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 2006-02-10 13:42:42.000000000 +0000 +++ stacks-0.2/src/categories.tex 2006-02-16 16:33:23.000000000 +0000 @@ -155,6 +155,12 @@ 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 $x=y$. +\end{example} + \smallskip\noindent A functor $F : \mathcal{A} \to \mathcal{B}$ between two categories $\mathcal{A}, \mathcal{B}$ is given by the following data: @@ -193,9 +199,22 @@ 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 $p_X\circ F = + p_{X'}$. +\end{example} + \smallskip\noindent -A transformation of functors $t : F \to G$ (or simply a morphism of funtors) +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 data: \begin{enumerate} @@ -256,10 +275,36 @@ and $g\in \text{Mor}_{\mathcal C}(y,z)$. \end{definition} +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 +$\text{Mor}_{\mathcal{C}^{\text{opp}}}(x,y) = +\text{Mor}_{\mathcal{C}}(y,x)$. A contravariant functor $F\colon +\mathcal{C}\to \mathcal{S}$ is a functor $\mathcal{C}^{\text{opp}}\to +\mathcal{S}$. Concretely, if $F$ is contravariant then for composable +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 $U$. +\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} + \noindent -FIXME: Contravariant functors, representable functors, -representable morphisms (namely morphisms $x\to y$ such that for every -$w \to y$ the fibre product $w\times_y x$ exists), etc. +FIXME: representable morphisms (namely morphisms +$x\to y$ such that for every $w \to y$ the fibre product $w\times_y x$ +exists), etc. \subsection{2-Categories} \label{subsection-2-categories} @@ -455,7 +500,7 @@ Suppose that $g : W \to V$ and $f : V \to U$ are morphisms in $\mathcal{C}$. Let $x \in \text{Ob}(\mathcal{S}_U)$. By the first condition we can lift $f$ to $ \phi : y \to x$ and then we can lift $g$ to $\psi : z \to y$. -Instead of doing this two step proces we can directly lift $g \circ f$ to +Instead of doing this two step process we can directly lift $g \circ f$ to $\gamma : z' \to x$. This gives the solid arrows in the diagram below. $$ \xymatrix{ @@ -492,7 +537,7 @@ f^\ast x \ar[r]^{f^\ast \phi} \ar[d] & f^\ast x' \ar[d] \\ x \ar[r]^{\phi} & x' } $$ -commutes. Again uniqueness of this arrow garantees that $f^\ast$ is a +commutes. Again uniqueness of this arrow guarantees that $f^\ast$ is a functor $ f^\ast : \mathcal{S}_U \to \mathcal{S}_V$. \begin{lemma} @@ -604,7 +649,7 @@ From this we can construct a category fibred in groupoids over $\mathcal{C}$ as follows. Define $$ -\text{ob}(\mathcal{S}) = +\text{Ob}(\mathcal{S}) = \{(U,x) \mid U\in \text{Ob}(\mathcal{C}), x\in \text{Ob}(F(U)\}. $$ For $(U,x), (V,y) \in \text{Ob}(\mathcal{S})$ we define @@ -619,7 +664,7 @@ 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 $U$ garantees that $\mathcal{S}$ is +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 $f$. @@ -639,7 +684,13 @@ 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 fibed in groupoids. +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 @@ -656,8 +707,8 @@ as follows. Its objects will be functors $p : \mathcal{S} \to \mathcal{C}$ (belonging to some set, see Sets, \autoref{sets-section-reflection-principle}). Its -$1$-morphisms will be functors $F : \mathcal{S} \to \mathcal{S}'$ -such that $p' \circ F = p$, and its $2$-morphisms $t : F \to G$ +$1$-morphisms will be functors $G : \mathcal{S} \to \mathcal{S}'$ +such that $p' \circ G = p$, and its $2$-morphisms $t : G \to H$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \text{Ob}(\mathcal{S})$. \end{definition} @@ -667,6 +718,54 @@ is the full sub-$2$-category of this $2$-category whose objects are categories fibred in groupoids. +\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). +\end{lemma} + +\begin{proof} +Clearly if $G$ is fully faithful (resp.\ an equivalence) then so is $G_U$. So +suppose that $G_U$ is fully faithful for all $U\in\text{Ob}(\mathcal C)$. To +show that $G$ is fully faithful we have to show for any objects +$x,y\in\text{Ob}(\mathcal{S})$ that $G$ induces a bijection between +$\text{Mor}_{\mathcal{S}}(x,y)$ and $\text{Mor}_{\mathcal{S}'}(G(x),G(y))$. +To this end let $\phi'\colon G(x)\to G(y)$ and set $U=p(x)$ and $V=p(y)$. +As $\mathcal{S}$ is fibred in groupoids there is a lift $z\to y$ of +$p'(\phi')$ in $\mathcal{S}$, and any morphisms $x\to y$ factors uniquely +as $x\to z\to y$, where the map $x\to z$ lifts $\text{id}_U$, as in the +following diagram +$$ +\xymatrix{ +x \ar@{-->}[d] \ar[rd] \\ +z \ar[r]^\psi \ar[d] & y \ar[d] \\ +U \ar[r]^{p'(\phi')} &V} +$$ +Now in $\mathcal{S}'$, $G(\psi)\colon G(z)\to G(y)$ is the pullback of +$G(y)$, so any morphism $G(x)\to G(y)$ factors uniquely +as $G(x)\to G(z)\to G(y)$, where again the map +$G(x)\to G(z)$ lifts $\text{id}_U$. Since $G_U$ +induces a bijection between $\text{Mor}_{\mathcal{S}_U}(x,z)$ and +$\text{Mor}_{\mathcal{S}'_U}(G(x),G(z))$ we get that +$G$ induces a bijection between $\text{Mor}_{\mathcal{S}}(x,y)$ +and $\text{Mor }_{\mathcal{S}'}(G(x),G(y))$, hence $G$ +is fully faithful. + +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 +it is essentially surjective. This is clear, for if $z'\in +\text{Ob}(\mathcal{S}')$ then $z'\in \text{Ob}(\mathcal{S}'_U)$ where +$U=p'(z')$. Since $G_U$ is essentially surjective we know that +$z'$ is isomorphic, in $\mathcal{S}'_U$, to an object of the form +$G_U(z)$ for some $z\in \text{Ob}(\mathcal{S}_U)$. But morphisms +in $\mathcal{S}'_U$ are morphisms in $\mathcal{S}'$ and hence $z'$ is +isomorphic to $G(z)$ in $\mathcal{S}'$. +\end{proof} + \begin{lemma} \label{lemma-fibred-strict} Let $ p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. @@ -698,7 +797,7 @@ FIXME. We need to check that $p'$ makes $\mathcal{S}'$ into a category fibred in groupoids over $\mathcal{C}$, and we need to check that $\mathcal{S} \to \mathcal{S}'$ is an equivalence of categories over -$\mathcal{C}$. +$\mathcal{C}$ (hopefully the lemma above helps!). \smallskip\noindent Finally, we can define pullback functors on $\mathcal{S}'$ @@ -714,6 +813,54 @@ is split as desired. \end{proof} +\begin{proof}[Alternate proof] +We define a contravariant functor $F$ from $\mathcal{C}$ to the +category of groupoids as follows: for $U\in \text{Ob}(\mathcal{C})$ +set $F(U) = \text{Mor}(\mathcal{S}/U,\mathcal{S})$ to be the set of +base preserving natural transformations. If $f\colon U\to V$ the +induced functor $\mathcal{S}/U\to \mathcal{S}/V$ induces the +morphism $F(f)\colon F(V)\to F(U)$. Clearly $F$ is a functor, and +we will see below that it is a functor into groupoids. Let +$\mathcal{S}'$ be the associated category fibred in groupoids from Example +\ref{example-functor-groupoids}. + +\smallskip\noindent +There is an obvious functor $G\colon \mathcal{S}'\to \mathcal{S}$ +over $\mathcal{C}$ given by taking the pair $(U,x)$, where +$U\in\text{Ob}(\mathcal{C})$ and $x\in F(U)$, to +$x(U\stackrel{\text{id}_U}{\to} U) \in \mathcal{S}$. Now Lemma +\ref{lemma-yoneda-2category} implies that for each $U$, +$$ +G_U\colon \mathcal{S}'_U = F(U)= +\text{Mor}(\mathcal{C}/U,\mathcal{S}) \to \mathcal{S}_U +$$ +is an equivalence, and thus $G$ equivalence between $\mathcal{S}$ and +$\mathcal{S}'$ by Lemma \ref{lemma-equivalence-fibred-categories}. +\end{proof} + +\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 +$$ +G\colon \text{Mor}(\mathcal{C}/U,\mathcal{S}) \to \mathcal{S}_U +$$ +given by $G(x) = x(U\stackrel{\text{id}_U}{\to} U)$ is an equivalence. +\end{lemma} + +\noindent +FIXME: Do we have notation for base preserving transformations already? +Say what $G$ does on arrows. + +\begin{proof} +We define a functor $H\colon \mathcal{S}_U \to +\text{Mor}(\mathcal{C}/U,\mathcal{S})$ as follows. Given $x\in +\text{Ob}(\mathcal{S}_U)$ and $f\colon X\to U$ set $H(x)(f) = f^*x$. +(FIXME: say what this does on arrows and prove this gives an +equivalence). +\end{proof} + +\noindent {\bf Biographical notes:} Parts of this have been taken from +Vistoli's notes \cite{Vis2}. \smallskip\noindent To continue reading, @@ -727,9 +874,7 @@ \end{enumerate} - - -\bibliography{my} \bibliographystyle{alpha} +\bibliography{my} \end{document} diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/documentation/dontdiff stacks-0.2/src/documentation/dontdiff --- stacks-0.2.orig/src/documentation/dontdiff 2005-10-17 14:59:55.000000000 +0000 +++ stacks-0.2/src/documentation/dontdiff 2006-02-16 16:33:23.000000000 +0000 @@ -13,3 +13,4 @@ *.patch *.gz *.bz2 +auto \ No newline at end of file diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/documentation/submitting-patches stacks-0.2/src/documentation/submitting-patches --- stacks-0.2.orig/src/documentation/submitting-patches 2005-10-17 14:59:55.000000000 +0000 +++ stacks-0.2/src/documentation/submitting-patches 2006-02-16 16:33:23.000000000 +0000 @@ -39,7 +39,7 @@ done, produce a patch as follows: cd ~ - diff -ruN -X stacks-0.2.orig/src/documentation/dontdiff \ + diff -ruNb -X stacks-0.2.orig/src/documentation/dontdiff \ stacks-0.2.orig stacks-0.2 > patchfile Submit the patchfile to the mailing list; please include PATCH in the subject. diff -urN -X stacks-0.2/src/documentation/dontdiff stacks-0.2.orig/src/my.bib stacks-0.2/src/my.bib --- stacks-0.2.orig/src/my.bib 2005-08-01 20:29:37.000000000 +0000 +++ stacks-0.2/src/my.bib 2006-02-16 16:33:23.000000000 +0000 @@ -1651,6 +1651,13 @@ PAGES = "613--670" } +@UNPUBLISHED{Vis2, + AUTHOR = "Angelo Vistoli", + TITLE = "Notes on Grothendieck topologies, fibered categories and descent theory", + EPRINT = "arXiv:math.AG/0412512", + URL = "http://www.dm.unibo.it/~vistoli/descent.pdf" + } + @INCOLLECTION{VHilb, AUTHOR = "Angelo Vistoli", TITLE = "The {H}ilbert stack and the theory of moduli of