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\begin{document}

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\section*{Grothendieck topology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Saturation}{Saturation}\dotfill \pageref*{Saturation} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{topology_on_open_subsets_of_a_topological_space}{Topology on open subsets of a topological space}\dotfill \pageref*{topology_on_open_subsets_of_a_topological_space} \linebreak
\noindent\hyperlink{regular_topology}{Regular topology}\dotfill \pageref*{regular_topology} \linebreak
\noindent\hyperlink{extensive_topology}{Extensive topology}\dotfill \pageref*{extensive_topology} \linebreak
\noindent\hyperlink{coherent_topology}{Coherent topology}\dotfill \pageref*{coherent_topology} \linebreak
\noindent\hyperlink{canonical_topology}{Canonical topology}\dotfill \pageref*{canonical_topology} \linebreak
\noindent\hyperlink{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{Grothendieck topology} on a category is a choice of morphisms in that category which are regarded as [[cover|covers]].

A category equipped with a Grothendieck topology is a [[site]]. Sometimes all sites are required to be [[small category|small]].

Probably the main point of having a site is so that one can define [[sheaf|sheaves]], or more generally [[stack|stacks]], on it. In particular, the category of sheaves on a (small) site is a [[Grothendieck topos]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{Grothendieck topology} $J$ on a [[category]] $C$ is an assignment to each object $c \in C$ of a collection of [[sieves]] on $c$ which are called \emph{covering sieves}, satisfying the following axioms:

\begin{enumerate}%
\item If $F$ is a [[sieve]] that covers $c$ and $g: d \to c$ is any morphism, then the pullback sieve $g^* F$ covers $d$.


\item The [[maximal sieve]] $id: \hom(-, c) \hookrightarrow \hom(-, c)$ is always a covering sieve;


\item Two [[sieve|sieves]] $F, G$ of $c$ cover $c$ if and only if their intersection $F \cap G$ covers $c$. (Here the saturation condition is important.)


\item If $F$ is a sieve on $c$ such that the sieve $\bigcup_d \{g: d \to c| g^* F \; covers \; d\}$ is a covering sieve of $c$, then $F$ itself covers $c$.



\end{enumerate}
\end{defn}
The set of covering sieves of an object $c$ is denoted $J(c)$.

A category equipped with a Grothendieck topology is called a \textbf{[[site]]} .

\begin{remark}
\label{}\hypertarget{}{}
The first axiom guarantees that we have a functor $J: C^{op} \to Set$. Thus $J$ itself can be regarded as an object of the [[presheaf topos]] $[C^{op},Set]$; in this way Grothendieck topologies on $C$ are identified with [[Lawvere-Tierney topologies]] on $[C^{op},Set]$.

\end{remark}
Given a Grothendieck topology $J$ on a [[small category]] $C$, one can define the category $Sh(C,J)$ of [[sheaves]] on $C$ relative to $J$, which is a [[reflective subcategory]] of the category $[C^{op},Set]$ of [[presheaves]] on $C$. Thus we have a functor $C\to Sh(C,J)$ given by the composite of the [[Yoneda embedding]] with the reflection (or ``sheafification''). This composite functor is [[full and faithful functor|fully faithful]] if and only if all representable presheaves are sheaves for $J$; a topology with this property is called [[subcanonical site|subcanonical]].

\hypertarget{Saturation}{}\subsection*{{Saturation}}\label{Saturation}

Grothendieck topologies may be and in practice quite often are obtained as closures of collections of morphisms that are not yet closed under the operations above (that are not yet sieves, not yet pullback stable, etc.).

Two notions of such unsaturated collections of morphisms inducing Grothendieck topologies are

\begin{itemize}%
\item [[Grothendieck pretopology]],


\item [[coverage]].



\end{itemize}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{topology_on_open_subsets_of_a_topological_space}{}\subsubsection*{{Topology on open subsets of a topological space}}\label{topology_on_open_subsets_of_a_topological_space}

The archetypical example of a Grothendieck topology is that on a [[category of open subsets]] $Op(X)$ of a [[topological space]] $X$. A covering family of an open subset $U \subset X$ is a collection of open subsets $V_i \subset U$ that cover $U$ in the ordinary sense of the word, i.e. which are such that every point $x \in U$ is in at least one of the $V_i$.

\hypertarget{regular_topology}{}\subsubsection*{{Regular topology}}\label{regular_topology}

Any [[regular category]] $C$ admits a [[subcanonical site|subcanonical]] Grothendieck topology whose covering families are generated by single [[regular epimorphisms]]. If $C$ is [[exact category|exact]] or has pullback-stable [[reflexive coequalizer]]s, then its [[codomain fibration]] is a [[stack]] for this topology (the necessary and sufficient condition is that any pullback of a kernel pair is again a kernel pair).

\hypertarget{extensive_topology}{}\subsubsection*{{Extensive topology}}\label{extensive_topology}

Any [[extensive category]] admits a [[Grothendieck topology]] whose [[cover|covering families]] are (generated by) the families of inclusions into a [[coproduct]] (finite or small, as appropriate). We call this the \textbf{extensive [[coverage]]} or \textbf{extensive topology}. The [[codomain fibration]] of any extensive category is a [[stack]] for its extensive topology.

\hypertarget{coherent_topology}{}\subsubsection*{{Coherent topology}}\label{coherent_topology}

Any [[coherent category]] $C$ admits a [[subcanonical site|subcanonical]] Grothendieck topology in which the covering families are generated by finite, jointly [[regular epimorphism|regular-epimorphic]] families. Equivalently, they are generated by single [[regular epimorphisms]] and by finite unions of [[subobject]]s. If $C$ is [[extensive category|extensive]], then its coherent topology is generated by the regular topology together with the extensive topology. (In fact, the coherent topology is [[superextensive site|superextensive]].)

\hypertarget{canonical_topology}{}\subsubsection*{{Canonical topology}}\label{canonical_topology}

On any category there is a largest subcanonical topology. This is called the \emph{[[canonical topology]]}, with ``subcanonical'' a back-formation from this (since a topology is subcanonical iff it is contained in the canonical topology). On a [[Grothendieck topos]], the covering families in the canonical topology are those which are jointly epimorphic.

\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

A more general notion is simply a collection of ``covering families,'' not necessarily sieves, satisfying only pullback-stability; this suffices to define an equivalent notion of sheaf. Following the [[Elephant]], we call such a system a [[coverage]]. A Grothendieck topology may then be defined as a coverage that consists of sieves (which the Elephant calls ``sifted'') and satisfies certain extra saturation conditions; see [[coverage]] for details.

An intermediate notion is that of a [[Grothendieck pretopology]], which consists of covering families that satisfy some, but not all, of the closure conditions for a Grothendieck topology. Many examples are ``naturally'' pretopologies, but must be ``saturated'' under the remaining closure conditions to produce Grothendieck topologies.

As remarked above, Grothendieck topologies on a small category $C$ are also in bijective correspondence with [[Lawvere-Tierney topology|Lawvere-Tierney topologies]] on the [[presheaf]] [[topos]] $[C^{op},Set]$. See [[Lawvere-Tierney topology]] for a description of the correspondence.

See also

\begin{itemize}%
\item [[coverage]], [[site]], [[Lawvere-Tierney topology]], [[Grothendieck pretopology]], [[Q-category]], [[cd-structure]]


\item [[Weiss topology]]


\item [[(∞,1)-Grothendieck topology]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Standard texbooks inlcude

\begin{itemize}%
\item [[Saunders MacLane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]}


\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}



\end{itemize}
Discussions of variants of the notion and its variants is at \emph{[[historical notes on Grothendieck topology]]}.

[[!redirects Grothendieck topologies]]



\end{document}