hypercovering-lemma-injective-trivial-cech-variant

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}$. Let $K$ be a hypercovering of $\mathcal{G}$. Let $\mathcal{I}$ be an injective sheaf of abelian groups on $\mathcal{C}$. Then $$ \check{H}^p(K, \mathcal{I}) = \left\{ \begin{matrix} H^0(\mathcal{G}, \mathcal{I}) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right. $$


hypercovering-lemma-cech-spectral-sequence-variant

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}$. Let $K$ be a hypercovering of $\mathcal{G}$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. There is a map $$ s(\mathcal{F}(K)) \longrightarrow R\Gamma(\mathcal{G}, \mathcal{F}) $$ in $D^{+}(\textit{Ab})$ functorial in $\mathcal{F}$, which induces a natural transformation $$ \check{H}^i(K, -) \longrightarrow H^i(\mathcal{G}, -) $$ of functors $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$. Moreover, there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$ E_2^{p, q} = \check{H}^p(K, \underline{H}^q(\mathcal{F})) $$ converging to $H^{p + q}(\mathcal{G}, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$ and in the hypercovering $K$.


hypercovering-lemma-funny-gamma

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K, L, M$ be simplicial objects of $\text{SR}(\mathcal{C}, X)$. Let $a : K \to L$, $b : M \to L$ be morphisms. Assume \begin{enumerate} \item $K$ is a hypercovering of $X$, \item the morphism $M_0 \to L_0$ is a covering, and \item for all $n \geq 0$ in the diagram $$ \xymatrix{ M_{n + 1} \ar[dd] \ar[rr] \ar[rd]^\gamma & & (\text{cosk}_n \text{sk}_n M)_{n + 1} \ar[dd] \\ & L_{n + 1} \times_{(\text{cosk}_n \text{sk}_n L)_{n + 1}} (\text{cosk}_n \text{sk}_n M)_{n + 1} \ar[ld] \ar[ru] & \\ L_{n + 1} \ar[rr] & & (\text{cosk}_n \text{sk}_n L)_{n + 1} } $$ the arrow $\gamma$ is a covering. \end{enumerate} Then the fibre product $K \times_L M$ is a hypercovering of $X$.


hypercovering-lemma-covering

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $k \geq 0$ be an integer. Let $u : Z \to K_k$ be a covering in $\text{SR}(\mathcal{C}, X)$. Then there exists a morphism of hypercoverings $f: L \to K$ such that $L_k \to K_k$ factors through $u$.


hypercovering-lemma-one-more-simplex

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $U \subset V$ be simplicial sets, with $U_n, V_n$ finite nonempty for all $n$. Assume that $U$ has finitely many nondegenerate simplices. Suppose $n \geq 0$ and $x \in V_n$, $x \not \in U_n$ are such that \begin{enumerate} \item $V_i = U_i$ for $i < n$, \item $V_n = U_n \cup \{x\}$, \item any $z \in V_j$, $z \not \in U_j$ for $j > n$ is degenerate. \end{enumerate} Then the morphism $$ \Hom(V, K)_0 \longrightarrow \Hom(U, K)_0 $$ of $\text{SR}(\mathcal{C}, X)$ is a covering.


hypercovering-lemma-add-simplices

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $U \subset V$ be simplicial sets, with $U_n, V_n$ finite nonempty for all $n$. Assume that $U$ and $V$ have finitely many nondegenerate simplices. Then the morphism $$ \Hom(V, K)_0 \longrightarrow \Hom(U, K)_0 $$ of $\text{SR}(\mathcal{C}, X)$ is a covering.


hypercovering-lemma-degeneracy-maps-coverings

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Then \begin{enumerate} \item $K_n$ is a covering of $X$ for each $n \geq 0$, \item $d^n_i : K_n \to K_{n - 1}$ is a covering for all $n \geq 1$ and $0 \leq i \leq n$. \end{enumerate}


hypercovering-lemma-hom-hypercovering

Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $L$ be a simplicial object of $\text{SR}(\mathcal{C}, X)$. Let $n \geq 0$. Consider the commutative diagram \begin{equation} \label{equation-diagram} \xymatrix{ \Hom(\Delta[1], L)_{n + 1} \ar[r] \ar[d] & (\text{cosk}_n \text{sk}_n \Hom(\Delta[1], L))_{n + 1} \ar[d] \\ (L \times L)_{n + 1} \ar[r] & (\text{cosk}_n \text{sk}_n (L \times L))_{n + 1} } \end{equation} coming from the morphism defined above. We can identify the terms in this diagram as follows, where $\partial \Delta[n + 1] = i_{n!}\text{sk}_n \Delta[n + 1]$ is the $n$-skeleton of the $(n + 1)$-simplex: \begin{eqnarray*} \Hom(\Delta[1], L)_{n + 1} & = & \Hom(\Delta[1] \times \Delta[n + 1], L)_0 \\ (\text{cosk}_n \text{sk}_n \Hom(\Delta[1], L))_{n + 1} & = & \Hom(\Delta[1] \times \partial \Delta[n + 1], L)_0 \\ (L \times L)_{n + 1} & = & \Hom( (\Delta[n + 1] \amalg \Delta[n + 1], L)_0 \\ (\text{cosk}_n \text{sk}_n (L \times L))_{n + 1} & = & \Hom( \partial \Delta[n + 1] \amalg \partial \Delta[n + 1], L)_0 \end{eqnarray*} and the morphism between these objects of $\text{SR}(\mathcal{C}, X)$ come from the commutative diagram of simplicial sets \begin{equation} \label{equation-dual-diagram} \xymatrix{ \Delta[1] \times \Delta[n + 1] & \Delta[1] \times \partial\Delta[n + 1] \ar[l] \\ \Delta[n + 1] \amalg \Delta[n + 1] \ar[u] & \partial\Delta[n + 1] \amalg \partial\Delta[n + 1] \ar[l] \ar[u] } \end{equation} Moreover the fibre product of the bottom arrow and the right arrow in (\ref{equation-diagram}) is equal to $$ \Hom(U, L)_0 $$ where $U \subset \Delta[1] \times \Delta[n + 1]$ is the smallest simplicial subset such that both $\Delta[n + 1] \amalg \Delta[n + 1]$ and $\Delta[1] \times \partial\Delta[n + 1]$ map into it.


hypercovering-lemma-basis-hypercovering

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology of $X$. There exists a hypercovering $(I, \{U_i\})$ of $X$ such that each $U_i$ is an element of $\mathcal{B}$.


hypercovering-lemma-quasi-separated-quasi-compact-hypercovering

Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology of $X$. Assume that \begin{enumerate} \item $X$ is quasi-compact, \item each $U \in \mathcal{B}$ is quasi-compact open, and \item the intersection of any two quasi-compact opens in $X$ is quasi-compact. \end{enumerate} Then there exists a hypercovering $(I, \{U_i\})$ of $X$ with the following properties \begin{enumerate} \item each $U_i$ is an element of the basis $\mathcal{B}$, \item each of the $I_n$ is a finite set, and in particular \item each of the coverings  (\ref{equation-covering-X}), (\ref{equation-covering-two}), and (\ref{equation-covering-general}) is finite. \end{enumerate}


hypercovering-lemma-split

Let $\mathcal{C}$ be a site. Let $K$ be an $r$-truncated simplicial object of $\text{SR}(\mathcal{C})$. The following are equivalent \begin{enumerate} \item $K$ is split (Simplicial, Definition \ref{simplicial-definition-split}), \item $f_{\varphi, i} : U_{n, i} \to U_{m, \alpha(\varphi)(i)}$ is an isomorphism for $r \geq n \geq 0$, $\varphi : [m] \to [n]$ surjective, $i \in I_n$, and \item $f_{\sigma^n_j, i} : U_{n, i} \to U_{n + 1, \alpha(\sigma^n_j)(i)}$ is an isomorphism for $0 \leq j \leq n < r$, $i \in I_n$. \end{enumerate} The same holds for simplicial objects if in (2) and (3) we set $r = \infty$.


hypercovering-lemma-hypercovering-object

Let $\mathcal{C}$ be a site with fibre products. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume \begin{enumerate} \item any object $U$ of $\mathcal{C}$ has a covering $\{U_j \to U\}_{j \in J}$ with $U_j \in \mathcal{B}$, and \item if $\{U_j \to U\}_{j \in J}$ is a covering with $U_j \in \mathcal{B}$ and $\{U' \to U\}$ is a morphism with $U' \in \mathcal{B}$, then $\{U_j \to U\}_{j \in J} \amalg \{U' \to U\}$ is a covering. \end{enumerate} Then for any $X$ in $\mathcal{C}$ there is a hypercovering $K$ of $X$ such that $K_n = \{U_{n, i}\}_{i \in I_n}$ with $U_{n, i} \in \mathcal{B}$ for all $i \in I_n$.


hypercovering-lemma-hypercovering-site

Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume that any object of $\mathcal{C}$ has a covering whose members are elements of $\mathcal{B}$. Then there is a hypercovering $K$ such that $K_n = \{U_i\}_{i \in I_n}$ with $U_i \in \mathcal{B}$ for all $i \in I_n$.


hypercovering-lemma-hypercovering-morphism-sites

Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites given by the functor $u : \mathcal{D} \to \mathcal{C}$. Assume $\mathcal{D}$ and $\mathcal{C}$ have equalizers and fibre products and $u$ commutes with them. If a simplicial object $K$ of $\text{SR}(\mathcal{D})$ is a hypercovering, then $u(K)$ is a hypercovering.


hypercovering-lemma-hypercovering-continuous-functor

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{D} \to \mathcal{C}$ be a continuous functor. Assume $\mathcal{D}$ and $\mathcal{C}$ have fibre products and $u$ commutes with them. Let $Y \in \mathcal{D}$ and $K \in \text{SR}(\mathcal{D}, Y)$ a hypercovering of $Y$. Then $u(K)$ is a hypercovering of $u(Y)$.


hypercovering-lemma-w-contractible

Let $\mathcal{C}$ be a site. Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset. Assume \begin{enumerate} \item $\mathcal{C}$ has fibre products, \item for all $X \in \Ob(\mathcal{C})$ there exists a finite covering $\{U_i \to X\}_{i \in I}$ with $U_i \in \mathcal{B}$, \item if $\{U_i \to X\}_{i \in I}$ is a finite covering with $U_i \in \mathcal{B}$ and $U \to X$ is a morphism with $U \in \mathcal{B}$, then $\{U_i \to X\}_{i \in I} \amalg \{U \to X\}$ is a covering. \end{enumerate} Then for every $X$ there exists a hypercovering $K$ of $X$ such that each $K_n = \{U_{n, i} \to X\}_{i \in I_n}$ with $I_n$ finite and $U_{n, i} \in \mathcal{B}$.


algebraic-theorem-smooth-groupoid-gives-algebraic-stack

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$. Then the quotient stack $[U/R]$ is an algebraic stack over $S$.


algebraic-lemma-morphism-schemes-gives-representable-transformation

Let $S$, $X$, $Y$ be objects of $\Sch_{fppf}$. Let $f : X \to Y$ be a morphism of schemes. Then the $1$-morphism induced by $f$ $$ (\Sch/X)_{fppf} \longrightarrow (\Sch/Y)_{fppf} $$ is a representable $1$-morphism.


algebraic-lemma-representable-morphism-equivalent

Let $S$ be an object of $\Sch_{fppf}$. Consider a $2$-commutative diagram $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ of $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume the horizontal arrows are equivalences. Then $f$ is representable if and only if $f'$ is representable.


algebraic-lemma-composition-representable-transformations

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$ Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Y} \to \mathcal{Z}$ be representable $1$-morphisms. Then $$ g \circ f : \mathcal{X} \longrightarrow \mathcal{Z} $$ is a representable $1$-morphism.


algebraic-lemma-base-change-representable-transformations

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$ Let $f : \mathcal{X} \to \mathcal{Y}$ be a representable $1$-morphism. Let $g : \mathcal{Z} \to \mathcal{Y}$ be any $1$-morphism. Consider the fibre product diagram $$ \xymatrix{ \mathcal{Z} \times_{g, \mathcal{Y}, f} \mathcal{X} \ar[r]_-{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Z} \ar[r]^g & \mathcal{Y} } $$ Then the base change $f'$ is a representable $1$-morphism.


algebraic-lemma-characterize-representable-by-space

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $p : \mathcal{X} \to (\Sch/S)_{fppf}$ be a category fibred in groupoids. Then $\mathcal{X}$ is representable by an algebraic space over $S$ if and only if the following conditions are satisfied: \begin{enumerate} \item $\mathcal{X}$ is fibred in setoids\footnote{This means that it is fibred in groupoids and objects in the fibre categories have no nontrivial automorphisms, see Categories, Definition \ref{categories-definition-category-fibred-sets}.}, and \item the presheaf $U \mapsto \Ob(\mathcal{X}_U)/\!\!\cong$ is an algebraic space. \end{enumerate}


algebraic-lemma-criterion-map-representable-spaces-fibred-in-groupoids

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. The following are necessary and sufficient conditions for $f$ to be representable by algebraic spaces: \begin{enumerate} \item for each scheme $U/S$ the functor $f_U : \mathcal{X}_U \longrightarrow \mathcal{Y}_U$ between fibre categories is faithful, and \item for each $U$ and each $y \in \Ob(\mathcal{Y}_U)$ the presheaf $$ (h : V \to U) \longmapsto \{(x, \phi) \mid x \in \Ob(\mathcal{X}_V), \phi : h^*y \to f(x)\}/\cong $$ is an algebraic space over $U$. \end{enumerate} Here we have made a choice of pullbacks for $\mathcal{Y}$.


algebraic-lemma-representable-by-spaces-morphism-equivalent

Let $S$ be an object of $\Sch_{fppf}$. Consider a $2$-commutative diagram $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ of $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume the horizontal arrows are equivalences. Then $f$ is representable by algebraic spaces if and only if $f'$ is representable by algebraic spaces.


algebraic-lemma-morphism-spaces-gives-representable-by-spaces

Let $S$ be an object of $\Sch_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $S$. If $\mathcal{X}$ and $\mathcal{Y}$ are representable by algebraic spaces over $S$, then the $1$-morphism $f$ is representable by algebraic spaces.


algebraic-lemma-map-presheaves-representable-by-algebraic-spaces

Let $S$ be an object of $\Sch_{fppf}$. Let $a : F \to G$ be a map of presheaves of sets on $(\Sch/S)_{fppf}$. Denote $a' : \mathcal{S}_F  \to \mathcal{S}_G$ the associated map of categories fibred in sets. Then $a$ is representable by algebraic spaces (see Bootstrap, Definition \ref{bootstrap-definition-morphism-representable-by-spaces}) if and only if $a'$ is representable by algebraic spaces.


algebraic-lemma-map-fibred-setoids-representable-algebraic-spaces

Let $S$ be an object of $\Sch_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in setoids over $(\Sch/S)_{fppf}$. Let $F$, resp.\ $G$ be the presheaf which to $T$ associates the set of isomorphism classes of objects of $\mathcal{X}_T$, resp.\ $\mathcal{Y}_T$. Let $a : F \to G$ be the map of presheaves corresponding to $f$. Then $a$ is representable by algebraic spaces (see Bootstrap, Definition \ref{bootstrap-definition-morphism-representable-by-spaces}) if and only if $f$ is representable by algebraic spaces.


algebraic-lemma-base-change-representable-by-spaces

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $g : \mathcal{Z} \to \mathcal{Y}$ be any $1$-morphism. Consider the fibre product diagram $$ \xymatrix{ \mathcal{Z} \times_{g, \mathcal{Y}, f} \mathcal{X} \ar[r]_-{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Z} \ar[r]^g & \mathcal{Y} } $$ Then the base change $f'$ is a $1$-morphism representable by algebraic spaces.


algebraic-lemma-base-change-by-space-representable-by-space

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$ Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Z} \to \mathcal{Y}$ be $1$-morphisms. Assume \begin{enumerate} \item $f$ is representable by algebraic spaces, and \item $\mathcal{Z}$ is representable by an algebraic space over $S$. \end{enumerate} Then the $2$-fibre product $\mathcal{Z} \times_{g, \mathcal{Y}, f} \mathcal{X}$ is representable by an algebraic space.


algebraic-lemma-composition-representable-by-spaces

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$. If $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Y} \to \mathcal{Z}$ are $1$-morphisms representable by algebraic spaces, then $$ g \circ f : \mathcal{X} \longrightarrow \mathcal{Z} $$ is a $1$-morphism representable by algebraic spaces.


algebraic-lemma-product-representable-by-spaces

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}_i, \mathcal{Y}_i$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$, $i = 1, 2$. Let $f_i : \mathcal{X}_i \to \mathcal{Y}_i$, $i = 1, 2$ be $1$-morphisms representable by algebraic spaces. Then $$ f_1 \times f_2 : \mathcal{X}_1 \times \mathcal{X}_2 \longrightarrow \mathcal{Y}_1 \times \mathcal{Y}_2 $$ is a $1$-morphism representable by algebraic spaces.


algebraic-lemma-property-morphism-equivalent

Let $S$ be an object of $\Sch_{fppf}$. Let $\mathcal{P}$ be as in Definition \ref{definition-relative-representable-property}. Consider a $2$-commutative diagram $$ \xymatrix{ \mathcal{X}' \ar[r] \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Y}' \ar[r] & \mathcal{Y} } $$ of $1$-morphisms of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume the horizontal arrows are equivalences and $f$ (or equivalently $f'$) is representably by algebraic spaces. Then $f$ has $\mathcal{P}$ if and only if $f'$ has $\mathcal{P}$.


algebraic-lemma-map-presheaves-representable-by-spaces-transformation-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $a : F \to G$ be a map of presheaves on $(\Sch/S)_{fppf}$. Let $\mathcal{P}$ be as in Definition \ref{definition-relative-representable-property}. Assume $a$ is representable by algebraic spaces. Then $a : F \to G$ has property $\mathcal{P}$ (see Bootstrap, Definition \ref{bootstrap-definition-property-transformation}) if and only if the corresponding morphism $\mathcal{S}_F \to \mathcal{S}_G$ of categories fibred in groupoids has property $\mathcal{P}$.


algebraic-lemma-map-fibred-setoids-property

Let $S$ be an object of $\Sch_{fppf}$. Let $\mathcal{P}$ be as in Definition \ref{definition-relative-representable-property}. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in setoids over $(\Sch/S)_{fppf}$. Let $F$, resp.\ $G$ be the presheaf which to $T$ associates the set of isomorphism classes of objects of $\mathcal{X}_T$, resp.\ $\mathcal{Y}_T$. Let $a : F \to G$ be the map of presheaves corresponding to $f$. Then $a$ has $\mathcal{P}$ if and only if $f$ has $\mathcal{P}$.


algebraic-lemma-composition-representable-transformations-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$, $\mathcal{Z}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property} which is stable under composition. Let $f : \mathcal{X} \to \mathcal{Y}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms which are representable by algebraic spaces. If $f$ and $g$ have property $\mathcal{P}$ so does $g \circ f : \mathcal{X} \to \mathcal{Z}$.


algebraic-lemma-base-change-representable-transformations-property

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $\mathcal{P}$ be a property as in Definition \ref{definition-relative-representable-property}. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $g : \mathcal{Z} \to \mathcal{Y}$ be any $1$-morphism. Consider the $2$-fibre product diagram $$ \xymatrix{ \mathcal{Z} \times_{g, \mathcal{Y}, f} \mathcal{X} \ar[r]_-{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^f \\ \mathcal{Z} \ar[r]^g & \mathcal{Y} } $$ If $f$ has $\mathcal{P}$, then the base change $f'$ has $\mathcal{P}$.


algebraic-lemma-representable-transformations-property-implication

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $(\Sch/S)_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism representable by algebraic spaces. Let $\mathcal{P}$, $\mathcal{P}'$ be properties as in Definition \ref{definition-relative-representable-property}. Suppose that for any morphism of algebraic spaces $a : F \to G$ we have $\mathcal{P}(a) \Rightarrow \mathcal{P}'(a)$. If $f$ has property $\mathcal{P}$ then $f$ has property $\mathcal{P}'$.


algebraic-lemma-representable-diagonal

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$ be a category fibred in groupoids over $(\Sch/S)_{fppf}$. The following are equivalent: \begin{enumerate} \item the diagonal $\mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces, \item for every scheme $U$ over $S$, and any $x, y \in \Ob(\mathcal{X}_U)$ the sheaf $\mathit{Isom}(x, y)$ is an algebraic space over $U$, \item for every scheme $U$ over $S$, and any $x \in \Ob(\mathcal{X}_U)$ the associated $1$-morphism $x : (\Sch/U)_{fppf} \to \mathcal{X}$ is representable by algebraic spaces, \item for every pair of schemes $T_1, T_2$ over $S$, and any $x_i \in \Ob(\mathcal{X}_{T_i})$, $i = 1, 2$ the $2$-fibre product $(\Sch/T_1)_{fppf} \times_{x_1, \mathcal{X}, x_2} (\Sch/T_2)_{fppf}$ is representable by an algebraic space, \item for every representable category fibred in groupoids $\mathcal{U}$ over $(\Sch/S)_{fppf}$ every $1$-morphism $\mathcal{U} \to \mathcal{X}$ is representable by algebraic spaces, \item for every pair $\mathcal{T}_1, \mathcal{T}_2$ of representable categories fibred in groupoids over $(\Sch/S)_{fppf}$ and any $1$-morphisms $x_i : \mathcal{T}_i \to \mathcal{X}$, $i = 1, 2$ the $2$-fibre product $\mathcal{T}_1 \times_{x_1, \mathcal{X}, x_2} \mathcal{T}_2$ is representable by an algebraic space, \item for every category fibred in groupoids $\mathcal{U}$ over $(\Sch/S)_{fppf}$ which is representable by an algebraic space every $1$-morphism $\mathcal{U} \to \mathcal{X}$ is representable by algebraic spaces, \item for every pair $\mathcal{T}_1, \mathcal{T}_2$ of categories fibred in groupoids over $(\Sch/S)_{fppf}$ which are representable by algebraic spaces, and any $1$-morphisms $x_i : \mathcal{T}_i \to \mathcal{X}$ the $2$-fibre product $\mathcal{T}_1 \times_{x_1, \mathcal{X}, x_2} \mathcal{T}_2$ is representable by an algebraic space. \end{enumerate}


algebraic-lemma-equivalent

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$, $\mathcal{Y}$ be categories over $(\Sch/S)_{fppf}$. Assume $\mathcal{X}$, $\mathcal{Y}$ are equivalent as categories over $(\Sch/S)_{fppf}$. Then $\mathcal{X}$ is an algebraic stack if and only if $\mathcal{Y}$ is an algebraic stack. Similarly, $\mathcal{X}$ is a Deligne-Mumford stack if and only if $\mathcal{Y}$ is a Deligne-Mumford stack.


algebraic-lemma-representable-algebraic

Let $S$ be a scheme contained in $\Sch_{fppf}$. \begin{enumerate} \item A category fibred in groupoids $p : \mathcal{X} \to (\Sch/S)_{fppf}$ which is representable by an algebraic space is a Deligne-Mumford stack. \item If $F$ is an algebraic space over $S$, then the associated category fibred in groupoids $p : \mathcal{S}_F \to (\Sch/S)_{fppf}$ is a Deligne-Mumford stack. \item If $X \in \Ob((\Sch/S)_{fppf})$, then $(\Sch/X)_{fppf} \to (\Sch/S)_{fppf}$ is a Deligne-Mumford stack. \end{enumerate}


algebraic-lemma-algebraic-stack-no-automorphisms

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. The following are equivalent \begin{enumerate} \item $\mathcal{X}$ is a Deligne-Mumford stack and is a stack in setoids, \item $\mathcal{X}$ is a Deligne-Mumford stack such that the canonical $1$-morphism $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is an equivalence, and \item $\mathcal{X}$ is representable by an algebraic space. \end{enumerate}


algebraic-lemma-2-fibre-product-general

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{Z}$ be a stack in groupoids over $(\Sch/S)_{fppf}$ whose diagonal is representable by algebraic spaces. Let $\mathcal{X}$, $\mathcal{Y}$ be algebraic stacks over $S$. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of stacks in groupoids. Then the $2$-fibre product $\mathcal{X} \times_{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack.


algebraic-lemma-2-fibre-product

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be algebraic stacks over $S$. Let $f : \mathcal{X} \to \mathcal{Z}$, $g : \mathcal{Y} \to \mathcal{Z}$ be $1$-morphisms of algebraic stacks. Then the $2$-fibre product $\mathcal{X} \times_{f, \mathcal{Z}, g} \mathcal{Y}$ is an algebraic stack. It is also the $2$-fibre product in the $2$-category of algebraic stacks over $(\Sch/S)_{fppf}$.


algebraic-lemma-lift-morphism-presentations

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. Let $V \in \Ob((\Sch/S)_{fppf})$. Let $y : (\Sch/V)_{fppf} \to \mathcal{Y}$ be surjective and smooth. Then there exists an object $U \in \Ob((\Sch/S)_{fppf})$ and a $2$-commutative diagram $$ \xymatrix{ (\Sch/U)_{fppf} \ar[r]_a \ar[d]_x & (\Sch/V)_{fppf} \ar[d]^y \\ \mathcal{X} \ar[r]^f & \mathcal{Y} } $$ with $x$ surjective and smooth.


algebraic-lemma-characterize-representable-by-algebraic-spaces

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. The following are equivalent: \begin{enumerate} \item for $U \in \Ob((\Sch/S)_{fppf})$ the functor $f : \mathcal{X}_U \to \mathcal{Y}_U$ is faithful, \item the functor $f$ is faithful, and \item $f$ is representable by algebraic spaces. \end{enumerate}


algebraic-lemma-smooth-surjective-morphism-implies-algebraic

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $u : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. If \begin{enumerate} \item $\mathcal{U}$ is representable by an algebraic space, and \item $u$ is representable by algebraic spaces, surjective and smooth, \end{enumerate} then $\mathcal X$ is an algebraic stack over $S$.


algebraic-lemma-representable-morphism-to-algebraic

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X} \to \mathcal{Y}$ be a morphism of stacks in groupoids over $(\Sch/S)_{fppf}$. Assume that \begin{enumerate} \item $\mathcal{X} \to \mathcal{Y}$ is representable by algebraic spaces, and \item $\mathcal{Y}$ is an algebraic stack over $S$. \end{enumerate} Then $\mathcal{X}$ is an algebraic stack over $S$.


algebraic-lemma-open-fibred-category-is-algebraic

\begin{reference} Removing the hypothesis that $j$ is a monomorphism was observed in an email from Matthew Emerton dates June 15, 2016 \end{reference} Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $j : \mathcal X \to \mathcal Y$ be a $1$-morphism of categories fibred in groupoids over $(\Sch/S)_{fppf}$. Assume $j$ is representable by algebraic spaces. Then, if $\mathcal{Y}$ is a stack in groupoids (resp.\ an algebraic stack), so is $\mathcal{X}$.


algebraic-lemma-map-space-into-stack

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. Let $\mathcal{U}$ be an algebraic stack over $S$ which is representable by an algebraic space. Let $f : \mathcal{U} \to \mathcal{X}$ be a 1-morphism. Then \begin{enumerate} \item the $2$-fibre product $\mathcal{R} = \mathcal{U} \times_{f, \mathcal{X}, f} \mathcal{U}$ is representable by an algebraic space, \item there is a canonical equivalence $$ \mathcal{U} \times_{f, \mathcal{X}, f} \mathcal{U} \times_{f, \mathcal{X}, f} \mathcal{U} = \mathcal{R} \times_{\text{pr}_1, \mathcal{U}, \text{pr}_0} \mathcal{R}, $$ \item the projection $\text{pr}_{02}$ induces via (2) a $1$-morphism $$ \text{pr}_{02} : \mathcal{R} \times_{\text{pr}_1, \mathcal{U}, \text{pr}_0} \mathcal{R} \longrightarrow \mathcal{R} $$ \item let $U$, $R$ be the algebraic spaces representing $\mathcal{U}, \mathcal{R}$ and $t, s : R \to U$ and $c : R \times_{s, U, t} R \to R$ are the morphisms corresponding to the $1$-morphisms $\text{pr}_0, \text{pr}_1 : \mathcal{R} \to \mathcal{U}$ and $\text{pr}_{02} : \mathcal{R} \times_{\text{pr}_1, \mathcal{U}, \text{pr}_0} \mathcal{R} \to \mathcal{R}$ above, then the quintuple $(U, R, s, t, c)$ is a groupoid in algebraic spaces over $S$, \item the morphism $f$ induces a canonical $1$-morphism $f_{can} : [U/R] \to \mathcal{X}$ of stacks in groupoids over $(\Sch/S)_{fppf}$, and \item the $1$-morphism $f_{can} : [U/R] \to \mathcal{X}$ is fully faithful. \end{enumerate}


algebraic-lemma-stack-presentation

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. Let $U$ be an algebraic space over $S$. Let $f : \mathcal{S}_U \to \mathcal{X}$ be a surjective smooth morphism. Let $(U, R, s, t, c)$ be the groupoid in algebraic spaces and $f_{can} : [U/R] \to \mathcal{X}$ be the result of applying Lemma \ref{lemma-map-space-into-stack} to $U$ and $f$. Then \begin{enumerate} \item the morphisms $s$, $t$ are smooth, and \item the $1$-morphism $f_{can} : [U/R] \to \mathcal{X}$ is an equivalence. \end{enumerate}


algebraic-lemma-diagonal-quotient-stack

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Then the diagonal of $[U/R]$ is representable by algebraic spaces.


algebraic-lemma-smooth-quotient-smooth-presentation

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$. Then the morphism $\mathcal{S}_U \to [U/R]$ is smooth and surjective.


algebraic-lemma-change-big-site

Suppose given big sites $\Sch_{fppf}$ and $\Sch'_{fppf}$. Assume that $\Sch_{fppf}$ is contained in $\Sch'_{fppf}$, see Topologies, Section \ref{topologies-section-change-alpha}. Let $S$ be an object of $\Sch_{fppf}$. Let $f : (\Sch'/S)_{fppf} \to (\Sch/S)_{fppf}$ the morphism of sites corresponding to the inclusion functor $u : (\Sch/S)_{fppf} \to (\Sch'/S)_{fppf}$. Let $\mathcal{X}$ be a stack in groupoids over $(\Sch/S)_{fppf}$. \begin{enumerate} \item if $\mathcal{X}$ is representable by some $X \in \Ob((\Sch/S)_{fppf})$, then $f^{-1}\mathcal{X}$ is representable too, in fact it is representable by the same scheme $X$, now viewed as an object of $(\Sch'/S)_{fppf}$, \item if $\mathcal{X}$ is representable by $F \in \Sh((\Sch/S)_{fppf})$ which is an algebraic space, then $f^{-1}\mathcal{X}$ is representable by the algebraic space $f^{-1}F$, \item if $\mathcal{X}$ is an algebraic stack, then $f^{-1}\mathcal{X}$ is an algebraic stack, and \item if $\mathcal{X}$ is a Deligne-Mumford stack, then $f^{-1}\mathcal{X}$ is a Deligne-Mumford stack too. \end{enumerate}


algebraic-lemma-category-of-spaces-over-smaller-base-scheme

Let $\Sch_{fppf}$ be a big fppf site. Let $S \to S'$ be a morphism of this site. The constructions A and B of Stacks, Section \ref{stacks-section-localize} above give isomorphisms of $2$-categories $$ \left\{ \begin{matrix} 2\text{-category of algebraic}\\ \text{stacks }\mathcal{X}\text{ over }S \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} 2\text{-category of pairs }(\mathcal{X}', f)\text{ consisting of an}\\ \text{algebraic stack }\mathcal{X}'\text{ over }S'\text{ and a morphism}\\ f : \mathcal{X}' \to (\Sch/S)_{fppf}\text{ of algebraic stacks over }S' \end{matrix} \right\} $$


algebraic-proposition-algebraic-stack-no-automorphisms

Let $S$ be a scheme contained in $\Sch_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. The following are equivalent \begin{enumerate} \item $\mathcal{X}$ is a stack in setoids, \item the canonical $1$-morphism $\mathcal{I}_\mathcal{X} \to \mathcal{X}$ is an equivalence, and \item $\mathcal{X}$ is representable by an algebraic space. \end{enumerate}


sites-theorem-plus

With $\mathcal{F}$ as above \begin{enumerate} \item \label{item-sep} The presheaf $\mathcal{F}^+$ is separated. \item \label{item-sheaf} If $\mathcal{F}$ is separated, then $\mathcal{F}^+$ is a sheaf and the map of presheaves $\mathcal{F} \to \mathcal{F}^+$ is injective. \item \label{item-plus-iso} If $\mathcal{F}$ is a sheaf, then $\mathcal{F} \to \mathcal{F}^+$ is an isomorphism. \item \label{item-plusplus} The presheaf $\mathcal{F}^{++}$ is always a sheaf. \end{enumerate}


sites-theorem-topology-and-topos

Let $\mathcal{C}$ be a category. Let $J$, $J'$ be topologies on $\mathcal{C}$. The following are equivalent \begin{enumerate} \item $J = J'$, \item sheaves for the topology $J$ are the same as sheaves for the topology $J'$. \end{enumerate}


sites-lemma-mono-epi

The injective (resp.\ surjective) maps defined above are exactly the monomorphisms (resp.\ epimorphisms) of $\textit{PSh}(\mathcal{C})$. A map is an isomorphism if and only if it is both injective and surjective.


sites-lemma-image

Let $\mathcal{C}$ be a category. Suppose that $\varphi : \mathcal{F} \to \mathcal{G}$ is a morphism of presheaves of sets on $\mathcal{C}$. There exists a unique subpresheaf $\mathcal{G}' \subset \mathcal{G}$ such that $\varphi$ factors as $\mathcal{F} \to \mathcal{G}' \to \mathcal{G}$ and such that the first map is surjective.


sites-lemma-almost-directed

Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. Suppose that $\mathcal{C}$ has fibre products and equalizers, and that $u$ commutes with them. Then the categories $(\mathcal{I}_V)^{opp}$ satisfy the hypotheses of Categories, Lemma \ref{categories-lemma-split-into-directed}.


sites-lemma-directed

Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. Assume \begin{enumerate} \item the category $\mathcal{C}$ has a final object $X$ and $u(X)$ is a final object of $\mathcal{D}$ , and \item the category $\mathcal{C}$ has fibre products and $u$ commutes with them. \end{enumerate} Then the index categories $(\mathcal{I}^u_V)^{opp}$ are filtered (see Categories, Definition \ref{categories-definition-directed}).


sites-lemma-recover

There is a canonical map $\mathcal{F}(U) \to u_p\mathcal{F}(u(U))$, which is compatible with restriction maps (on $\mathcal{F}$ and on $u_p\mathcal{F}$).


sites-lemma-adjoints-u

The functor $u_p$ is a left adjoint to the functor $u^p$. In other words the formula $$ \Mor_{\textit{PSh}(\mathcal{C})}(\mathcal{F}, u^p\mathcal{G}) = \Mor_{\textit{PSh}(\mathcal{D})}(u_p\mathcal{F}, \mathcal{G}) $$ holds bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.


sites-lemma-pullback-representable-presheaf

Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. For any object $U$ of $\mathcal{C}$ we have $u_ph_U = h_{u(U)}$.


sites-lemma-tautological-combinatorial

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$, and $\mathcal{V} = \{\psi_j : V_j \to U\}_{j\in J}$ be two families of morphisms with the same fixed target. \begin{enumerate} \item If $\mathcal{U}$ and $\mathcal{V}$ are combinatorially equivalent then they are tautologically equivalent. \item If $\mathcal{U}$ and $\mathcal{V}$ are tautologically equivalent then $\mathcal{U}$ is a refinement of $\mathcal{V}$ and $\mathcal{V}$ is a refinement of $\mathcal{U}$. \item The relation ``being combinatorially equivalent'' is an equivalence relation on all families of morphisms with fixed target. \item The relation ``being tautologically equivalent'' is an equivalence relation on all families of morphisms with fixed target. \item The relation ``$\mathcal{U}$ refines $\mathcal{V}$ and $\mathcal{V}$ refines $\mathcal{U}$'' is an equivalence relation on all families of morphisms with fixed target. \end{enumerate}


sites-lemma-tautological-same-sheaf

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{\varphi_i : U_i \to U\}_{i\in I}$, and $\mathcal{V} = \{\psi_j : V_j \to U\}_{j\in J}$ be two families of morphisms with the same fixed target. Assume that the fibre products $U_i \times_U U_{i'}$ and $V_j \times_U V_{j'}$ exist. If $\mathcal{U}$ and $\mathcal{V}$ are tautologically equivalent, then for any presheaf $\mathcal{F}$ on $\mathcal{C}$ the sheaf condition for $\mathcal{F}$ with respect to $\mathcal{U}$ is equivalent to the sheaf condition for $\mathcal{F}$ with respect to $\mathcal{V}$.


sites-lemma-compare-separated-presheaf-condition

Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{U_i \to U\}_{i \in I} \to \mathcal{V} = \{V_j \to U\}_{j \in J}$ be a morphism of families of maps with fixed target of $\mathcal{C}$ given by $\text{id} : U \to U$, $\alpha : J \to I$ and $f_j : V_j \to U_{\alpha(j)}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. If $\mathcal{F}(U) \to \prod_{j \in J} \mathcal{F}(V_j)$ is injective then $\mathcal{F}(U) \to \prod_{i \in I} \mathcal{F}(U_i)$ is injective.


sites-lemma-compare-sheaf-condition

Let $\mathcal{C}$ be a category. Let $\mathcal{V} = \{V_j \to U\}_{j \in J} \to \mathcal{U} = \{U_i \to U\}_{i \in I}$ be a morphism of families of maps with fixed target of $\mathcal{C}$ given by $\text{id} : U \to U$, $\alpha : J \to I$ and $f_j : V_j \to U_{\alpha(j)}$. Let $\mathcal{F}$ be a presheaf on $\mathcal{C}$. If \begin{enumerate} \item the fibre products $U_i \times_U U_{i'}$, $U_i \times_U V_j$, $V_j \times_U V_{j'}$ exist, \item $\mathcal{F}$ satisfies the sheaf condition with respect to $\mathcal{V}$, and \item for every $i \in I$ the map $\mathcal{F}(U_i) \to \prod_{j \in J} \mathcal{F}(V_j \times_U U_i)$ is injective. \end{enumerate} Then $\mathcal{F}$ satisfies the sheaf condition with respect to $\mathcal{U}$.


sites-lemma-refine-same-topology

Let $\mathcal{C}$ be a category. Let $\text{Cov}_i$, $i = 1, 2$ be two sets of families of morphisms with fixed target which each define the structure of a site on $\mathcal{C}$. \begin{enumerate} \item If every $\mathcal{U} \in \text{Cov}_1$ is tautologically equivalent to some $\mathcal{V} \in \text{Cov}_2$, then $\Sh(\mathcal{C}, \text{Cov}_2) \subset \Sh(\mathcal{C}, \text{Cov}_1)$. If also, every $\mathcal{U} \in \text{Cov}_2$ is tautologically equivalent to some $\mathcal{V} \in \text{Cov}_1$ then the category of sheaves are equal. \item Suppose that for each $\mathcal{U} \in \text{Cov}_1$ there exists a $\mathcal{V} \in \text{Cov}_2$ such that $\mathcal{V}$ refines $\mathcal{U}$. In this case $\Sh(\mathcal{C}, \text{Cov}_2) \subset \Sh(\mathcal{C}, \text{Cov}_1)$. If also for every $\mathcal{U} \in \text{Cov}_2$ there exists a $\mathcal{V} \in \text{Cov}_1$ such that $\mathcal{V}$ refines $\mathcal{U}$, then the categories of sheaves are equal. \end{enumerate}


sites-lemma-choice-set-coverings-immaterial

Let $\mathcal{C}$ be a category. Let $\text{Cov}(\mathcal{C})$ be a proper class of coverings satisfying conditions (1), (2) and (3) of Definition \ref{definition-site}. Let $\text{Cov}_1, \text{Cov}_2 \subset \text{Cov}(\mathcal{C})$ be two subsets of $\text{Cov}(\mathcal{C})$ which endow $\mathcal{C}$ with the structure of a site. If every covering $\mathcal{U} \in \text{Cov}(\mathcal{C})$ is combinatorially equivalent to a covering in $\text{Cov}_1$ and combinatorially equivalent to a covering in $\text{Cov}_2$, then $\Sh(\mathcal{C}, \text{Cov}_1) = \Sh(\mathcal{C}, \text{Cov}_2)$.


sites-lemma-limit-sheaf

Let $\mathcal{F} : \mathcal{I} \to \Sh(\mathcal{C})$ be a diagram. Then $\lim_\mathcal{I} \mathcal{F}$ exists and is equal to the limit in the category of presheaves.


sites-lemma-plus-presheaf

The constructions above define a presheaf $\mathcal{F}^+$ together with a canonical map of presheaves $\mathcal{F} \to \mathcal{F}^+$.


sites-lemma-plus-functorial

The association $\mathcal{F} \mapsto (\mathcal{F} \to \mathcal{F}^+)$ is a functor.


sites-lemma-common-refinement

Given a pair of coverings $\{U_i \to U\}$ and $\{V_j \to U\}$ of a given object $U$ of the site $\mathcal{C}$, there exists a covering which is a common refinement.


sites-lemma-independent-refinement

Any two morphisms $f, g: \mathcal{U} \to \mathcal{V}$ of coverings inducing the same morphism $U \to V$ induce the same map $H^0(\mathcal{V}, \mathcal{F}) \to  H^0(\mathcal{U}, \mathcal{F})$.


sites-lemma-plus-surjective

The map $\theta : \mathcal{F} \to \mathcal{F}^+$ has the following property: For every object $U$ of $\mathcal{C}$ and every section $s \in \mathcal{F}^+(U)$ there exists a covering $\{U_i \to U\}$ such that $s|_{U_i}$ is in the image of $\theta : \mathcal{F}(U_i) \to \mathcal{F}^{+}(U_i)$.


sites-lemma-colimit-sheaves

Let $\mathcal{F} : \mathcal{I} \to \Sh(\mathcal{C})$ be a diagram. Then $\colim_\mathcal{I} \mathcal{F}$ exists and is the sheafification of the colimit in the category of presheaves.


sites-lemma-sheafification-exact

The functor $\textit{PSh}(\mathcal{C}) \to \Sh(\mathcal{C})$, $\mathcal{F} \mapsto \mathcal{F}^\#$ is exact.


sites-lemma-sections-sheafification

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a presheaf of sets on $\mathcal{C}$. Denote $\theta^2 : \mathcal{F} \to \mathcal{F}^\#$ the canonical map of $\mathcal{F}$ into its sheafification. Let $U$ be an object of $\mathcal{C}$. Let $s \in \mathcal{F}^\#(U)$. There exists a covering $\{U_i \to U\}$ and sections $s_i \in \mathcal{F}(U_i)$ such that \begin{enumerate} \item $s|_{U_i} = \theta^2(s_i)$, and \item for every $i, j$ there exists a covering $\{U_{ijk} \to U_i \times_U U_j\}$ of $\mathcal{C}$ such that the pullback of $s_i$ and $s_j$ to each $U_{ijk}$ agree. \end{enumerate} Conversely, given any covering $\{U_i \to U\}$, elements $s_i \in \mathcal{F}(U_i)$ such that (2) holds, then there exists a unique section $s \in \mathcal{F}^\#(U)$ such that (1) holds.


sites-lemma-mono-epi-sheaves

The injective (resp.\ surjective) maps defined above are exactly the monomorphisms (resp.\ epimorphisms) of the category $\Sh(\mathcal{C})$. A map of sheaves is an isomorphism if and only if it is both injective and surjective.


sites-lemma-coequalizer-surjection

Let $\mathcal{C}$ be a site. Let $\mathcal{F} \to \mathcal{G}$ be a surjection of sheaves of sets. Then the diagram $$ \xymatrix{ \mathcal{F} \times_\mathcal{G} \mathcal{F} \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F} \ar[r] & \mathcal{G}} $$ represents $\mathcal{G}$ as a coequalizer.


sites-lemma-covering-surjective-after-sheafification

\begin{slogan} Coverings become surjective after sheafification. \end{slogan} Let $\mathcal{C}$ be a site. If $\{U_i \to U\}_{i \in I}$ is a covering of the site $\mathcal{C}$, then the morphism of presheaves of sets $$ \coprod\nolimits_{i \in I} h_{U_i} \to h_U $$ becomes surjective after sheafification.


sites-lemma-sheaf-coequalizer-representable

Let $\mathcal{C}$ be a site. Let $E \subset \Ob(\mathcal{C})$ be a subset such that every object of $\mathcal{C}$ has a covering by elements of $E$. Let $\mathcal{F}$ be a sheaf of sets. There exists a diagram of sheaves of sets $$ \xymatrix{ \mathcal{F}_1 \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}_0 \ar[r] & \mathcal{F} } $$ which represents $\mathcal{F}$ as a coequalizer, such that $\mathcal{F}_i$, $i = 0, 1$ are coproducts of sheaves of the form $h_U^\#$ with $U \in E$.


sites-lemma-pushforward-sheaf

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor. If $\mathcal{F}$ is a sheaf on $\mathcal{D}$ then $u^p\mathcal{F}$ is a sheaf as well.


sites-lemma-adjoint-sheaves

In the situation of Lemma \ref{lemma-pushforward-sheaf}. The functor $u_s : \mathcal{G} \mapsto (u_p \mathcal{G})^\#$ is a left adjoint to $u^s$.


sites-lemma-technical-up

In the situation of Lemma \ref{lemma-pushforward-sheaf}. For any presheaf $\mathcal{G}$ on $\mathcal{C}$ we have $(u_p\mathcal{G})^\# = (u_p(\mathcal{G}^\#))^\#$.


sites-lemma-pullback-representable-sheaf

Let $u : \mathcal{C} \to \mathcal{D}$ be a continuous functor between sites. For any object $U$ of $\mathcal{C}$ we have $u_sh_U^\# = h_{u(U)}^\#$.


sites-lemma-composition-morphisms-sites

Let $\mathcal{C}_i$, $i = 1, 2, 3$ be sites. Let $u : \mathcal{C}_2 \to \mathcal{C}_1$ and $v : \mathcal{C}_3 \to \mathcal{C}_2$ be continuous functors which induce morphisms of sites. Then the functor $u \circ v : \mathcal{C}_3 \to \mathcal{C}_1$ is continuous and defines a morphism of sites $\mathcal{C}_1 \to \mathcal{C}_3$.


sites-lemma-directed-morphism

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be continuous. Assume all the categories $(\mathcal{I}_V^u)^{opp}$ of Section \ref{section-functoriality-PSh} are filtered. Then $u$ defines a morphism of sites $\mathcal{D} \to \mathcal{C}$, in other words $u_s$ is exact.


sites-lemma-morphism-of-sites-covering

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by the functor $u : \mathcal{C} \to \mathcal{D}$. Given any object $V$ of $\mathcal{D}$ there exists a covering $\{V_j \to V\}$ such that for every $j$ there exists a morphism $V_j \to u(U_j)$ for some object $U_j$ of $\mathcal{C}$.


sites-lemma-morphism-sites-topoi

Given a morphism of sites $f : \mathcal{D} \to \mathcal{C}$ corresponding to the functor $u : \mathcal{C} \to \mathcal{D}$ the pair of functors $(f^{-1} = u_s, f_* = u^s)$ is a morphism of topoi.


sites-lemma-conclude-quasi-compact

Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. Consider the following conditions \begin{enumerate} \item $U$ is quasi-compact, \item for every covering $\{U_i \to U\}_{i \in I}$ in $\mathcal{C}$ there exists a finite covering $\{V_j \to U\}_{j = 1, \ldots, m}$ of $\mathcal{C}$ refining $\mathcal{U}$, and \item for every covering $\{U_i \to U\}_{i \in I}$ in $\mathcal{C}$ there exists a finite subset $I' \subset I$ such that $\{U_i \to U\}_{i \in I'}$ is a covering in $\mathcal{C}$. \end{enumerate} Then we always have (3) $\Rightarrow$ (2) $\Rightarrow$ (1) but the reverse implications do not hold in general.


sites-lemma-quasi-compact

Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. The following are equivalent \begin{enumerate} \item $U$ is quasi-compact, and \item for every surjection of sheaves $\coprod_{i \in I} \mathcal{F}_i \to h_U^\#$ there is a finite subset $J \subset I$ such that $\coprod_{i \in J} \mathcal{F}_i \to h_U^\#$ is surjective. \end{enumerate}


sites-lemma-directed-colimits-sections

Let $\mathcal{C}$ be a site. Let $\mathcal{I} \to \Sh(\mathcal{C})$, $i \mapsto \mathcal{F}_i$ be a filtered diagram of sheaves of sets. Let $U \in \Ob(\mathcal{C})$. Consider the canonical map $$ \Psi : \colim_i \mathcal{F}_i(U) \longrightarrow \left(\colim_i \mathcal{F}_i\right)(U) $$ With the terminology introduced above: \begin{enumerate} \item If all the transition maps are injective then $\Psi$ is injective for any $U$. \item If $U$ is quasi-compact, then $\Psi$ is injective. \item If $U$ is quasi-compact and all the transition maps are injective then $\Psi$ is an isomorphism. \item If $U$ has a cofinal system of coverings $\{U_j \to U\}_{j \in J}$ with $J$ finite and $U_j \times_U U_{j'}$ quasi-compact for all $j, j' \in J$, then $\Psi$ is bijective. \end{enumerate}


sites-lemma-colimit-sites

In Situation \ref{situation-inverse-limit-sites} we can construct a site $(\mathcal{C}, \text{Cov}(\mathcal{C}))$ as follows \begin{enumerate} \item as a category $\mathcal{C} = \colim \mathcal{C}_i$, and \item $\text{Cov}(\mathcal{C})$ is the union of the images of $\text{Cov}(\mathcal{C}_i)$ by $u_i : \mathcal{C}_i \to \mathcal{C}$. \end{enumerate}


sites-lemma-compute-pullback-to-limit

In Situation \ref{situation-inverse-limit-sites} let $u_i : \mathcal{C}_i \to \mathcal{C}$ be as constructed in Lemma \ref{lemma-colimit-sites}. Then $u_i$ defines a morphism of sites $f_i : \mathcal{C} \to \mathcal{C}_i$. For $U_i \in \Ob(\mathcal{C}_i)$ and sheaf $\mathcal{F}$ on $\mathcal{C}_i$ we have \begin{equation} \label{equation-compute-pullback-to-limit} f_i^{-1}\mathcal{F}(u_i(U_i)) = \colim_{a : j \to i} f_a^{-1}\mathcal{F}(u_a(U_i)) \end{equation}


sites-lemma-colimit

In Situation \ref{situation-inverse-limit-sites} assume given \begin{enumerate} \item a sheaf $\mathcal{F}_i$ on $\mathcal{C}_i$ for all $i \in \Ob(\mathcal{I})$, \item for $a : j \to i$ a map $\varphi_a : f_a^{-1}\mathcal{F}_i \to \mathcal{F}_j$ of sheaves on $\mathcal{C}_j$ \end{enumerate} such that $\varphi_c = \varphi_b \circ f_b^{-1}\varphi_a$ whenever $c = a \circ b$. Set $\mathcal{F} = \colim f_i^{-1}\mathcal{F}_i$ on the site $\mathcal{C}$ of Lemma \ref{lemma-colimit-sites}. Let $i \in \Ob(\mathcal{I})$ and $X_i \in \text{Ob}(\mathcal{C}_i)$. Then $$ \colim_{a : j \to i} \mathcal{F}_j(u_a(X_i)) = \mathcal{F}(u_i(X_i)) $$


sites-lemma-colimit-push-pull

In Situation \ref{situation-inverse-limit-sites} assume we have a sheaf $\mathcal{F}$ on $\mathcal{C}$. Then $$ \mathcal{F} = \colim f_i^{-1}f_{i, *}\mathcal{F} $$ where the transition maps are $f_j^{-1}\varphi_a$ for $a : j \to i$ where $\varphi_a : f_a^{-1}f_{i, *}\mathcal{F} \to f_{j, *}\mathcal{F}$ is a canonical map satisfying a cocycle condition as in Lemma \ref{lemma-colimit}.


sites-lemma-recover-pu

There is a canonical map ${}_pu\mathcal{F}(u(U)) \to \mathcal{F}(U)$, which is compatible with restriction maps.


sites-lemma-adjoints-pu

The functor ${}_pu$ is a right adjoint to the functor $u^p$. In other words the formula $$ \Mor_{\textit{PSh}(\mathcal{C})}(u^p\mathcal{G}, \mathcal{F}) = \Mor_{\textit{PSh}(\mathcal{D})}(\mathcal{G}, {}_pu\mathcal{F}) $$ holds bifunctorially in $\mathcal{F}$ and $\mathcal{G}$.


sites-lemma-adjoint-functors

Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors of categories. Assume that $v$ is right adjoint to $u$. Then we have \begin{enumerate} \item $u^ph_V = h_{v(V)}$ for any $V$ in $\mathcal{D}$, \item the category $\mathcal{I}^v_U$ has an initial object, \item the category ${}_V^u\mathcal{I}$ has a final object, \item ${}_pu = v^p$, and \item $u^p = v_p$. \end{enumerate}


sites-lemma-pu-sheaf

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be cocontinuous. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then ${}_pu\mathcal{F}$ is a sheaf on $\mathcal{D}$, which we will denote ${}_su\mathcal{F}$.


sites-lemma-exact-cocontinuous

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be cocontinuous. The functor $\Sh(\mathcal{D}) \to \Sh(\mathcal{C})$, $\mathcal{G} \mapsto (u^p\mathcal{G})^\#$ is a left adjoint to the functor ${}_su$ introduced in Lemma \ref{lemma-pu-sheaf} above. Moreover, it is exact.


sites-lemma-technical-pu

In the situation of Lemma \ref{lemma-exact-cocontinuous}. For any presheaf $\mathcal{G}$ on $\mathcal{D}$ we have $(u^p\mathcal{G})^\# = (u^p(\mathcal{G}^\#))^\#$.


sites-lemma-cocontinuous-morphism-topoi

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be cocontinuous. The functors $g_* = {}_su$ and $g^{-1} = (u^p\ )^\#$ define a morphism of topoi $g$ from  $\Sh(\mathcal{C})$ to $\Sh(\mathcal{D})$.


sites-lemma-composition-cocontinuous

Let $u : \mathcal{C} \to \mathcal{D}$, and $v : \mathcal{D} \to \mathcal{E}$ be cocontinuous functors. Then $v \circ u$ is cocontinuous and we have $h = g \circ f$ where $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$, resp.\ $g : \Sh(\mathcal{D}) \to \Sh(\mathcal{E})$, resp.\ $h : \Sh(\mathcal{C}) \to \Sh(\mathcal{E})$ is the morphism of topoi associated to $u$, resp.\ $v$, resp.\ $v \circ u$.


sites-lemma-when-shriek

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that \begin{enumerate} \item[(a)] $u$ is cocontinuous, and \item[(b)] $u$ is continuous. \end{enumerate} Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the associated morphism of topoi. Then \begin{enumerate} \item sheafification in the formula $g^{-1} = (u^p\ )^\#$ is unnecessary, in other words $g^{-1}(\mathcal{G})(U) = \mathcal{G}(u(U))$, \item $g^{-1}$ has a left adjoint $g_{!} = (u_p\ )^\#$, and \item $g^{-1}$ commutes with arbitrary limits and colimits. \end{enumerate}


sites-lemma-preserve-equalizers

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that \begin{enumerate} \item[(a)] $u$ is cocontinuous, \item[(b)] $u$ is continuous, and \item[(c)] fibre products and equalizers exist in $\mathcal{C}$ and $u$ commutes with them. \end{enumerate} In this case the functor $g_!$ above commutes with fibre products and equalizers (and more generally with finite connected limits).


sites-lemma-back-and-forth

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that \begin{enumerate} \item[(a)] $u$ is cocontinuous, \item[(b)] $u$ is continuous, and \item[(c)] $u$ is fully faithful. \end{enumerate} For $g_!, g^{-1}, g_*$ as above the canonical maps $\mathcal{F} \to g^{-1}g_!\mathcal{F}$ and $g^{-1}g_*\mathcal{F} \to \mathcal{F}$ are isomorphisms for all sheaves $\mathcal{F}$ on $\mathcal{C}$.


sites-lemma-bigger-site

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that \begin{enumerate} \item[(a)] $u$ is cocontinuous, \item[(b)] $u$ is continuous, \item[(c)] $u$ is fully faithful, \item[(d)] fibre products exist in $\mathcal{C}$ and $u$ commutes with them, and \item[(e)] there exist final objects $e_\mathcal{C} \in \Ob(\mathcal{C})$, $e_\mathcal{D} \in \Ob(\mathcal{D})$ such that $u(e_\mathcal{C}) = e_\mathcal{D}$. \end{enumerate} Let $g_!, g^{-1}, g_*$ be as above. Then, $u$ defines a morphism of sites $f : \mathcal{D} \to \mathcal{C}$ with $f_* = g^{-1}$, $f^{-1} = g_!$. The composition $$ \xymatrix{ \Sh(\mathcal{C}) \ar[r]^g & \Sh(\mathcal{D}) \ar[r]^f & \Sh(\mathcal{C}) } $$ is isomorphic to the identity morphism of the topos $\Sh(\mathcal{C})$. Moreover, the functor $f^{-1}$ is fully faithful.


sites-lemma-have-functor-other-way

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$, and $v : \mathcal{D} \to \mathcal{C}$ be functors. Assume that $u$ is cocontinuous, and that $v$ is a right adjoint to $u$. Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the morphism of topoi associated to $u$, see Lemma \ref{lemma-cocontinuous-morphism-topoi}. Then $g_*\mathcal{F}$ is equal to the presheaf $v^p\mathcal{F}$, in other words, $(g_*\mathcal{F})(V) = \mathcal{F}(v(V))$.


sites-lemma-have-functor-other-way-morphism

In the situation of Lemma \ref{lemma-have-functor-other-way}. We have $g_* = v^s = v^p$ and $g^{-1} = v_s = (v_p\ )^\#$. If $v$ is continuous then $v$ defines a morphism of sites $f$ from $\mathcal{C}$ to $\mathcal{D}$ whose associated morphism of topoi is equal to the morphism $g$ associated to the cocontinuous functor $u$. In other words, a continuous functor which has a cocontinuous left adjoint defines a morphism of sites.


sites-lemma-have-left-adjoint

Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be the morphism of topoi associated to a continuous and cocontinuous functor $u : \mathcal{C} \to \mathcal{D}$, see Lemmas \ref{lemma-cocontinuous-morphism-topoi} and \ref{lemma-when-shriek}. \begin{enumerate} \item If $w : \mathcal{D} \to \mathcal{C}$ is a left adjoint to $u$, then \begin{enumerate} \item $g_!\mathcal{F}$ is the sheaf associated to the presheaf $w^p\mathcal{F}$, and \item $g_!$ is exact. \end{enumerate} \item if $w$ is a continuous left adjoint, then $g_!$ has a left adjoint. \item If $w$ is a cocontinuous left adjoint, then $g_! = h^{-1}$ and $g^{-1} = h_*$ where $h : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ is the morphism of topoi associated to $w$. \end{enumerate}


sites-lemma-existence-lower-shriek

Let $\mathcal{C}$, $\mathcal{D}$ be two sites. Let $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a morphism of topoi. Let $E \subset \Ob(\mathcal{D})$ be a subset such that \begin{enumerate} \item for $V \in E$ there exists a sheaf $\mathcal{G}$ on $\mathcal{C}$ such that $f^{-1}\mathcal{F}(V) =  \Mor_{\Sh(\mathcal{C})}(\mathcal{G}, \mathcal{F})$ functorially for $\mathcal{F}$ in $\Sh(\mathcal{C})$, \item every object of $\mathcal{D}$ has a covering by objects of $E$. \end{enumerate} Then $f^{-1}$ has a left adjoint $f_!$.


sites-lemma-describe-j-shriek

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}/U$. Then $j_{U!}(\mathcal{G}^\#)$ is the sheaf associated to the presheaf $$ V \longmapsto \coprod\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U) $$ with obvious restriction mappings.


sites-lemma-describe-j-shriek-representable

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. Let $X/U$ be an object of $\mathcal{C}/U$. Then we have $j_{U!}(h_{X/U}^\#) = h_X^\#$.


sites-lemma-essential-image-j-shriek

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. The functor $j_{U!}$ gives an equivalence of categories $$ \Sh(\mathcal{C}/U) \longrightarrow \Sh(\mathcal{C})/h_U^\# $$


sites-lemma-j-shriek-commutes-equalizers-fibre-products

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. The functor $j_{U!}$ commutes with fibre products and equalizers (and more generally finite connected limits). In particular, if $\mathcal{F} \subset \mathcal{F}'$ in $\Sh(\mathcal{C}/U)$, then $j_{U!}\mathcal{F} \subset j_{U!}\mathcal{F}'$.


sites-lemma-compute-j-shriek-restrict

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. For any sheaf $\mathcal{F}$ on $\mathcal{C}$ we have $j_{U!}j_U^{-1}\mathcal{F} = \mathcal{F} \times h_U^\#$.


sites-lemma-relocalize

Let $\mathcal{C}$ be a site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram $$ \xymatrix{ \mathcal{C}/V \ar[rd]_{j_V} \ar[rr]_j & & \mathcal{C}/U \ar[ld]^{j_U} \\ & \mathcal{C} & } $$ of continuous and cocontinuous functors. The functor $j : \mathcal{C}/V \to \mathcal{C}/U$, $(a : W \to V) \mapsto (f \circ a : W \to U)$ is identified with the functor $j_{V/U} : (\mathcal{C}/U)/(V/U) \to \mathcal{C}/U$ via the identification $(\mathcal{C}/U)/(V/U) = \mathcal{C}/V$. Moreover we have $j_{V!} = j_{U!} \circ j_!$, $j_V^{-1} = j^{-1} \circ j_U^{-1}$, and $j_{V*} = j_{U*} \circ j_*$.


sites-lemma-relocalize-explicit

Notation $\mathcal{C}$, $f : V \to U$, $j_U$, $j_V$, and $j$ as in Lemma \ref{lemma-relocalize}. Via the identifications $\Sh(\mathcal{C}/V) = \Sh(\mathcal{C})/h_V^\#$ and $\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#$ of Lemma \ref{lemma-essential-image-j-shriek} we have \begin{enumerate} \item the functor $j^{-1}$ has the following description $$ j^{-1}(\mathcal{H} \xrightarrow{\varphi} h_U^\#) = (\mathcal{H} \times_{\varphi, h_U^\#, f} h_V^\# \to h_V^\#). $$ \item the functor $j_!$ has the following description $$ j_!(\mathcal{H} \xrightarrow{\varphi} h_V^\#) = (\mathcal{H} \xrightarrow{h_f \circ \varphi} h_U^\#) $$ \end{enumerate}


sites-lemma-glue-maps

\begin{slogan} Maps of sheaves glue. \end{slogan} Let $\mathcal{C}$ be a site. Let $\{U_i \to U\}$ be a covering of $\mathcal{C}$. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves on $\mathcal{C}$. Given a collection $$ \varphi_i : \mathcal{F}|_{\mathcal{C}/U_i} \longrightarrow \mathcal{G}|_{\mathcal{C}/U_i} $$ of maps of sheaves such that for all $i, j \in I$ the maps $\varphi_i, \varphi_j$ restrict to the same map $\varphi_{ij} : \mathcal{F}|_{\mathcal{C}/U_i \times_U U_j} \to \mathcal{G}|_{\mathcal{C}/U_i \times_U U_j}$ then there exists a unique map of sheaves $$ \varphi : \mathcal{F}|_{\mathcal{C}/U} \longrightarrow \mathcal{G}|_{\mathcal{C}/U} $$ whose restriction to each $\mathcal{C}/U_i$ agrees with $\varphi_i$.


sites-lemma-internal-hom-sheaf

\begin{slogan} The category of sheaves on a site is cartesian closed \end{slogan} Let $\mathcal{C}$ be a site. Let $\mathcal{F}$, $\mathcal{G}$ and $\mathcal{H}$ be sheaves on $\mathcal{C}$. There is a canonical bijection $$ \Mor_{\Sh(\mathcal{C})}(\mathcal{F}\times\mathcal{G},\mathcal{H}) = \Mor_{\Sh(\mathcal{C})}(\mathcal{F},\SheafHom(\mathcal{G},\mathcal{H})) $$ which is functorial in all three entries.


sites-lemma-hom-sheaf-hU

Let $\mathcal{C}$ be a site and $U \in \Ob(\mathcal{C})$. Then $\SheafHom(h_U^\#, \mathcal{F}) = j_*(\mathcal{F}|_{\mathcal{C}/U})$ for $\mathcal{F}$ in $\Sh(\mathcal{C})$.


sites-lemma-glue-sheaves

Let $\mathcal{C}$ be a site. Let $\{U_i \to U\}_{i \in I}$ be a covering of $\mathcal{C}$. Given any glueing data $(\mathcal{F}_i, \varphi_{ij})$ for sheaves of sets with respect to the covering $\{U_i \to U\}_{i \in I}$ there exists a sheaf of sets $\mathcal{F}$ on $\mathcal{C}/U$ together with isomorphisms $$ \varphi_i : \mathcal{F}|_{\mathcal{C}/U_i} \to \mathcal{F}_i $$ such that the diagrams $$ \xymatrix{ \mathcal{F}|_{\mathcal{C}/U_i \times_U U_j} \ar[d]_{\text{id}} \ar[r]_{\varphi_i} & \mathcal{F}_i|_{\mathcal{C}/U_i \times_U U_j} \ar[d]^{\varphi_{ij}} \\ \mathcal{F}|_{\mathcal{C}/U_i \times_U U_j} \ar[r]^{\varphi_j} & \mathcal{F}_j|_{\mathcal{C}/U_i \times_U U_j} } $$ are commutative.


sites-lemma-mapping-property-glue

Let $\mathcal{C}$ be a site. Let $\{U_i \to U\}_{i \in I}$ be a covering of $\mathcal{C}$. The category $\Sh(\mathcal{C}/U)$ is equivalent to the category of glueing data via the functor that associates to $\mathcal{F}$ on $\mathcal{C}/U$ the canonical glueing data.


sites-lemma-describe-j-shriek-good-site

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. If the topology on $\mathcal{C}$ is subcanonical, see Definition \ref{definition-weaker-than-canonical}, and if $\mathcal{G}$ is a sheaf on $\mathcal{C}/U$, then $$ j_{U!}(\mathcal{G})(V) = \coprod\nolimits_{\varphi \in \Mor_\mathcal{C}(V, U)} \mathcal{G}(V \xrightarrow{\varphi} U), $$ in other words sheafification is not necessary in Lemma \ref{lemma-describe-j-shriek}.


sites-lemma-localize-given-products

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. Assume $\mathcal{C}$ has products of pairs of objects. Then \begin{enumerate} \item the functor $j_U$ has a continuous right adjoint, namely the functor $v(X) = X \times U / U$, \item the functor $v$ defines a morphism of sites $\mathcal{C}/U \to \mathcal{C}$ whose associated morphism of topoi equals $j_U : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})$, and \item we have $j_{U*}\mathcal{F}(X) = \mathcal{F}(X \times U/U)$. \end{enumerate}


sites-lemma-relocalize-given-fibre-products

Let $\mathcal{C}$ be a site. Let $U \to V$ be a morphism of $\mathcal{C}$. Assume $\mathcal{C}$ has fibre products. Let $j$ be as in Lemma \ref{lemma-relocalize}. Then \begin{enumerate} \item the functor $j : \mathcal{C}/U \to \mathcal{C}/V$ has a continuous right adjoint, namely the functor $v : (X/V) \mapsto (X \times_V U/U)$, \item the functor $v$ defines a morphism of sites $\mathcal{C}/U \to \mathcal{C}/V$ whose associated morphism of topoi equals $j$, and \item we have $j_*\mathcal{F}(X/V) = \mathcal{F}(X \times_V U/U)$. \end{enumerate}


sites-lemma-restrict-back

Let $\mathcal{C}$ be a site. Let $U \in \Ob(\mathcal{C})$. Assume that every $X$ in $\mathcal{C}$ has at most one morphism to $U$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}/U$. The canonical maps $\mathcal{F} \to j_U^{-1}j_{U!}\mathcal{F}$ and $j_U^{-1}j_{U*}\mathcal{F} \to \mathcal{F}$ are isomorphisms.


sites-lemma-localize-cartesian-square

Let $\mathcal{C}$ be a site. Let $$ \xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ V' \ar[r] & V } $$ be a commutative diagram of $\mathcal{C}$. The morphisms of Lemma \ref{lemma-relocalize} produce commutative diagrams $$ \vcenter{ \xymatrix{ \mathcal{C}/U' \ar[d]_{j_{U'/V'}} \ar[r]_{j_{U'/U}} & \mathcal{C}/U \ar[d]^{j_{U/V}} \\ \mathcal{C}/V' \ar[r]^{j_{V'/V}} & \mathcal{C}/V } } \quad\text{and}\quad \vcenter{ \xymatrix{ \Sh(\mathcal{C}/U') \ar[d]_{j_{U'/V'}} \ar[r]_{j_{U'/U}} & \Sh(\mathcal{C}/U) \ar[d]^{j_{U/V}} \\ \Sh(\mathcal{C}/V') \ar[r]^{j_{V'/V}} & \Sh(\mathcal{C}/V) } } $$ of continuous and cocontinuous functors and of topoi. Moreover, if the initial diagram of $\mathcal{C}$ is cartesian, then we have $j_{V'/V}^{-1} \circ j_{U/V, *} = j_{U'/V', *} \circ j_{U'/U}^{-1}$.


sites-lemma-localize-morphism

Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \Ob(\mathcal{D})$ and set $U = u(V)$. Then the functor $u' : \mathcal{D}/V \to \mathcal{C}/U$, $V'/V \mapsto u(V')/U$ determines a morphism of sites $f' : \mathcal{C}/U \to \mathcal{D}/V$. The morphism $f'$ fits into a commutative diagram of topoi $$ \xymatrix{ \Sh(\mathcal{C}/U) \ar[r]_{j_U} \ar[d]_{f'} & \Sh(\mathcal{C}) \ar[d]^f \\ \Sh(\mathcal{D}/V) \ar[r]^{j_V} & \Sh(\mathcal{D}). } $$ Using the identifications $\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#$ and $\Sh(\mathcal{D}/V) = \Sh(\mathcal{D})/h_V^\#$ of Lemma \ref{lemma-essential-image-j-shriek} the functor $(f')^{-1}$ is described by the rule $$ (f')^{-1}(\mathcal{H} \xrightarrow{\varphi} h_V^\#) = (f^{-1}\mathcal{H} \xrightarrow{f^{-1}\varphi} h_U^\#). $$ Finally, we have $f'_*j_U^{-1} = j_V^{-1}f_*$.


sites-lemma-localize-morphism-strong

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{D} \to \mathcal{C}$ be a functor. Let $V \in \Ob(\mathcal{D})$. Set $U = u(V)$. Assume that \begin{enumerate} \item $\mathcal{C}$ and $\mathcal{D}$ have all finite limits, \item $u$ is continuous, and \item $u$ commutes with finite limits. \end{enumerate} There exists a commutative diagram of morphisms of sites $$ \xymatrix{ \mathcal{C}/U \ar[r]_{j_U} \ar[d]_{f'} & \mathcal{C} \ar[d]^f \\ \mathcal{D}/V \ar[r]^{j_V} & \mathcal{D} } $$ where the right vertical arrow corresponds to $u$, the left vertical arrow corresponds to the functor $u' : \mathcal{D}/V \to \mathcal{C}/U$, $V'/V \mapsto u(V')/u(V)$ and the horizontal arrows correspond to the functors $\mathcal{C} \to \mathcal{C}/U$, $X \mapsto X \times U$ and $\mathcal{D} \to \mathcal{D}/V$, $Y \mapsto Y \times V$ as in Lemma \ref{lemma-localize-given-products}. Moreover, the associated diagram of morphisms of topoi is equal to the diagram of Lemma \ref{lemma-localize-morphism}. In particular we have $f'_*j_U^{-1} = j_V^{-1}f_*$.


sites-lemma-relocalize-morphism

Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \Ob(\mathcal{D})$, $U \in \Ob(\mathcal{C})$ and $c : U \to u(V)$ a morphism of $\mathcal{C}$. There exists a commutative diagram of topoi $$ \xymatrix{ \Sh(\mathcal{C}/U) \ar[r]_{j_U} \ar[d]_{f_c} & \Sh(\mathcal{C}) \ar[d]^f \\ \Sh(\mathcal{D}/V) \ar[r]^{j_V} & \Sh(\mathcal{D}). } $$ We have $f_c = f' \circ j_{U/u(V)}$ where $f' : \Sh(\mathcal{C}/u(V)) \to \Sh(\mathcal{D}/V)$ is as in Lemma \ref{lemma-localize-morphism} and $j_{U/u(V)} : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C}/u(V))$ is as in Lemma \ref{lemma-relocalize}. Using the identifications $\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#$ and $\Sh(\mathcal{D}/V) = \Sh(\mathcal{D})/h_V^\#$ of Lemma \ref{lemma-essential-image-j-shriek} the functor $(f_c)^{-1}$ is described by the rule $$ (f_c)^{-1}(\mathcal{H} \xrightarrow{\varphi} h_V^\#) = (f^{-1}\mathcal{H} \times_{f^{-1}\varphi, h_{u(V)}^\#, c} h_U^\# \rightarrow h_U^\#). $$ Finally, given any morphisms $b : V' \to V$, $a : U' \to U$ and $c' : U' \to u(V')$ such that $$ \xymatrix{ U' \ar[r]_-{c'} \ar[d]_a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) } $$ commutes, then the diagram $$ \xymatrix{ \Sh(\mathcal{C}/U') \ar[r]_{j_{U'/U}} \ar[d]_{f_{c'}} & \Sh(\mathcal{C}/U) \ar[d]^{f_c} \\ \Sh(\mathcal{D}/V') \ar[r]^{j_{V'/V}} & \Sh(\mathcal{D}/V). } $$ commutes.


sites-lemma-localize-cocontinuous

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a cocontinuous functor. Let $U$ be an object of $\mathcal{C}$, and set $V = u(U)$. We have a commutative diagram $$ \xymatrix{ \mathcal{C}/U \ar[r]_{j_U} \ar[d]_{u'} & \mathcal{C} \ar[d]^u \\ \mathcal{D}/V \ar[r]^-{j_V} & \mathcal{D} } $$ where the left vertical arrow is $u' : \mathcal{C}/U \to \mathcal{D}/V$, $U'/U \mapsto V'/V$. Then $u'$ is cocontinuous also and we get a commutative diagram of topoi $$ \xymatrix{ \Sh(\mathcal{C}/U) \ar[r]_{j_U} \ar[d]_{f'} & \Sh(\mathcal{C}) \ar[d]^f \\ \Sh(\mathcal{D}/V) \ar[r]^-{j_V} & \Sh(\mathcal{D}) } $$ where $f$ (resp.\ $f'$) corresponds to $u$ (resp.\ $u'$).


sites-lemma-special-square-cocontinuous

Assume given sites $\mathcal{C}', \mathcal{C}, \mathcal{D}', \mathcal{D}$ and functors $$ \xymatrix{ \mathcal{C}' \ar[r]_{v'} \ar[d]_{u'} & \mathcal{C} \ar[d]^u \\ \mathcal{D}' \ar[r]^v & \mathcal{D} } $$ Assume \begin{enumerate} \item $u$, $u'$, $v$, and $v'$ are cocontinuous giving rise to morphisms of topoi $f$, $f'$, $g$, and $g'$ by Lemma \ref{lemma-cocontinuous-morphism-topoi}, \item $v \circ u' = u \circ v'$, \item $v$ and $v'$ are continuous as well as cocontinuous, and \item for any object $V'$ of $\mathcal{D}'$ the functor ${}^{u'}_{V'}\mathcal{I} \to {}^{\ \ \ u}_{v(V')}\mathcal{I}$ given by $v$ is cofinal. \end{enumerate} Then $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$.


sites-lemma-special-square-continuous

Assume given sites $\mathcal{C}', \mathcal{C}, \mathcal{D}', \mathcal{D}$ and functors $$ \xymatrix{ \mathcal{C}' \ar[r]_{v'} & \mathcal{C} \\ \mathcal{D}' \ar[r]^v \ar[u]^{u'} & \mathcal{D} \ar[u]_u } $$ With notation as in Sections \ref{section-morphism-sites} and \ref{section-cocontinuous-morphism-topoi} assume \begin{enumerate} \item $u$ and $u'$ are continuous giving rise to morphisms of sites $f$ and $f'$, \item $v$ and $v'$ are cocontinuous giving rise to morphisms of topoi $g$ and $g'$, \item $u \circ v = v' \circ u'$, and \item $v$ and $v'$ are continuous as well as cocontinuous. \end{enumerate} Then\footnote{In this generality we don't know $f \circ g'$ is equal to $g \circ f'$ as morphisms of topoi (there is a canonical $2$-arrow from the first to the second which may not be an isomorphism).} $f'_* \circ (g')^{-1} = g^{-1} \circ f_*$ and $g'_! \circ (f')^{-1} = f^{-1} \circ g_!$.


sites-lemma-equivalence

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that \begin{enumerate} \item $u$ is cocontinuous, \item $u$ is continuous, \item given $a, b : U' \to U$ in $\mathcal{C}$ such that $u(a) = u(b)$, then there exists a covering $\{f_i : U'_i \to U'\}$ in $\mathcal{C}$ such that $a \circ f_i = b \circ f_i$, \item given $U', U \in \Ob(\mathcal{C})$ and a morphism $c : u(U') \to u(U)$ in $\mathcal{D}$ there exists a covering $\{f_i : U_i' \to U'\}$ in $\mathcal{C}$ and morphisms $c_i : U_i' \to U$ such that $u(c_i) = c \circ u(f_i)$, and \item given $V \in \Ob(\mathcal{D})$ there exists a covering of $V$ in $\mathcal{D}$ of the form $\{u(U_i) \to V\}_{i \in I}$. \end{enumerate} Then the morphism of topoi $$ g : \Sh(\mathcal{C}) \longrightarrow \Sh(\mathcal{D}) $$ associated to the cocontinuous functor $u$ by Lemma \ref{lemma-cocontinuous-morphism-topoi} is an equivalence.


sites-lemma-localize-special-cocontinuous

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a special cocontinuous functor. For every object $U$ of $\mathcal{C}$ we have a commutative diagram $$ \xymatrix{ \mathcal{C}/U \ar[r]_{j_U} \ar[d] & \mathcal{C} \ar[d]^u \\ \mathcal{D}/u(U) \ar[r]^-{j_{u(U)}} & \mathcal{D} } $$ as in Lemma \ref{lemma-localize-cocontinuous}. The left vertical arrow is a special cocontinuous functor. Hence in the commutative diagram of topoi $$ \xymatrix{ \Sh(\mathcal{C}/U) \ar[r]_{j_U} \ar[d] & \Sh(\mathcal{C}) \ar[d]^u \\ \Sh(\mathcal{D}/u(U)) \ar[r]^-{j_{u(U)}} & \Sh(\mathcal{D}) } $$ the vertical arrows are equivalences.


sites-lemma-special-equivalence

Let $\mathcal{C}$ be a site. Let $\mathcal{C}' \subset \Sh(\mathcal{C})$ be a full subcategory (with a set of objects) such that \begin{enumerate} \item $h_U^\# \in \Ob(\mathcal{C}')$ for all $U \in \Ob(\mathcal{C})$, and \item $\mathcal{C}'$ is preserved under fibre products in $\Sh(\mathcal{C})$. \end{enumerate} Declare a covering of $\mathcal{C}'$ to be any family $\{\mathcal{F}_i \to \mathcal{F}\}_{i \in I}$ of maps such that $\coprod_{i \in I} \mathcal{F}_i \to \mathcal{F}$ is a surjective map of sheaves. Then \begin{enumerate} \item $\mathcal{C}'$ is a site (after choosing a set of coverings, see Sets, Lemma \ref{sets-lemma-coverings-site}), \item representable presheaves on $\mathcal{C}'$ are sheaves (i.e., the topology on $\mathcal{C}'$ is subcanonical, see Definition \ref{definition-weaker-than-canonical}), \item the functor $v : \mathcal{C} \to \mathcal{C}'$, $U \mapsto h_U^\#$ is a special cocontinuous functor, hence induces an equivalence $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$, \item for any $\mathcal{F} \in \Ob(\mathcal{C}')$ we have $g^{-1}h_\mathcal{F} = \mathcal{F}$, and \item for any $U \in \Ob(\mathcal{C})$ we have $g_*h_U^\# = h_{v(U)} = h_{h_U^\#}$. \end{enumerate}


sites-lemma-topos-good-site

Let $\Sh(\mathcal{C})$ be a topos. Let $\{\mathcal{F}_i\}_{i \in I}$ be a set of sheaves on $\mathcal{C}$. There exists an equivalence of topoi $g : \Sh(\mathcal{C}) \to \Sh(\mathcal{C}')$ induced by a special cocontinuous functor $u : \mathcal{C} \to \mathcal{C}'$ of sites such that \begin{enumerate} \item $\mathcal{C}'$ has a subcanonical topology, \item a family $\{V_j \to V\}$ of morphisms of $\mathcal{C}'$ is (combinatorially equivalent to) a covering of $\mathcal{C}'$ if and only if $\coprod h_{V_j} \to h_V$ is surjective, \item $\mathcal{C}'$ has fibre products and a final object (i.e., $\mathcal{C}'$ has all finite limits), \item every subsheaf of a representable sheaf on $\mathcal{C}'$ is representable, and \item each $g_*\mathcal{F}_i$ is a representable sheaf. \end{enumerate}


sites-lemma-morphism-topoi-comes-from-morphism-sites

\begin{reference} This statement is closely related to \cite[Proposition 4.9.4. Expos\'e IV]{SGA4}. In order to get the whole result, one should also use \cite[Remarque 4.7.4, Expos\'e IV]{SGA4}. \end{reference} Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Then there exists a site $\mathcal{C}'$ and a diagram of functors $$ \xymatrix{ \mathcal{C} \ar[r]_v & \mathcal{C}' & \mathcal{D} \ar[l]^u } $$ such that \begin{enumerate} \item the functor $v$ is a special cocontinuous functor, \item the functor $u$ commutes with fibre products, is continuous and defines a morphism of sites $\mathcal{C}' \to \mathcal{D}$, and \item the morphism of topoi $f$ agrees with the composition of morphisms of topoi $$ \Sh(\mathcal{C}) \longrightarrow \Sh(\mathcal{C}') \longrightarrow \Sh(\mathcal{D}) $$ where the first arrow comes from $v$ via Lemma \ref{lemma-equivalence} and the second arrow from $u$ via Lemma \ref{lemma-morphism-sites-topoi}. \end{enumerate}


sites-lemma-localize-topos

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Then the category $\Sh(\mathcal{C})/\mathcal{F}$ is a topos. There is a canonical morphism of topoi $$ j_\mathcal{F} : \Sh(\mathcal{C})/\mathcal{F} \longrightarrow \Sh(\mathcal{C}) $$ which is a localization as in Section \ref{section-localize} such that \begin{enumerate} \item the functor $j_\mathcal{F}^{-1}$ is the functor $\mathcal{H} \mapsto \mathcal{H} \times \mathcal{F}/\mathcal{F}$, and \item the functor $j_{\mathcal{F}!}$ is the forgetful functor $\mathcal{G}/\mathcal{F} \mapsto \mathcal{G}$. \end{enumerate}


sites-lemma-localize-topos-site

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $\mathcal{C}/\mathcal{F}$ be the category of pairs $(U, s)$ where $U \in \Ob(\mathcal{C})$ and $s \in \mathcal{F}(U)$. Let a covering in $\mathcal{C}/\mathcal{F}$ be a family $\{(U_i, s_i) \to (U, s)\}$ such that $\{U_i \to U\}$ is a covering of $\mathcal{C}$. Then $j : \mathcal{C}/\mathcal{F} \to \mathcal{C}$ is a continuous and cocontinuous functor of sites which induces a morphism of topoi $j : \Sh(\mathcal{C}/\mathcal{F}) \to \Sh(\mathcal{C})$. In fact, there is an equivalence $\Sh(\mathcal{C}/\mathcal{F}) = \Sh(\mathcal{C})/\mathcal{F}$ which turns $j$ into $j_\mathcal{F}$.


sites-lemma-localize-compare

Let $\mathcal{C}$ be a site. Let $\mathcal{F} = h_U^\#$ for some object $U$ of $\mathcal{C}$. Then $j_\mathcal{F} : \Sh(\mathcal{C})/\mathcal{F} \to \Sh(\mathcal{C})$ constructed in Lemma \ref{lemma-localize-topos} agrees with the morphism of topoi $j_U : \Sh(\mathcal{C}/U) \to \Sh(\mathcal{C})$ constructed in Section \ref{section-localize} via the identification $\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#$ of Lemma \ref{lemma-essential-image-j-shriek}.


sites-lemma-relocalize-topos

Let $\mathcal{C}$ be a site. If $s : \mathcal{G} \to \mathcal{F}$ is a morphism of sheaves on $\mathcal{C}$ then there exists a natural commutative diagram of morphisms of topoi $$ \xymatrix{ \Sh(\mathcal{C})/\mathcal{G} \ar[rd]_{j_\mathcal{G}} \ar[rr]_j & & \Sh(\mathcal{C})/\mathcal{F} \ar[ld]^{j_\mathcal{F}} \\ & \Sh(\mathcal{C}) & } $$ where $j = j_{\mathcal{G}/\mathcal{F}}$ is the localization of the topos $\Sh(\mathcal{C})/\mathcal{F}$ at the object $\mathcal{G}/\mathcal{F}$. In particular we have $$ j^{-1}(\mathcal{H} \to \mathcal{F}) = (\mathcal{H} \times_\mathcal{F} \mathcal{G} \to \mathcal{G}) $$ and $$ j_!(\mathcal{E} \xrightarrow{e} \mathcal{F}) = (\mathcal{E} \xrightarrow{s \circ e} \mathcal{G}). $$


sites-lemma-relocalize-compare

Assume $\mathcal{C}$ and $s : \mathcal{G} \to \mathcal{F}$ are as in Lemma \ref{lemma-relocalize-topos}. If $\mathcal{G} = h_V^\#$ and $\mathcal{F} = h_U^\#$ and $s : \mathcal{G} \to \mathcal{F}$ comes from a morphism $V \to U$ of $\mathcal{C}$ then the diagram in Lemma \ref{lemma-relocalize-topos} is identified with diagram (\ref{equation-relocalize}) via the identifications $\Sh(\mathcal{C}/V) = \Sh(\mathcal{C})/h_V^\#$ and $\Sh(\mathcal{C}/U) = \Sh(\mathcal{C})/h_U^\#$ of Lemma \ref{lemma-essential-image-j-shriek}.


sites-lemma-localize-morphism-topoi

Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Let $\mathcal{G}$ be a sheaf on $\mathcal{D}$. Set $\mathcal{F} = f^{-1}\mathcal{G}$. Then there exists a commutative diagram of topoi $$ \xymatrix{ \Sh(\mathcal{C})/\mathcal{F} \ar[r]_{j_\mathcal{F}} \ar[d]_{f'} & \Sh(\mathcal{C}) \ar[d]^f \\ \Sh(\mathcal{D})/\mathcal{G} \ar[r]^{j_\mathcal{G}} & \Sh(\mathcal{D}). } $$ The morphism $f'$ is characterized by the property that $$ (f')^{-1}(\mathcal{H} \xrightarrow{\varphi} \mathcal{G}) = (f^{-1}\mathcal{H} \xrightarrow{f^{-1}\varphi} \mathcal{F}) $$ and we have $f'_*j_\mathcal{F}^{-1} = j_\mathcal{G}^{-1}f_*$.


sites-lemma-localize-morphism-compare

Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V$ be an object of $\mathcal{D}$. Set $U = u(V)$. Set $\mathcal{G} = h_V^\#$, and $\mathcal{F} = h_U^\# = f^{-1}h_V^\#$ (see Lemma \ref{lemma-pullback-representable-sheaf}). Then the diagram of morphisms of topoi of Lemma \ref{lemma-localize-morphism-topoi} agrees with the diagram of morphisms of topoi of Lemma \ref{lemma-localize-morphism} via the identifications $j_\mathcal{F}= j_U$ and $j_\mathcal{G} = j_V$ of Lemma \ref{lemma-localize-compare}.


sites-lemma-relocalize-morphism-topoi

Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Let $\mathcal{G} \in \Sh(\mathcal{D})$, $\mathcal{F} \in \Sh(\mathcal{C})$ and $s : \mathcal{F} \to f^{-1}\mathcal{G}$ a morphism of sheaves. There exists a commutative diagram of topoi $$ \xymatrix{ \Sh(\mathcal{C})/\mathcal{F} \ar[r]_{j_\mathcal{F}} \ar[d]_{f_s} & \Sh(\mathcal{C}) \ar[d]^f \\ \Sh(\mathcal{D})/\mathcal{G} \ar[r]^{j_\mathcal{G}} & \Sh(\mathcal{D}). } $$ We have $f_s = f' \circ j_{\mathcal{F}/f^{-1}\mathcal{G}}$ where $f' : \Sh(\mathcal{C})/f^{-1}\mathcal{G} \to \Sh(\mathcal{D})/\mathcal{F}$ is as in Lemma \ref{lemma-localize-morphism-topoi} and $j_{\mathcal{F}/f^{-1}\mathcal{G}} : \Sh(\mathcal{C})/\mathcal{F} \to \Sh(\mathcal{C})/f^{-1}\mathcal{G}$ is as in Lemma \ref{lemma-relocalize-topos}. The functor $(f_s)^{-1}$ is described by the rule $$ (f_s)^{-1}(\mathcal{H} \xrightarrow{\varphi} \mathcal{G}) = (f^{-1}\mathcal{H} \times_{f^{-1}\varphi, f^{-1}\mathcal{G}, s} \mathcal{F} \rightarrow \mathcal{F}). $$ Finally, given any morphisms $b : \mathcal{G}' \to \mathcal{G}$, $a : \mathcal{F}' \to \mathcal{F}$ and $s' : \mathcal{F}' \to f^{-1}\mathcal{G}'$ such that $$ \xymatrix{ \mathcal{F}' \ar[r]_-{s'} \ar[d]_a & f^{-1}\mathcal{G}' \ar[d]^{f^{-1}b} \\ \mathcal{F} \ar[r]^-s & f^{-1}\mathcal{G} } $$ commutes, then the diagram $$ \xymatrix{ \Sh(\mathcal{C})/\mathcal{F}' \ar[r]_{j_{\mathcal{F}'/\mathcal{F}}} \ar[d]_{f_{s'}} & \Sh(\mathcal{C})/\mathcal{F} \ar[d]^{f_s} \\ \Sh(\mathcal{D})/\mathcal{G}' \ar[r]^{j_{\mathcal{G}'/\mathcal{G}}} & \Sh(\mathcal{D})/\mathcal{G}. } $$ commutes.


sites-lemma-points-recover

Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. There are functorial isomorphisms $(h_U)_p = u(U)$ for $U \in \Ob(\mathcal{C})$.


sites-lemma-adjoint-point-push-stalk

For any functor $u : \mathcal{C} \to \textit{Sets}$. The functor $u^p$ is a right adjoint to the stalk functor on presheaves.


sites-lemma-point-pushforward-sheaf

Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. Suppose that for every covering $\{U_i \to U\}$ of $\mathcal{C}$ \begin{enumerate} \item the map $\coprod u(U_i) \to u(U)$ is surjective, and \item the maps $u(U_i \times_U U_j) \to u(U_i) \times_{u(U)} u(U_j)$ are surjective. \end{enumerate} Then we have \begin{enumerate} \item the presheaf $u^pE$ is a sheaf for all sets $E$, denote it $u^sE$, \item the stalk functor $\Sh(\mathcal{C}) \to \textit{Sets}$ and the functor $u^s: \textit{Sets} \to \Sh(\mathcal{C})$ are adjoint, and \item we have $\mathcal{F}_p = \mathcal{F}^\#_p$ for every presheaf of sets $\mathcal{F}$. \end{enumerate}


sites-lemma-point-site-topos

Let $\mathcal{C}$ be a site. \begin{enumerate} \item Let $p$ be a point of the site $\mathcal{C}$. Then the pair of functors $(p_*, p^{-1})$ introduced above define a morphism of topoi $\Sh(pt) \to \Sh(\mathcal{C})$. \item Let $p = (p_*, p^{-1})$ be a point of the topos $\Sh(\mathcal{C})$. Then the functor $u : U \mapsto p^{-1}(h_U^\#)$ gives rise to a point $p'$ of the site $\mathcal{C}$ whose associated morphism of topoi $(p'_*, (p')^{-1})$ is equal to $p$. \end{enumerate}


sites-lemma-site-point-morphism

Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ given by $u : \mathcal{C} \to \textit{Sets}$. Let $S_0$ be an infinite set such that $u(U) \subset S_0$ for all $U \in \Ob(\mathcal{C})$. Let $\mathcal{S}$ be the site constructed out of the powerset $S = \mathcal{P}(S_0)$ in Remark \ref{remark-pt-topos}. Then \begin{enumerate} \item there is an equivalence $i : \Sh(pt) \to \Sh(\mathcal{S})$, \item the functor $u : \mathcal{C} \to \mathcal{S}$ induces a morphism of sites $f : \mathcal{S} \to \mathcal{C}$, and \item the composition $$ \Sh(pt) \to \Sh(\mathcal{S}) \to \Sh(\mathcal{C}) $$ is the morphism of topoi $(p_*, p^{-1})$ of Lemma \ref{lemma-point-site-topos}. \end{enumerate}


sites-lemma-stalk-skyscraper

Let $\mathcal{C}$ be a site. Let $p : \Sh(pt) \to \Sh(\mathcal{C})$ be a point of the topos associated to $\mathcal{C}$. For any set $E$ there are canonical maps $$ E \longrightarrow (p_*E)_p \longrightarrow E $$ whose composition is $\text{id}_E$.


sites-lemma-skyscraper-functor-exact

Let $\mathcal{C}$ be a site. Let $p : \Sh(pt) \to \Sh(\mathcal{C})$ be a point of the topos associated to $\mathcal{C}$. The functor $p_* : \textit{Sets} \to \Sh(\mathcal{C})$ has the following properties: It commutes with arbitrary limits, it is left exact, it is faithful, it transforms surjections into surjections, it commutes with coequalizers, it reflects injections, it reflects surjections, and it reflects isomorphisms.


sites-lemma-neighbourhoods-cofiltered

Let $\mathcal{C}$ be a site. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor. If the category of neighbourhoods of $p$ is cofiltered, then the stalk functor (\ref{equation-stalk}) is left exact.


sites-lemma-neighbourhoods-directed

Let $\mathcal{C}$ be a site. Assume that $\mathcal{C}$ has a final object $X$ and fibred products. Let $p = u : \mathcal{C} \to \textit{Sets}$ be a functor such that \begin{enumerate} \item $u(X)$ is a singleton set, and \item for every pair of morphisms $U \to W$ and $V \to W$ with the same target the map $u(U \times_W V) \to u(U) \times_{u(W)} u(V)$ is bijective. \end{enumerate} Then the the category of neighbourhoods of $p$ is cofiltered and consequently the stalk functor $\Sh(\mathcal{C}) \to \textit{Sets}$, $\mathcal{F} \to \mathcal{F}_p$ commutes with finite limits.


sites-lemma-point-functor

Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Let $v : \mathcal{D} \to \textit{Sets}$ be a functor and set $w = v \circ u$. Denote $q$, resp., $p$ the stalk functor (\ref{equation-stalk}) associated to $v$, resp.\ $w$. Then $(u_p\mathcal{F})_q = \mathcal{F}_p$ functorially in the presheaf $\mathcal{F}$ on $\mathcal{C}$.


sites-lemma-point-morphism-sites

\begin{slogan} A map of sites defines a map on points, and pullback respects stalks. \end{slogan} Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by a continuous functor $u : \mathcal{C} \to \mathcal{D}$. Let $q$ be a point of $\mathcal{D}$ given by the functor $v : \mathcal{D} \to \textit{Sets}$, see Definition \ref{definition-point}. Then the functor $v \circ u : \mathcal{C} \to \textit{Sets}$ defines a point $p$ of $\mathcal{C}$ and moreover there is a canonical identification $$ (f^{-1}\mathcal{F})_q = \mathcal{F}_p $$ for any sheaf $\mathcal{F}$ on $\mathcal{C}$.


sites-lemma-point-morphism-topoi

Let $f : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a morphism of topoi. Let $q : \Sh(pt) \to \Sh(\mathcal{D})$ be a point. Then $p = f \circ q$ is a point of the topos $\Sh(\mathcal{C})$ and we have a canonical identification $$ (f^{-1}\mathcal{F})_q = \mathcal{F}_p $$ for any sheaf $\mathcal{F}$ on $\mathcal{C}$.


sites-lemma-point-localize

Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ given by $u : \mathcal{C} \to \textit{Sets}$. Let $U$ be an object of $\mathcal{C}$ and let $x \in u(U)$. The functor $$ v : \mathcal{C}/U \longrightarrow \textit{Sets}, \quad (\varphi : V \to U) \longmapsto \{y \in u(V) \mid u(\varphi)(y) = x\} $$ defines a point $q$ of the site $\mathcal{C}/U$ such that the diagram $$ \xymatrix{ & \Sh(pt) \ar[d]^p \ar[ld]_q \\ \Sh(\mathcal{C}/U) \ar[r]^{j_U} & \Sh(\mathcal{C}) } $$ commutes. In other words $\mathcal{F}_p = (j_U^{-1}\mathcal{F})_q$ for any sheaf on $\mathcal{C}$.


sites-lemma-points-above-point

Let $\mathcal{C}$, $p$, $u$, $U$ be as in Lemma \ref{lemma-point-localize}. The construction of Lemma \ref{lemma-point-localize} gives a one to one correspondence between points $q$ of $\mathcal{C}/U$ lying over $p$ and elements $x$ of $u(U)$.


sites-lemma-stalk-j-shriek

Let $\mathcal{C}$ be a site. Let $p$ be a point of $\mathcal{C}$ given by $u : \mathcal{C} \to \textit{Sets}$. Let $U$ be an object of $\mathcal{C}$. For any sheaf $\mathcal{G}$ on $\mathcal{C}/U$ we have $$ (j_{U!}\mathcal{G})_p = \coprod\nolimits_q \mathcal{G}_q $$ where the coproduct is over the points $q$ of $\mathcal{C}/U$ associated to elements $x \in u(U)$ as in Lemma \ref{lemma-point-localize}.


sites-lemma-maps-u-points

Let $\mathcal{C}$ be a site. Let $u, u' : \mathcal{C} \to \textit{Sets}$ be two functors, and let $t : u' \to u$ be a transformation of functors. Then we obtain a canonical transformation of stalk functors $t_{stalk} : \mathcal{F}_{p'} \to \mathcal{F}_p$ which agrees with $t$ via the identifications of Lemma \ref{lemma-points-recover}.


sites-lemma-exactness-stalks

Let $\mathcal{C}$ be a site and let $\{p_i\}_{i\in I}$ be a conservative family of points. Then \begin{enumerate} \item Given any map of sheaves $\varphi : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi_{p_i}$ injective implies $\varphi$ injective. \item Given any map of sheaves $\varphi : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi_{p_i}$ surjective implies $\varphi$ surjective. \item Given any pair of maps of sheaves $\varphi_1, \varphi_2 : \mathcal{F} \to \mathcal{G}$ we have $\forall i, \varphi_{1, p_i} = \varphi_{2, p_i}$ implies $\varphi_1 = \varphi_2$. \item Given a finite diagram $\mathcal{G} : \mathcal{J} \to \Sh(\mathcal{C})$, a sheaf $\mathcal{F}$ and morphisms $q_j : \mathcal{F} \to \mathcal{G}_j$ then $(\mathcal{F}, q_j)$ is a limit of the diagram if and only if for each $i$ the stalk $(\mathcal{F}_{p_i}, (q_j)_{p_i})$ is one. \item Given a finite diagram $\mathcal{F} : \mathcal{J} \to \Sh(\mathcal{C})$, a sheaf $\mathcal{G}$ and morphisms $e_j : \mathcal{F}_j \to \mathcal{G}$ then $(\mathcal{G}, e_j)$ is a colimit of the diagram if and only if for each $i$ the stalk $(\mathcal{G}_{p_i}, (e_j)_{p_i})$ is one. \end{enumerate}


sites-lemma-enough

Let $\mathcal{C}$ be a site and let $\{(p_i, u_i)\}_{i\in I}$ be a family of points. The family is conservative if and only if for every sheaf $\mathcal{F}$ and every $U\in \Ob(\mathcal{C})$ and every pair of distinct sections $s, s' \in \mathcal{F}(U)$, $s \not = s'$ there exists an $i$ and $x\in u_i(U)$ such that the triples $(U, x, s)$ and $(U, x, s')$ define distinct elements of $\mathcal{F}_{p_i}$.


sites-lemma-enough-points-local

Let $\mathcal{C}$ be a site. Let $\{U_i\}_{i \in I}$ be a family of objects of $\mathcal{C}$. Assume \begin{enumerate} \item $\coprod h_{U_i}^\# \to *$ is a surjective map of sheaves, and \item each localization $\mathcal{C}/U_i$ has enough points. \end{enumerate} Then $\mathcal{C}$ has enough points.


sites-lemma-refine

Let $\mathcal{C}$ be a site. Let $(J, \geq, V_j, g_{jj'})$ be a system as above with associated pair of functors $(u', p')$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $s, s' \in \mathcal{F}_{p'}$ be distinct elements. Let $\{W_k \to W\}$ be a finite covering of $\mathcal{C}$. Let $f \in u'(W)$. There exists a refinement $(I, \geq, U_i, f_{ii'})$ of $(J, \geq, V_j, g_{jj'})$ such that $s, s'$ map to distinct elements of $\mathcal{F}_p$ and that the image of $f$ in $u(W)$ is in the image of one of the $u(W_k)$.


sites-lemma-refine-all-at-once

Let $\mathcal{C}$ be a site. Let $(J, \geq, V_j, g_{jj'})$ be a system as above with associated pair of functors $(u', p')$. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. Let $s, s' \in \mathcal{F}_{p'}$ be distinct elements. There exists a refinement $(I, \geq, U_i, f_{ii'})$ of $(J, \geq, V_j, g_{jj'})$ such that $s, s'$ map to distinct elements of $\mathcal{F}_p$ and such that for every finite covering $\{W_k \to W\}$ of the site $\mathcal{C}$, and any $f \in u'(W)$ the image of $f$ in $u(W)$ is in the image of one of the $u(W_k)$.


sites-lemma-criterion-points

Let $\mathcal{C}$ be a site. Let $I$ be a set and for $i \in I$ let $U_i$ be an object of $\mathcal{C}$ such that \begin{enumerate} \item $\coprod h_{U_i}$ surjects onto the final object of $\Sh(\mathcal{C})$, and \item $\mathcal{C}/U_i$ satisfies the hypotheses of Proposition \ref{proposition-criterion-points}. \end{enumerate} Then $\mathcal{C}$ has enough points.


sites-lemma-w-contractible

Let $\mathcal{C}$ be a site. Let $U$ be an object of $\mathcal{C}$. The following conditions are equivalent \begin{enumerate} \item For every covering $\{U_i \to U\}$ there exists a map of sheaves $h_U^\# \to \coprod h_{U_i}^\#$ right inverse to the sheafification of $\coprod h_{U_i} \to h_U$. \item For every surjection of sheaves of sets $\mathcal{F} \to \mathcal{G}$ the map $\mathcal{F}(U) \to \mathcal{G}(U)$ is surjective. \end{enumerate}


sites-lemma-exactness-properties

Let $f : \Sh(\mathcal{C}) \to \Sh(\mathcal{D})$ be a morphism of topoi. Consider the following properties (on sheaves of sets): \begin{enumerate} \item $f_*$ is faithful, \item $f_*$ is fully faithful, \item $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is surjective for all $\mathcal{F}$ in $\Sh(\mathcal{C})$, \item $f_*$ transforms surjections into surjections, \item $f_*$ commutes with coequalizers, \item $f_*$ commutes with pushouts, \item $f^{-1}f_*\mathcal{F} \to \mathcal{F}$ is an isomorphism for all $\mathcal{F}$ in $\Sh(\mathcal{C})$, \item $f_*$ reflects injections, \item $f_*$ reflects surjections, \item $f_*$ reflects bijections, and \item for any surjection $\mathcal{F} \to f^{-1}\mathcal{G}$ there exists a surjection $\mathcal{G}' \to \mathcal{G}$ such that $f^{-1}\mathcal{G}' \to f^{-1}\mathcal{G}$ factors through $\mathcal{F} \to f^{-1}\mathcal{G}$. \end{enumerate} Then we have the following implications \begin{enumerate} \item[(a)] (2) $\Rightarrow$ (1), \item[(b)] (3) $\Rightarrow$ (1), \item[(c)] (7) $\Rightarrow$ (1), (2), (3), (8), (9), (10). \item[(d)] (3) $\Leftrightarrow$ (9), \item[(e)] (6) $\Rightarrow$ (4) and (5) $\Rightarrow$ (4), \item[(f)] (4) $\Leftrightarrow$ (11), \item[(g)] (9) $\Rightarrow$ (8), (10), and \item[(h)] (2) $\Leftrightarrow$ (7). \end{enumerate} Picture $$ \xymatrix{ (6) \ar@{=>}[rd] & & & & & (9) \ar@{=>}[r] \ar@{=>}[rd] & (8) \\ & (4) \ar@{<=>}[r] & (11) & (2) \ar@{<=>}[r] & (7) \ar@{=>}[ru] \ar@{=>}[rd] & & (10) \\ (5) \ar@{=>}[ur] & & & & & (3) \ar@{=>}[r] & (1) } $$


sites-lemma-weaker

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$. Assume that for any object $U$ of $\mathcal{C}$ and any covering $\{V_j \to u(U)\}$ in $\mathcal{D}$ there exists a covering $\{U_i \to U\}$ in $\mathcal{C}$ such that the map of sheaves $$ \coprod h_{u(U_i)}^\# \to h_{u(U)}^\# $$ factors through the map of sheaves $$ \coprod h_{V_j}^\# \to h_{u(U)}^\#. $$ Then $f_*$ transforms surjective maps of sheaves into surjective maps of sheaves.


sites-lemma-cover-from-below

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites given by the functor $u : \mathcal{C} \to \mathcal{D}$. Assume that for every object $V$ of $\mathcal{D}$ there exist objects $U_i$ of $\mathcal{C}$ and morphisms $u(U_i) \to V$ such that $\{u(U_i) \to V\}$ is a covering of $\mathcal{D}$. In this case the functor $f_* : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ reflects injections and surjections.


sites-lemma-almost-cocontinuous-sheafification

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that $u$ is continuous and almost cocontinuous. Let $\mathcal{G}$ be a presheaf on $\mathcal{D}$ such that $\mathcal{G}(V)$ is a singleton whenever $V$ is sheaf theoretically empty. Then $(u^p\mathcal{G})^\# = u^p(\mathcal{G}^\#)$.


sites-lemma-continuous-almost-cocontinuous

Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that $u$ is continuous and almost cocontinuous. Then $u^s = u^p : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ commutes with pushouts and coequalizers (and more generally finite connected colimits).


sites-lemma-morphism-of-sites-almost-cocontinuous

Let $f : \mathcal{D} \to \mathcal{C}$ be a morphism of sites associated to the continuous functor $u : \mathcal{C} \to \mathcal{D}$. If $u$ is almost cocontinuous then $f_*$ commutes with pushouts and coequalizers (and more generally finite connected colimits).


sites-lemma-open-subtopos

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf on $\mathcal{C}$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is a subobject of the final object of $\Sh(\mathcal{C})$, and \item the topos $\Sh(\mathcal{C})/\mathcal{F}$ is a subtopos of $\Sh(\mathcal{C})$. \end{enumerate}


sites-lemma-closed-subtopos

Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a subsheaf of the final object $*$ of $\Sh(\mathcal{C})$. The full subcategory of sheaves $\mathcal{G}$ such that $\mathcal{F} \times \mathcal{G} \to \mathcal{F}$ is an isomorphism is a subtopos of $\Sh(\mathcal{C})$.


sites-lemma-closed-immersion

Let $i : \Sh(\mathcal{D}) \to \Sh(\mathcal{C})$ be a closed immersion of topoi. Then $i_*$ is fully faithful, transforms surjections into surjections, commutes with coequalizers, commutes with pushouts, reflects injections, reflects surjections, and reflects bijections.


sites-lemma-compute-global-sections

Let $\mathcal{C}$ be a site. Let $a, b : V \to U$ be objects of $\mathcal{C}$ such that $$ \xymatrix{ h_V^\# \ar@<1ex>[r] \ar@<-1ex>[r] & h_U^\# \ar[r] & {*} } $$ is a coequalizer in $\Sh(\mathcal{C})$. Then $\Gamma(\mathcal{C}, \mathcal{F})$ is the equalizer of $a^*, b^* : \mathcal{F}(U) \to \mathcal{F}(V)$.


sites-lemma-sieves-set

Let $\mathcal{C}$ be a category. Let $U \in \Ob(\mathcal{C})$. \begin{enumerate} \item The collection of sieves on $U$ is a set. \item Inclusion defines a partial ordering on this set. \item Unions and intersections of sieves are sieves. \item \label{item-sieve-generated} 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 $S(U_i)$. \item The sieve $S = h_U$ is the maximal sieve. \item The empty subpresheaf is the minimal sieve. \end{enumerate}


sites-lemma-pullback-sieve-section

Let $\mathcal{C}$ be a category. Let $U \in \Ob(\mathcal{C})$. Let $S$ be a sieve on $U$. If $f : V \to U$ is in $S$, then $S \times_U V = h_V$ is maximal.


sites-lemma-topology-basic

Let $\mathcal{C}$ be a category. Let $J$ be a topology on $\mathcal{C}$. Let $U \in \Ob(\mathcal{C})$. \begin{enumerate} \item Finite intersections of elements of $J(U)$ are in $J(U)$. \item If $S \in J(U)$ and $S' \supset S$, then $S' \in J(U)$. \end{enumerate}


sites-lemma-play-with-topologies

Let $\mathcal{C}$ be a category. Let $\{J_i\}_{i\in I}$ be a set of topologies. \begin{enumerate} \item The rule $J(U) = \bigcap J_i(U)$ defines a topology on $\mathcal{C}$. \item There is a coarsest topology finer than all of the topologies $J_i$. \end{enumerate}


sites-lemma-topology-presheaves-sheaves

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 $U \in \Ob(\mathcal{C})$ denote $J(U)$ the set of sieves $S$ with the following property: For every morphism $V \to U$, the maps $$ \Mor_{\textit{PSh}(\mathcal{C})}(h_V, \mathcal{F}_i) \longrightarrow \Mor_{\textit{PSh}(\mathcal{C})}(S \times_U V, \mathcal{F}_i) $$ 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.