : E \to D.
\]
We call an element $x \in P(X)$ an **amalgamation** of the matching family if 
\[
P(f)(x) = x_f 
\]
for each $f \in S$. A **sheaf** for $J$ is a presheaf where every matching family has a unique amalgamation.

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