Grothendieck Topos

SiteSieveGrothendieck topology and site

 

Site

Sieve

Definition. A sieve on an object $c$$c$ of a small category $\mathcal{C}$$\mathcal C$ is a set of arrow in $\mathcal{C}$$\mathcal C$ with codomain $c$$c$ such that for any

$$\begin{array}{}\text{(1)}& (f:X\to c)\in S,g:Y\to X⟹f\circ g:(Y\to c)\in S\end{array}$$

Example.

Let $(P,\le )$$(P,\le)$ be a partial order set, hence a small category. Then $c\downarrow :=\{x\in P:x\le c\}$$c\downarrow:=\{x\in P:x\le c\}$ is a sieve on $c$$c$.

Indeed, for any object $c\in \mathcal{C}$$c\in\mathcal C$, we can define the maximal sieve $M_c:=\{f|\mathrm{cod}(f)=c\}$$M_c:=\{f|\mathrm{cod}(f)=c\}$, i.e. $\bigcup _{X\in \mathcal{C}}{\mathrm{Hom}}_{\mathcal{C}}(X,c)$$\bigcup_{X\in\mathcal C}\mathrm{Hom}_{\mathcal C}(X,c)$.

We will use the maximal sieve to definr Grothendicek topology latter.

Definition. A sieve $S$$S$ is said to be generated by a family of $\mathcal{F}$$\mathcal F$ of arrows contained in it if every arrow in $S$$S$ factor through an arrow in $\mathcal{F}$$\mathcal F$​.

Example.

The maximal sieve $M_c:=\{f|\mathrm{cod}(f)=c\}$$M_c:=\{f|\mathrm{cod}(f)=c\}$ is generated by ${\mathrm{id}}_c$$\mathrm{id}_c$.

Let $(X,\tau )$$(X,\tau)$ be a topological space and $\mathcal{O}(X)$$\mathcal O(X)$ be the open set category of $X$$X$. For $U\in \mathcal{O}(X)$$U\in\mathcal O(X)$, we can define a sieve on $U$$U$ as follows: Let $\{U_i\to U{\}}_{i\in I}$$\{U_i\rightarrow U\}_{i\in I}$ be a covering of $U$$U$ we can generate a sieve $s(U)$$s(U)$,

$$\begin{array}{}\text{(2)}& s(U):=\{V\to U|\mathrm{\exists }i\in I,V\to U_i\}\end{array}$$

Definition. Let $\mathcal{C}$$\mathcal C$ be a small category and $f:d\to c$$f:d\to c$ be a morphism. For a sieve $S$$S$ on $C$$C$, we can define sieve on $d$$d$ as

$$\begin{array}{}\text{(3)}& f^{\ast }(S):=\{g:e\to d|f\circ g\in S\}\end{array}$$

Hence we get a functor $\mathcal{J}:{\mathcal{C}}^{\mathrm{op}}\to \mathrm{Set}$$\mathcal J:\mathcal C^{\mathrm{op}}\to\mathrm{Set}$, $\mathcal{J}(c)$$\mathcal J(c)$ is the set of all the sieves on $c$$c$, and $\mathcal{J}(f)(S)=f^{\ast }(S)$$\mathcal J(f)(S)=f^*(S)$.

Grothendieck topology and site

The primary motivation for defining a Grothendieck topology is to introduce a notion of “local data,” enabling the definition of sheaves on general categories. While in traditional topological spaces, sheaves can be defined directly using open covers, in broader categorical contexts, we rely on Grothendieck topologies to achieve this goal.

The object $c$$c$ in $\mathcal{C}$$\mathcal C$ should be imaged as open set and an element in $J(c)$$J(c)$ should be imaged as covering of $c$$c$.

Grothendieck topology

A Grothendieck topology on a small category $\mathcal{C}$$\mathcal C$ is a functor $J:{\mathcal{C}}^{\mathrm{op}}\to \mathrm{Set}$$J:\mathcal C^{\mathrm{op}}\to\mathrm{Set}$ such that:

For each $c\in C$$c\in C$, $J(c)$$J(c)$ is a set of sieves on $c$$c$ satisfies $M_c\in J(c)$$M_c\in J(c)$, and for $S\in J(c),J(f)(S)=f^{\ast }(S)\in J(d)$$S\in J(c),J(f)(S)=f^*(S)\in J(d)$.

Also, if let $S$$S$ be a sieve on $c$$c$ and $T\in J(c)$$T\in J(c)$. If $\mathrm{\forall }f\in T,f^{\ast }(S)\in J(\mathrm{dom}(f))$$\forall f\in T,f^*(S)\in J(\mathrm{dom}(f))$ then $S\in J(c)$$S\in J(c)$​.

A sieve on an object $c$$c$ is said to be$J$$J$-covering, for a Grothendieck topology $J$$J$ on $\mathcal{C}$$\mathcal C$, if it belongs to $J(c)$$J(c)$.

Example.

For a topological space $(X,\tau )$$(X,\tau)$, we can define a Grothendieck topology on $\mathcal{O}(X)$$\mathcal O(X)$ as follows:

Let $J(U)$$J(U)$ be the set of all the sieves generated by some open covering of $U$$U$.

Let $\mathcal{F}$$\mathcal F$ be a sheaf on $X$$X$, and consider $\mathcal{F}(U)$$\mathcal F(U)$, you can get all the local informations from $J(U)$$J(U)$ since it contains all the open covering of $U$$U$ and all the sub open set $V\subseteq U$$V\subseteq U$.

Site

A site is a pair $(\mathcal{C},J)$$(\mathcal C, J)$ consisting of a small category $\mathcal{C}$$\mathcal C$ and a Grothendieck topology $J$$J$ on $C$$C$.