diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/categories.tex stacks-0.2/src/categories.tex --- stacks-0.2.orig/src/categories.tex 2006-02-16 16:33:23.000000000 +0000 +++ stacks-0.2/src/categories.tex 2006-03-14 02:49:38.000000000 +0000 @@ -149,16 +149,18 @@ A groupoid is a category where every morphism is an isomorphism. \end{definition} -\begin{example}\label{example-group} +\begin{example} +\label{example-group-groupoid} A group $G$ can be thought of as a groupoid with a single object $x$ and morphisms $\text{Mor}(x,x)=G$, with the composition rule given by the group law in $G$. \end{example} \begin{example} - Any set is $C$ the set of objects of a groupoid $\mathcal{C}$ if we - let $\text{Ob}(\mathcal{C})=C$ and declare $\text{Mor}(x,y)$ to be empty if - $x\neq y$ and to be $\{\text{id}_x\}$ if $x=y$. +\label{example-set-groupoid} +Any set $C$ the set of objects of a groupoid $\mathcal{C}$ if we +let $\text{Ob}(\mathcal{C})=C$ and declare $\text{Mor}(x,y)$ to be empty if +$x\neq y$ and to be $\{\text{id}_x\}$ if $x=y$. \end{example} \smallskip\noindent A functor $F : \mathcal{A} \to \mathcal{B}$ @@ -192,27 +194,29 @@ $x \in \text{Ob}(\mathcal{A})$ such that $F(x)$ is isomorphic to $z$ in $\mathcal{B}$. -\begin{example}\label{example-group-homorphism-functor} +\begin{example} +\label{example-group-homorphism-functor} A homomorphism $p\colon G\to H$ of groups gives rise to a functor -between the associated groupoids in Example \ref{example-group}. It is +between the associated groupoids in Example \ref{example-group-groupoid}. It is faithful (resp.\ fully faithful) if and only if $p$ is injective (resp.\ an isomorphism). \end{example} -\begin{example}\label{example-comma-category} - Given $X\in \text{Ob}(\mathcal{C})$ we define the comma category - $\mathcal{C}/X$ to be the category whose objects are morphisms $Y\to - X$ for some $Y\in \text{Ob}(\mathcal{C})$, and morphisms between objects - $Y\to X$ and $Y'\to X$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that - make the obvious diagram commute. Note that there is a functor - $p_X\colon \mathcal{C}/X\to \mathcal{C}$ which simply forgets the - morphism for $X$. Moreover given a morphism $f\colon X'\to X$ in - $\mathcal{C}$ there is an induced functor $F\colon \mathcal{C}/X' \to - \mathcal{C}/X$ obtained by composition with $f$, and $p_X\circ F = - p_{X'}$. +\begin{example} +\label{example-comma-category} +Given a category $\mathcal{C}$ and an object $X\in \text{Ob}(\mathcal{C})$ +we define the category of objects over $X$, denoted $\mathcal{C}/X$ as follows. +The objects of $\mathcal{C}/X$ are morphisms $Y\to X$ for +some $Y\in \text{Ob}(\mathcal{C})$. Morphisms between objects +$Y\to X$ and $Y'\to X$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that +make the obvious diagram commute. Note that there is a functor +$p_X\colon \mathcal{C}/X\to \mathcal{C}$ which simply forgets the +morphism for $X$. Moreover given a morphism $f\colon X'\to X$ in +$\mathcal{C}$ there is an induced functor +$F\colon \mathcal{C}/X' \to \mathcal{C}/X$ obtained by composition with $f$, +and $p_X\circ F = p_{X'}$. \end{example} - \smallskip\noindent A transformation of functors $t : F \to G$ (or simply a morphism of functors) between functors $F, G : \mathcal{A} \to \mathcal{B}$ is given by the following @@ -246,7 +250,9 @@ \subsubsection{Additional notions} \label{subsubsection-categories-additional} -\begin{definition}\label{definition-fibre-products} + +\begin{definition} +\label{definition-fibre-products} Let $x,y\in \text{Ob}(\mathcal{C})$ and $f\in \text{Mor}_{\mathcal{C}}(x,z)$ and $g\in \text{Mor}_{\mathcal C}(y,z)$. The fibre product of $f$ and $g$ is an object $x\times_z y\in \text{Ob}(\mathcal{C})$ together with morphisms @@ -275,6 +281,7 @@ and $g\in \text{Mor}_{\mathcal C}(y,z)$. \end{definition} +\noindent Given a category $\mathcal{C}$ we can form the opposite category $\mathcal{C}^{\text{opp}}$ which has the same objects as $\mathcal{C}$ but all morphisms reversed, so @@ -285,28 +292,37 @@ morphisms $f$ and $g$ in $\mathcal{C}$, $F(f\circ g) = F(g)\circ F(f)$. -\begin{example}\label{example-hom-functor} - For any $U\in \text{Ob}(\mathcal{C})$ there is a contravariant - functor $$\text{Mor}(-,U) \colon\mathcal{C} \to \text{Sets}$$ which - takes an object $X$ to the set $\text{Mor}_{\mathcal{C}}(X,U)$. - Given a morphism $f\colon X\to Y$ $\text{Mor}(-,U)(f)\colon - \text{Mor}(Y,U)\to \text{Mor}(X,U)$ takes $\phi$ to $\phi\circ f$ . - If $\mathcal{C}$ is a category of schemes this functor is sometimes - referred to as the \emph{functor of points} of $U$. +\begin{example} +\label{example-hom-functor} +For any $U\in \text{Ob}(\mathcal{C})$ there is a contravariant +functor +$$ +\text{Mor}(-,U) \colon\mathcal{C} \to \text{Sets} +$$ +which takes an object $X$ to the set $\text{Mor}_{\mathcal{C}}(X,U)$. +Given a morphism $f\colon X\to Y$ the corresponding map +$\text{Mor}(-,U)(f)\colon \text{Mor}(Y,U)\to \text{Mor}(X,U)$ takes +$\phi$ to $\phi\circ f$. More commonly this functor is denoted +$h_U : \mathcal{C}^{\text{opp}} \to \text{Sets}$. If $\mathcal{C}$ is the +category of schemes this functor is sometimes referred to as the +\emph{functor of points} of $U$. \end{example} -\begin{example}\label{example-representable-functor} - A contravariant functor $F\colon \mathcal{C}\to \text{Sets}$ is said - to be representable if it is isomorphic to the functor - $\text{Mor}(-,U)$ for some object $U$ of $\mathcal{C}$. -\end{example} +\begin{definition} +\label{definition-representable-functor} +A contravariant functor $F\colon \mathcal{C}\to \text{Sets}$ is said +to be representable if it is isomorphic to the functor +$h_U(-) = \text{Mor}(-,U)$ for some object $U$ of $\mathcal{C}$. +\end{definition} -\noindent -FIXME: representable morphisms (namely morphisms -$x\to y$ such that for every $w \to y$ the fibre product $w\times_y x$ -exists), etc. +\begin{definition} +\label{definition-representable-morphism} +A morphism $f : x \to y$ of a category $\mathcal{C}$ is said to be +representable, if and only if for every morphism $z \to y$ in $\mathcal{C}$ +the fibre product $z\times_y x$ exists. +\end{definition} -\subsection{2-Categories} +\subsection{2-categories} \label{subsection-2-categories} \noindent @@ -358,7 +374,7 @@ \item Every $2$-morphism is an isomorphism. This makes sense since the conditions sofar imply that $\text{Mor}_\mathcal{C}(x,y)$ is a category with $1$-morphisms as objects and $2$-morphisms as morphisms. So this -condition just means it is a groupoid. +condition means every $\text{Mor}_\mathcal{C}(x,y)$ is a groupoid. \item Let $x,y,z\in \text{Ob}(\mathcal{C})$ and let $G \in \text{Mor}_\mathcal{C}(y,z)$. The map $\text{Mor}_\mathcal{C}(x,y) \to @@ -383,8 +399,10 @@ \noindent This is obviously not a very pleasant type of object to work with. On the other hand, there are lots of examples where it is quite clear -how you work with it. Note that we require the $2$-morphisms to all -be isomorphisms. FIXME: Remove this definition? Replace by a better one? +how you work with it. Note that we require the $2$-morphisms to be +isomorphisms. As far as this text is concerned all 2-categories occuring +is this document are (full) sub 2-categories of the example below. +FIXME: Remove this definition? Replace by a better one? \begin{example} \label{example-category-of-categories} @@ -413,23 +431,220 @@ \begin{remark} \label{remark-other-2-categories} -There are variants of the construction of \ref{remark-category-of-categories} +There are variants of the construction of \ref{example-category-of-categories} above where we look at the $2$-category of groupoids (contained in some $\alpha$), or categories fibred in groupoids over a fixed category, or stacks. And so on. \end{remark} -\begin{remark} -\label{functor-into-2-category} -A functor from an ordinary category into a $2$-category will ignore the -$2$-morphisms unless mentioned otherwise. -\end{remark} +\begin{remarks} +\label{remarks-functor-into-2-category} +(1) A functor from an ordinary category into a $2$-category will ignore the +$2$-morphisms unless mentioned otherwise. In other words, it will be a +``usual'' functor into the category formed out of 2-category by forgetting +all the 2-morpshisms. + +\smallskip\noindent +(2) Another notion of a functor from a category $\mathcal{A}$ into a +2-category $\mathcal{C}$ would be to say that it is given by a map +$F : \text{Ob}(\mathcal{A}) \to \text{Ob}(\mathcal{C})$ together with a +family of maps +$F : \text{Mor}_{\mathcal{A}}(x,y) \to \text{Mor}_{\mathcal{C}}(F(x),F(y))$ +such that for every composable pair of morphisms $f,g$ of $\mathcal{A}$ +the morphisms $F(g \circ f)$ and $F(g) \circ F(f)$ are 2-isomorphic. This is +not a very good notion, since for example it does not require $F(\text{id}_x)$ +to be isomorphic to $\text{id}_{F(x)}$. Even if you do then +there may be a problem: see the conditions in (3) below. + +\smallskip\noindent +(3) A better notion is the following. A weak functor (or a pseudo-functor) +from a category $\mathcal{A}$ into a 2-category $\mathcal{C}$ is given by +\begin{enumerate} +\item a map $F : \text{Ob}(\mathcal{A}) \to \text{Ob}(\mathcal{C})$, +\item for every pair $x,y\in \text{Ob}(\mathcal{A})$ a map +$F : \text{Mor}_{\mathcal{A}}(x,y) \to \text{Mor}_{\mathcal{C}}(F(x),F(y))$, +\item for every $x\in \text{ob}(C)$ a $2$-morphism +$\alpha_x : \text{id}_x \to F(\text{id}_{x})$, and +\item for every pair of composable morphisms $f,g$ of $\mathcal{A}$ a +$2$-morphism $\alpha_{f,g} : F(g \circ f) \to F(g) \circ F(f)$. +\end{enumerate} +Now these data are subject to the following conditions: +(with notations as in Definition \ref{definition-2-category}) +\begin{enumerate} +\item for any morphism $f : x \to y$ in $\mathcal{A}$ the morphism +$\alpha_{f,\text{id}_y} : F(f) \to F(f) \circ F(\text{id}_y)$ +equals the composition of $F(f) \circ \text{id}_{F(y)} = F(f)$ with +$F(f)(\alpha_y)$, and similary for $\alpha_{\text{id}_x,f}$ and +$\alpha_x$, and +\item for any triple of composable morphisms $f,g,h$ the +compositions $F(h)(\alpha_{f,g}) \circ \alpha_{g\circ f, h}$ and +$F(f)(\alpha_{g,h}) \circ \alpha_{g,f\circ h}$ should be equal. +\end{enumerate} +Again this is not a very workable notion, but it does sometimes come up. +There is a theorem that says that any pseudo-functor is isomorphic to +a functor. FIXME: Add more as needed. +\end{remarks} + +\subsection{2-fibre products} +\label{subsection-2-fibre-products} + +\noindent +In this subsection we introduce $2$-fibre products. Suppose that $\mathcal{C}$ +is a 2-category. We say that a diagram +$$ +\xymatrix{ +w \ar[r] \ar[d] & y \ar[d] \\ +x \ar[r] & z } +$$ +2-commutes if the two 1-morphisms $w \to y \to z$ and $w \to x \to z$ are +2-isomorphic. In a 2-category it is more natural to ask for 2-commutativity +of diagrams than for actually commuting diagrams. (Indeed, some may say that +we should not work with strict 2-categories at all, and in a ``weak'' +2-category the notion of a commutative diagram of 1-morphisms does not even +make sense.) Correspondingly the notion of a fibre product has to be adjusted. + +\smallskip\noindent +Let $\mathcal{C}$ be a $2$-category. Let $x,y,z\in \text{Ob}(\mathcal{C})$ and +$f\in \text{Mor}_{\mathcal{C}}(x,z)$ and $g\in \text{Mor}_{\mathcal C}(y,z)$. +In order to define the 2-fibre product of $f$ and $g$ we are going to look at +2-commutative diagrams +$$ +\xymatrix{ +&w \ar[r]^{a} \ar[d]_{b} & x \ar[d]^{f} \\ +&y \ar[r]^{g} & z. } +$$ +Now in the case of categories, the fibre product is a final object in the +category of such diagrams. Correspondingly a 2-fibre product is a final object +in a 2-category (see definition below). The 2-category we will consider is +the 2-category of 2-commutative diagrams defined as follows: +\begin{enumerate} +\item Objects are quadruples $(w,a,b,\phi)$ as above where $\phi$ +is a 2-morphism $\phi : f \circ a \to g \circ b$, +\item 1-morphisms from $(w,a,b,\phi)$ to $(w',a',b',\phi')$ +are given by $(k : w \to w', \alpha : a' \to a \circ k, +\beta : b \circ k \to b')$ such that $\phi'$ equals +$$ +\xymatrix{ +f \circ a' \ar[r]^{f(\alpha)} & +f \circ a \circ k \ar[r]^{k(\phi)} & +g \circ b \circ k \ar[r]^{g(\beta)} & +g \circ b'. } +$$ +\item a 2-morphism between $(k_i, \alpha_i, \beta_i)$, $i=1,2$ is given +by a 2-morphism $\delta : k_1 \to k_2$ such that +$$ +\xymatrix{ +a' \ar[rd]_{\alpha_2} \ar[r]^{\alpha_1} & +a \circ k_1 \ar[d]^{a(\delta)} & +& +b \circ k_1 \ar[r]^{\beta_1} \ar[d]_{b(\delta)} & +b' +\\ +& +a \circ k_2 & +& +b \circ k_2 \ar[ru]_{\beta_2} +& +} +$$ +commute. +\end{enumerate} + +\begin{definition} +\label{definition-final-object-2-category} +A final object of a 2-category $\mathcal{C}$ is an object $x$ such that +(1) for every $y \in \text{Ob}(\mathcal{C})$ there is a morphism $y \to x$, +and (2) every two morphisms $y \to x$ are isomorphic by a unique 2-morphism. +\end{definition} + +\begin{definition} +\label{definition-2-fibre-products} +Let $\mathcal{C}$ be a $2$-category. +Let $x,y,z\in \text{Ob}(\mathcal{C})$ and $f\in \text{Mor}_{\mathcal{C}}(x,z)$ +and $g\in \text{Mor}_{\mathcal C}(y,z)$. A 2-fibre product of $f$ and $g$ is +a final object in the category of 2-commutative diagrams described above. If +a 2-fibre product exists we +will denote it $x\times_z y\in \text{Ob}(\mathcal{C})$, and denote the +required morphisms $p\in \text{Mor}_{\mathcal C}(x\times_z y,x)$ and +$q\in \text{Mor}_{\mathcal C}(x\times_z y,y)$ making the diagram +$$ +\xymatrix{ +&x\times_y z \ar[r]^{p} \ar[d]_{q} & x \ar[d]^{f} \\ +&y \ar[r]^{g} & z } +$$ +2-commute and we will denote the given 2-morphism exhibiting this by +$\psi : f \circ p \to g \circ q$. +\end{definition} \noindent -FIXME. Perhaps define $2$-functors. +Thus the following universal property holds: for any +$w\in \text{Ob}(\mathcal{C})$ and morphisms +$a \in \text{Mor}_{\mathcal C}(w,x)$ and +$b \in \text{Mor}_{\mathcal{C}}(w,y)$ with a given 2-morphism +$\phi : f \circ a \to g\circ b$ +there is a $\gamma \in \text{Mor}_{\mathcal C}(w,x\times_z y)$ +making the diagram +$$ +\xymatrix{ +w\ar[rrrd]^a \ar@{-->}[rrd]_\gamma \ar[rrdd]_b &&\\ +&&x\times_y z \ar[r]_{p} \ar[d]_{q} & x \ar[d]^{f} \\ +&&y \ar[r]^{g} & z } +$$ +2-commute such that for suitable choices of $q \circ \gamma \to b$ +and $a \to p \circ \gamma$ the composition +$$ +\xymatrix{ +f \circ a \ar[r] & +f \circ p \circ \gamma \ar[r]^{\gamma(\psi)} & +g \circ q \circ \gamma \ar[r] & +g\circ b } +$$ +equals $\phi$. Of course the exact properties are finer than this. All of the +cases of 2-fibre products that we will need later on come from the following +example of 2-fibre products in the 2-category of categories. + +\begin{example} +\label{example-2-fibre-product-categories} +In this example we switch notations and we let $\mathcal{A}$, $\mathcal{B}$, +and $\mathcal{C}$ be categories and we let $F : \mathcal{A} \to \mathcal{C}$ +and $G : \mathcal{B} \to \mathcal{C}$ be functors. In this case the 2-fibre +product $\mathcal{A}\times_\mathcal{C} \mathcal{B}$ exists and is given by +the following: +\begin{enumerate} +\item an object of $\mathcal{A}\times_\mathcal{C} \mathcal{B}$ is a triple +$(A,B,f)$, where $A\in \text{Ob}(\mathcal{A})$, $B\in \text{Ob}(\mathcal{B})$, +and $f : F(A) \to G(B)$ is an isomorphism in $\mathcal{C}$, +\item a morphism $(A,B,f) \to (A',B', f')$ is given by a pair $(a,b)$, where +$a : A \to A'$ is a morphism in $\mathcal{A}$, and $b : B \to B'$ is a +morphism in $\mathcal{B}$ such that the diagram +$$ +\xymatrix{ +F(A) \ar[r]^f \ar[d]^{F(a)} & G(B) \ar[d]^{G(b)} \\ +F(A') \ar[r]^{f'} & G(B') +} +$$ +is commutative. +\end{enumerate} +The functors $p : \mathcal{A}\times_\mathcal{C}\mathcal{B} \to \mathcal{A}$ +and $q : \mathcal{A}\times_\mathcal{C}\mathcal{B} \to \mathcal{A}$ are the +forgetfull functors in this case. The transformation $\psi : F \circ p \to +G \circ q$ is given on the object $\xi = (A,B,f)$ by +$\psi_\xi = f : F(p(\xi)) = F(A) \to G(B) = G(q(\xi))$. \smallskip\noindent -FIXME. Introduce $2$-fibre product here. +Let us check the universal property: let $\mathcal{W}$ be a category, let +$X : \mathcal{W} \to \mathcal{A}$ and $Y : \mathcal{W} \to \mathcal{B}$ be +functors, and let $t : F \circ X \to G \circ Y$ be an isomorphism of functors. +The desired functor $\gamma : \mathcal{W} \to +\mathcal{A}\times_\mathcal{C}\mathcal{B}$ +is given by $W \mapsto (X(W), Y(W), t_W)$. What else could it be? +(A meta-argument for uniqueness.) FIXME: write this out. + +\smallskip\noindent +Note that the functor $\gamma$ constructed above actually has the property +that $p \circ \gamma = X$ and $q \circ \gamma = Y$. In general this need not +be the case. +\end{example} \section{Categories fibred in groupoids} \label{subsection-fibred-groupoids} @@ -461,6 +676,7 @@ p(\phi) = \text{id}_U\}. $$ +\smallskip\noindent In order to discuss the notion of ``category fibred in groupoids'' we temporarily introduce the notion of lifting. A {\it lift} of an object $U \in \text{Ob}(\mathcal{C})$ is an object @@ -523,7 +739,7 @@ $p\colon \mathcal{S}\to\mathcal{C}$ is fibred in groupoids if and only if $p$ is surjective. The fibre category $\mathcal{S}_{U}$ over the (unique) object $U\in \text{Ob}(\mathcal{C})$ is the category associated to the -kernel of $p$ as in Example \ref{example-group}. +kernel of $p$ as in Example \ref{example-group-groupoid}. \end{example} \smallskip\noindent Suppose that for every $f : V \to U$ and $x\in @@ -552,7 +768,6 @@ $$ such that for every $y\in \text{Ob}(\mathcal{S}_W)$ the following diagram commutes - \begin{equation} \xymatrix{ f^\ast g^\ast y \ar[r] \ar[d]_{t_y} & g^\ast y \ar[d] \\ @@ -638,65 +853,11 @@ $$ \end{example} -\begin{example} -\label{example-functor-groupoids} -Suppose that $F : \mathcal{C} \to \text{Groupoids}$ is a contravariant functor -to the category of groupoids (see -\hyperref[remark-functor-into-sets]{Remark~\ref*{remark-functor-into-sets}} -and \hyperref[functor-into-2-category]{Remark~\ref{functor-into-2-category}}). -For $f : V \to U$ in $\mathcal{C}$ we will -suggestively write $F(f) = f^\ast$ for the functor from $F(U)$ to $F(V)$. -From this we can construct a category fibred in groupoids over $\mathcal{C}$ -as follows. Define -$$ -\text{Ob}(\mathcal{S}) = -\{(U,x) \mid U\in \text{Ob}(\mathcal{C}), x\in \text{Ob}(F(U)\}. -$$ -For $(U,x), (V,y) \in \text{Ob}(\mathcal{S})$ we define -$$ -\text{Mor}_\mathcal{S}((V,y),(U,x)) = -\{ (f, \phi) \mid f\in \text{Mor}_\mathcal{C}(V,U), -\phi \in \text{Mor}_{F(V)}(y, f^\ast x)\}. -$$ -In order to define composition we use that $g^\ast \circ f^\ast = -(f \circ g)^\ast$ for a pair of composable morphisms of $\mathcal{C}$ -(by definition of a functor into a $2$-category). -Namely, we define the composition of $\psi : z \to g^\ast y$ and -$ \phi : y \to f^\ast x$ to be $ g^\ast(\phi) \circ \psi$. It is clear -what the functor $p : \mathcal{S} \to \mathcal{C}$ is. The condition -that $F(U)$ is a groupoid for every $U$ guarantees that $\mathcal{S}$ is -fibred in groupoids over $\mathcal{C}$. Lifts of morphisms exist: given -$f: V \to U$ in $\mathcal{C}$ and $(U,x)$ a lift of $U$, then -$(f, id_{f^\ast x}): (V, {f^\ast x}) \to (U,x)$ is a lift of $f$. -Uniqueness means $h$ in the diagram on the left determines $(h,\nu)$ on -the right: -$$ -\xymatrix{ -V \ar[r]^f & U & (V,y) \ar[r]^{(f, \phi)} & (U,x) \\ -W \ar@{-->}[u]^h \ar[ru]_g & & -(W,z) \ar@{-->}[u]^{(h,\nu)} \ar[ru]_{(g, \psi)} & \\ -} -$$ -Then $\nu = (h^\ast \phi)^{-1} \circ \psi $ and the uniqueness of inverses -guarantees this is the only lift making the diagram commute. - \noindent -We will write $\mathcal{S}_F \to \mathcal{C}$ for the resulting functor -if we want to indicate the dependence on $F$. Because we can think of -objects of $\mathcal{S}_F$ as pairs $(U,x)$, we sometimes say $\mathcal{S}_F$ -is a {\it split} category fibred in groupoids. -\end{example} - -\begin{example}\label{example-fibred-category-from-functor-of-points} - When $F=\text{Mor}(-,X)$ for some $X \in \text{Ob}(\mathcal{C})$, - $\mathcal{S}_F\to \mathcal{C}$ is the comma category $\mathcal{C}/X - \to \mathcal{C}$ from Example \ref{example-comma-category}. -\end{example} - -\noindent -We would like to assert that any category fibred in groupoids -over $\mathcal{C}$ is equivalent (over $\mathcal{C}$) to some -$\mathcal{S}_F$ as in the example. The notion of equivalence +Later we would like to make assertions such as ``any category fibred in +groupoids over $\mathcal{C}$ is equivalent to a split one'', or +``any category fibred in groupoids whose fibre categories are setlike +is equivalent to a category fibred in sets''. The notion of equivalence depends on the $2$-category we are working with. To make sure that everybody knows what we are talking about we define the $2$-category of categories over $\mathcal{C}$. @@ -718,13 +879,15 @@ is the full sub-$2$-category of this $2$-category whose objects are categories fibred in groupoids. -\begin{lemma}\label{lemma-equivalence-fibred-categories} +\begin{lemma} +\label{lemma-equivalence-fibred-categories} Let $p\colon \mathcal{S}\to \mathcal{C}$ and $p'\colon \mathcal{S'}\to \mathcal{C}$ be categories fibred in groupoids, and suppose that $G\colon \mathcal{S}\to \mathcal {S}'$ is a functor over $\mathcal{C}$. Then $G$ is fully faithful (resp.\ an equivalence) if and only if for each $U\in\text{Ob}(\mathcal{C})$ the induced functor -$G_U\colon \mathcal{S}_U\to \mathcal{S}'_U$ is fully faithful (resp.\ an equivalence). +$G_U\colon \mathcal{S}_U\to \mathcal{S}'_U$ is fully faithful (resp.\ an +equivalence). \end{lemma} \begin{proof} @@ -754,6 +917,7 @@ and $\text{Mor }_{\mathcal{S}'}(G(x),G(y))$, hence $G$ is fully faithful. +\smallskip\noindent Finally suppose for all $G_U$ is an equivalence for all $U$, so it is fully faithful and essentially surjective. We have seen this implies $G$ is fully faithful, and thus to prove it is an equivalence we have to prove that @@ -766,6 +930,151 @@ isomorphic to $G(z)$ in $\mathcal{S}'$. \end{proof} +\subsection{Categories fibred in sets} +\label{subsection-fibred-in-sets} + +\noindent +Let us call a category setlike if it is a groupoid where every object +has exactly one automorphism: the identity. If $C$ is a set with an +equivalence relation $\sim$, then we can make a setlike category +$\mathcal{C}$ as follows: $\text{Ob}(\mathcal{C}) = C$ and +$\text{Mor}_\mathcal{C}(x,y) = \emptyset$ unless $x \sim y$ in which +case we set $\text{Mor}_\mathcal{C}(x,y) = \{1\}$. Transitivity of +$\sim$ means that we can compose morphisms. Conversely any setlike +category defines an equivalence relation on its objects (isomorphism) +such that you recover the category (up to unique isomorphism -- not +equivalence) from the procedure just described. This is why these categories +are sometimes simply called equivalence relations. + +\smallskip\noindent +A category is called discrete if the only morphisms are the identity +morphisms. Sometimes discrete categories are called sets (reasons as above). +Discrete categories are setlike. For any setlike category $\mathcal{C}$ +there is a canonical procedure to make a discrete category equivalent to it, +namely one replaces $\text{Ob}(\mathcal{C})$ by the set of isomorphism +classes, and adds identity morphisms. + +\begin{definition} +\label{definition-category-fibred-sets} +A category fibred in sets $p : \mathcal{S} \to \mathcal{C}$ +is a category fibred in sets if all fibre categories are discrete. +\end{definition} + +\noindent +We discuss briefly the relationship between categories fibred in sets +and presheaves (see Sites, \hyperref[sites-definition-presheaf]% +{Definition~\ref*{sites-definition-presheaf}}). Suppose that $p : +\mathcal{S} \to \mathcal{C}$ is fibred in sets. Let $f : V \to U$ +be a morphism in $\mathcal{C}$ and let $x \in \text{Ob}(\mathcal{S}_U)$. +Then there is exactly one choice for the object $f^\ast x$. Thus we see that +$(f \circ g)^\ast x = g^\ast(f^\ast x)$ for $f,g$ as in Lemma +\ref{lemma-fibred-groupoids}. It follows that we may think of the +assigments $U \mapsto \text{Ob}(\mathcal{S}_U)$ and $f \mapsto f^\ast$ +as a presheaf on $\mathcal{C}$. + +\smallskip\noindent +Conversely, given a presheaf of sets +$F : \mathcal{C}^{\text{opp}} \to \text{Sets}$ +we can construct a category $\mathcal{S}_F$ fibred in sets +over $\mathcal{C}$ by taking as fibre category $\mathcal{S}_{F,U}$ +the discrete category whose underlying set is $F(U)$. This is explained +more generally, and in more detail in Example \ref{example-functor-groupoids} +below. + +\begin{lemma} +Suppose that $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in +groupoids all of whose fibre categories $\mathcal{S}_U$ are setlike. +Then there exists a category fibred in sets $p' : \mathcal{S}' \to +\mathcal{C}$ and an equivalence $\mathcal{S} \to \mathcal{S}'$ +of categories over $\mathcal{C}$. +\end{lemma} + +\begin{proof} +An object of the category $\mathcal{S}'$ will be a pair $(U, \xi)$, where +$U \in \text{Ob}(\mathcal{C})$ and $\xi$ is an isomorphism class of objects +of $\mathcal{S}_U$. A morphism $(U,\xi) \to (V , \psi)$ is given by a +morphism $x \to y$, where $x \in \xi$ and $y \in \psi$. Here we identify +two morphisms $x \to y$ and $x' \to y'$ if they induce the same morphism +$U \to V$, and if for some choices of isomorphisms $x \to x'$ in +$\mathcal{S}_U$ and $y \to y'$ in $\mathcal{S}_V$ the compositions +$x \to x' \to y'$ and $x \to y \to y'$ agree. By construction there are +surjective maps on objects and morphisms from $\mathcal{S} \to +\mathcal{S}'$. We define composition of morphisms in $\mathcal{S}'$ to +be the unique law that turns $\mathcal{S} \to \mathcal{S}'$ into a functor. +FIXME: check this is well-defined. + +\smallskip\noindent +By construction the rule $(U,\xi) \mapsto U$ is a functor. FIXME: check the +other properties. +\end{proof} + +\subsection{Presheaves of groupoids} +\label{subsection-presheaves-groupoids} + +\noindent +In this subsection we compare the notion of categories fibred in groupoids +with the closely related notion of a ``presheaf of groupoids''. The basic +construction is explained in the following example. + +\begin{example} +\label{example-functor-groupoids} +Suppose that $F : \mathcal{C} \to \text{Groupoids}$ is a contravariant functor +to the category of groupoids (see +\hyperref[remark-functor-into-sets]{Remark~\ref*{remark-functor-into-sets}} and +\hyperref[remarks-functor-into-2-category]% +{Remark~\ref{remarks-functor-into-2-category}}). +For $f : V \to U$ in $\mathcal{C}$ we will +suggestively write $F(f) = f^\ast$ for the functor from $F(U)$ to $F(V)$. +From this we can construct a category fibred in groupoids over $\mathcal{C}$ +as follows. Define +$$ +\text{Ob}(\mathcal{S}) = +\{(U,x) \mid U\in \text{Ob}(\mathcal{C}), x\in \text{Ob}(F(U)\}. +$$ +For $(U,x), (V,y) \in \text{Ob}(\mathcal{S})$ we define +$$ +\text{Mor}_\mathcal{S}((V,y),(U,x)) = +\{ (f, \phi) \mid f\in \text{Mor}_\mathcal{C}(V,U), +\phi \in \text{Mor}_{F(V)}(y, f^\ast x)\}. +$$ +In order to define composition we use that $g^\ast \circ f^\ast = +(f \circ g)^\ast$ for a pair of composable morphisms of $\mathcal{C}$ +(by definition of a functor into a $2$-category). +Namely, we define the composition of $\psi : z \to g^\ast y$ and +$ \phi : y \to f^\ast x$ to be $ g^\ast(\phi) \circ \psi$. It is clear +what the functor $p : \mathcal{S} \to \mathcal{C}$ is. The condition +that $F(U)$ is a groupoid for every $U$ guarantees that $\mathcal{S}$ is +fibred in groupoids over $\mathcal{C}$. Lifts of morphisms exist: given +$f: V \to U$ in $\mathcal{C}$ and $(U,x)$ a lift of $U$, then +$(f, id_{f^\ast x}): (V, {f^\ast x}) \to (U,x)$ is a lift of $f$. +Uniqueness means $h$ in the diagram on the left determines $(h,\nu)$ on +the right: +$$ +\xymatrix{ +V \ar[r]^f & U & (V,y) \ar[r]^{(f, \phi)} & (U,x) \\ +W \ar@{-->}[u]^h \ar[ru]_g & & +(W,z) \ar@{-->}[u]^{(h,\nu)} \ar[ru]_{(g, \psi)} & \\ +} +$$ +Then $\nu = (h^\ast \phi)^{-1} \circ \psi $ and the uniqueness of inverses +guarantees this is the only lift making the diagram commute. + +\noindent +We will write $\mathcal{S}_F \to \mathcal{C}$ for the resulting functor +if we want to indicate the dependence on $F$. Because we can think of +objects of $\mathcal{S}_F$ as pairs $(U,x)$, we sometimes say $\mathcal{S}_F$ +is a {\it split} category fibred in groupoids. +\end{example} + +\begin{example} +\label{example-fibred-category-from-functor-of-points} +When $F=\text{Mor}(-,X)$ for some $X \in \text{Ob}(\mathcal{C})$, +$\mathcal{S}_F\to \mathcal{C}$ is the category +$\mathcal{C}/X \to \mathcal{C}$ from Example \ref{example-comma-category}. +\end{example} + + + \begin{lemma} \label{lemma-fibred-strict} Let $ p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. @@ -784,7 +1093,7 @@ and an object $x$ of $\mathcal{S}$ over $U$, i.e., $x\in \text{Ob}(\mathcal{S}_U)$. The functor $p' : \mathcal{S}' \to \mathcal{C}$ will map the pair $(x,f)$ to the source -of the morphism $f$, in on other words $p'(x,f:V\to U) = V$. A morphism +of the morphism $f$, in other words $p'(x,f:V\to U) = V$. A morphism $\varphi : (x_1,f_1: V_1 \to U_1) \to (x_2, f_2 : V_2 \to U_2)$ is given by a pair $(\varphi,g)$ consisting of a morphism $g : V_1 \to V_2$ and a morphism $\varphi : f_1^\ast x_1 \to f_2^\ast x_2$ with $p(\varphi) = g$. It is no @@ -838,7 +1147,8 @@ $\mathcal{S}'$ by Lemma \ref{lemma-equivalence-fibred-categories}. \end{proof} -\begin{lemma}\label{lemma-yoneda-2category} +\begin{lemma} +\label{lemma-yoneda-2category} Let $\mathcal{S}\to \mathcal{C}$ be fibred in groupoids. Then for any $U\in \text{Ob}(\mathcal{C})$ the functor $$ diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/stacks.tex stacks-0.2/src/stacks.tex --- stacks-0.2.orig/src/stacks.tex 2006-03-07 19:44:10.000000000 +0000 +++ stacks-0.2/src/stacks.tex 2006-03-14 02:49:38.000000000 +0000 @@ -116,28 +116,30 @@ \noindent Let $\mathcal{C}$ be a site. The $2$-category of stacks over $\mathcal{C}$ will be a full sub-$2$-category of the $2$-category -of categories over $\mathcal{C}$, see Categories, -\autoref{categories-definition-categories-over-C}. Thus a stack will be -given by a functor of categories $p : \mathcal{S} \to \mathcal{C}$ -which has certain additional properties. Loosely speaking -the conditions are the following: +of categories over $\mathcal{C}$, see Categories, +\hyperref[categories-definition-categories-over-C]% +{Definition~\ref*{categories-definition-categories-over-C}}. +Thus a stack will be given by a functor of categories +$p : \mathcal{S} \to \mathcal{C}$ which satisfies the following +conditions: \begin{enumerate} \item $p : \mathcal{S} \to \mathcal{C}$ is a category fibred -in groupoids, see Categories, -\autoref{categories-definition-fibred-groupoids}, +in groupoids, see Categories, +\hyperref[categories-definition-fibred-groupoids]% +{Definition~\ref*{categories-definition-fibred-groupoids}}, \item descent for morphisms holds, and \item descent data for objects are effective. \end{enumerate} -\subsection{Explanation and examples} -\label{subsection-defintion-explanation} +\subsection{Explanation} +\label{subsection-definition-explanation} \noindent To explain this, we choose a collection of pullback functors as in Categories, \hyperref[categories-lemma-fibred-groupoids]% {Lemma~\ref*{categories-lemma-fibred-groupoids}}. Another approach is to use -Categories, \hyperref[categories-lemma-fibred-groupoids]% -{Lemma~\ref*{categories-lemma-fibred-groupoids}}. +Categories, \hyperref[categories-lemma-fibred-strict]% +{Lemma~\ref*{categories-lemma-fibred-strict}}. \smallskip\noindent First, suppose that $x,y\in \text{Ob}(\mathcal{S}_U)$ are @@ -209,6 +211,7 @@ & = & [(\text{Isom}(x,y))(g'')]([(\text{Isom}(x,y))(g')](\phi)). \end{eqnarray*} +\noindent Alternatively we can argue with \hyperref[categories-lemma-fibred-strict]{Lemma~\ref*{categories-lemma-fibred-strict}} which says that $\mathcal{S}\to\mathcal{C}$ is equivalent to the @@ -219,40 +222,36 @@ presheaf. \end{proof} - -\noindent -This lemma says that there is no harm in thinking of -$g^\ast \phi$ as a morphism ${f'}^\ast x \to {f'}^\ast y$. - \smallskip\noindent -OK, so the second condition listed above is simply the condition that -$\text{Isom}(x,y)$ is a sheaf on the site $\mathcal{C}/U$! +OK, so the second condition listed in Section \ref{section-definition} +is simply the condition that $\text{Isom}(x,y)$ is a sheaf on the site +$\mathcal{C}/U$! \smallskip\noindent In order to explain the meaning of the third condition, we must define a -descent datum. First, we introduce some notation. -Suppose that $\{f_i : U_i \to U\}_{i\in I}$ -is a covering in the site $\mathcal{C}$. We will be looking at -fibre products $U_{ij} = U_i \times_U U_j$ and -$U_{ijk} = U_i \times_U U_j \times_U U_k$. -When three indices, say $i,j$, and $k$, are in play, the projection maps from -the twofold fibre products will be denoted as follows: $\text{pr}_{1/2}$ is -the projection $U_{ij} \to U_i$, $\text{pr}_{3/1}$ is the projection -$U_{ik} \to U_k$, and so on. If only two indices, say $i$ and $j$, are in -play, the projection maps $U_{ij} \to U_i$ and $U_{ij} \to U_j$ will be -denoted by $\text{pr}_{1}$ and $\text{pr}_{2}$, resp.; this should cause no -confusion. The projection maps from $U_{ijk}$ to the twofold fibre products -are $\text{pr}_{12} : U_{ijk} \to U_{ij}$, +descent datum. First, we introduce some notation. Suppose that +$\{f_i : U_i \to U\}_{i\in I}$ is a covering in the site $\mathcal{C}$. +We will be looking at fibre products $U_{ij} = U_i \times_U U_j$ and +$U_{ijk} = U_i \times_U U_j \times_U U_k$. It is very important to allow +$i=j$ and even $i=j=k$ in the following. The projection maps from +$U_{ij}$ to $U_i$, resp.\ $U_j$ are denoted $\text{pr}_1$, resp.\ +$\text{pr}_2$. The projection maps from $U_{ijk}$ to the twofold fibre +products are $\text{pr}_{12} : U_{ijk} \to U_{ij}$, $\text{pr}_{13} : U_{ijk} \to U_{ik}$, and -$\text{pr}_{23} : U_{ijk} \to U_{jk}$. Finally, the projection maps from -$U_{ijk}$ to $U_{i}$, $U_{j}$, and $U_{k}$ are ${\widetilde{pr}}_1$, -${\widetilde{pr}}_2$, and ${\widetilde{pr}}_3$, respectively. -(Note the relations ${\widetilde{pr}}_1 = \text{pr}_{1/2} \circ \text{pr}_{12} -= \text{pr}_{1/3} \circ \text{pr}_{13}$, -${\widetilde{pr}}_2 = \text{pr}_{2/1} \circ \text{pr}_{12} = -\text{pr}_{2/3} \circ \text{pr}_{23}$, and -${\widetilde{pr}}_3 = \text{pr}_{3/1} \circ \text{pr}_{13} = -\text{pr}_{3/2} \circ \text{pr}_{23}$.) +$\text{pr}_{23} : U_{ijk} \to U_{jk}$. The projection maps from +$U_{ijk}$ to $U_{i}$, $U_{j}$, and $U_{k}$ are $\text{pr}_1$, +$\text{pr}_2$, and $\text{pr}_3$, respectively. + +\smallskip\noindent +This notation is potentially ambiguous, but it is standard in the +literature. For an example of the ambiguity: note the relations +$\text{pr}_1 = \text{pr}_{1} \circ \text{pr}_{12} = +\text{pr}_1 \circ \text{pr}_{13}$, $\text{pr}_2 = +\text{pr}_2 \circ \text{pr}_{12} = \text{pr}_1 \circ \text{pr}_{23}$, +and $\text{pr}_3 = \text{pr}_2 \circ \text{pr}_{13} = +\text{pr}_3 \circ \text{pr}_{23}$. + +\smallskip\noindent Let $x_i \in \text{Ob}(\mathcal{S}_{U_i})$, $i\in I$ be a collection of objects, and let $$ @@ -260,10 +259,10 @@ \text{pr}_{2}^\ast x_j \qquad (i,j \in I) $$ be a collection of morphisms in the fibre categories -$\mathcal{S}_{U_{ij}}$. In the definition that follows we will abuse notation -by identifying, for instance, ${\widetilde{pr}}_1^\ast x_i$ with both -$\text{pr}_{12}^\ast \text{pr}_{1/2}^\ast x_i$ and -$\text{pr}_{13}^\ast \text{pr}_{1/3}^\ast x_i$. +$\mathcal{S}_{U_{ij}}$. In the definition that follows we will +identify, for instance, $\text{pr}_1^\ast x_i$ on $U_{ijk}$ with both +$\text{pr}_{12}^\ast \text{pr}_{1}^\ast x_i$ and +$\text{pr}_{13}^\ast \text{pr}_{1}^\ast x_i$. \begin{definition} \label{definition-descent-data} @@ -272,39 +271,39 @@ triple $(i,j,k)\in I^3$ the diagram $$ \xymatrix{ -{\widetilde{pr}}_1^\ast x_i +\text{pr}_1^\ast x_i \ar[rr]^{\text{pr}_{13}^\ast \phi_{ik}} \ar[rd]_{\text{pr}_{12}^\ast \phi_{ij}} & & -{\widetilde{pr}}_3^\ast x_k \\ -& {\widetilde{pr}}_2^\ast x_j \ar[ru]_{\text{pr}_{23}^\ast \phi_{jk}} +\text{pr}_3^\ast x_k \\ +& \text{pr}_2^\ast x_j \ar[ru]_{\text{pr}_{23}^\ast \phi_{jk}} } $$ -in $\mathcal{S}_{U_{ijk}}$ is commutative. +in the fibre category $\mathcal{S}_{U_{ijk}}$ is commutative. \end{definition} -\noindent +\begin{remarks} +\label{remarks-definition-descent-datum} Two remarks about this definition are in order. -\begin{itemize} -\item{There is a diagonal morphism $\Delta_i : U_i \to U_{ii}$. We can pull back +\smallskip\noindent +(1) There is a diagonal morphism $\Delta_i : U_i \to U_{ii}$. We can pull back $\phi_{ii}$ via this morphism to get an automorphism ${\Delta_i}^\ast \phi_{ii} \in \text{Aut}_{U_i}(x_i)$. On pulling back the cocycle condition for the triple $(i,i,i)$ by $\Delta_{123} : U_i \to U_{iii}$ we deduce that ${\Delta_i}^\ast \phi_{ii} \circ {\Delta_i}^\ast \phi_{ii} = {\Delta_i}^\ast \phi_{ii}$; thus ${\Delta_i}^\ast \phi_{ii} = -\text{id}_{x_i}$.} +\text{id}_{x_i}$. -\medskip - -\item{There is a morphism +\smallskip\noindent +(2) There is a morphism $\Delta_{13}: U_{ij} \to U_{iji}$ and we can pull back the cocycle condition for the triple $(i,j,i)$ to get the identity $(\sigma^\ast \phi_{ji}) \circ \phi_{ij} = \text{id}_{\text{pr}_{1/2}^\ast x_i}$, where $\sigma: U_{ij} \to U_{ji}$ is the -switching morphism.} -\end{itemize} +switching morphism. +\end{remarks} \smallskip\noindent A morphism of descent data @@ -385,9 +384,11 @@ \noindent Usually the hardest part to check is the third condition. -\begin{example} +\subsection{Examples} +\label{subsection-examples} + +\noindent FIXME: Need lots of examples. -\end{example} \noindent FIXME: To be continued.