# Friday, July 30 References: - Course website: - [Gómez 99: Expository article on algebraic stacks](https://arxiv.org/pdf/math/9911199.pdf) ::: {.remark} Stated goal of the course: prove that the moduli space \( \mkern 1.5mu\overline{\mkern-1.5mu{ \mathcal{M}_{g} }\mkern-1.5mu}\mkern 1.5mu \) of stable curves (for \( g\geq 2 \)) is a smooth, proper, irreducible Deligne-Mumford stack of dimension \( 3g-3 \). Moreover, it admits a projective coarse moduli space. In the process we'll define **algebraic spaces** and **stacks**. Prerequisites: - Schemes - Existence of Hilbert schemes - Artin approximation - Resolution of singularities for surfaces - Deformation theory ::: # Lecture 3: Groupoids and Prestacks (Monday, September 06) ## Groupoids ::: {.remark} Last time: functors, sheaves on sites, descent, and Artin approximation. Today: groupoids and stacks. Recall that a **site** \( \mathsf{S} \) is a category such that for all \( U\in {\operatorname{Ob}}(\mathsf{S}) \), there exists a set \( {\mathsf{Cov}}(U) \coloneqq\left\{{U_i \to U}\right\}_{i\in I} \) (a *covering family*) such that - \( \operatorname{id}_U \in {\mathsf{Cov}}(U) \), - \( {\mathsf{Cov}}(U) \) is closed under composition. - \( {\mathsf{Cov}}(U) \) is closed under pullbacks: ```{=tex} \begin{tikzcd} {\exists U_i{ \underset{\scriptscriptstyle {U} }{\times} }V} && {U_i} \\ \\ V && U \arrow["{\in {\mathsf{Cov}}(U)}", from=1-3, to=3-3] \arrow[from=3-1, to=3-3] \arrow[dashed, from=1-1, to=1-3] \arrow[dashed, from=1-1, to=3-1] \arrow["\lrcorner"{anchor=center, pos=0.025}, draw=none, from=1-1, to=3-3] \arrow["{\in{\mathsf{Cov}}(U)}"{description}, curve={height=-12pt}, dashed, from=1-1, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMiwyLCJVIl0sWzAsMiwiViJdLFsyLDAsIlVfaSJdLFswLDAsIlxcZXhpc3RzIFVfaVxcZmliZXJwcm9ke1V9ViJdLFsyLDAsIlxcaW4gXFxDb3YoVSkiXSxbMSwwXSxbMywyLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMywxLCIiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMywwLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XSxbMywwLCJcXGluXFxDb3YoVSkiLDEseyJjdXJ2ZSI6LTIsInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==) ::: ::: {.example title="The big étale site"} Take \( \mathsf{S} \coloneqq{\mathsf{Sch}}_{\text{Ét}} \) to be the big étale site: the category of all schemes, with covering families given by étale morphisms \( \left\{{U_i\to U}\right\}_{i\in I} \) such that \( \displaystyle\coprod_i U_i \twoheadrightarrow U \). Note that there is a special covering family given by *surjective* etale morphisms. ```{=tex} \todo[inline]{Reducing to case of single surjective etale cover somehow?} ``` ::: ::: {.definition title="Sheaves on sites"} Let \( \mathsf{C} \) be a category (e.g. \( \mathsf{C} \coloneqq{\mathsf{Set}} \)) and recall that a *presheaf* on a category \( \mathsf{S} \) is a contravariant functor \( \mathsf{S}\to \mathsf{C} \). A \( \mathsf{C}{\hbox{-}} \)valued **sheaf** on a site \( \mathsf{S} \) is a presheaf \[ {\mathcal{F}}:\mathsf{S} \to \mathsf{C} \] such that for all \( U_i, U_j\in {\mathsf{Cov}}(U) \), the following equalizer diagram is exact in \( \mathsf{C} \) ```{=tex} \begin{tikzcd} 0 \stackarr{1}[r] & F(U) \stackarr{3}[r] & \prod\limits_{i} F(U_i) \stackarr{5}[r] & \prod\limits_{i, j} F(U_i { \underset{\scriptscriptstyle {U} }{\times} } U_j) \end{tikzcd} ``` ::: ::: {.exercise title="Criterion for sheaves on the big etale site"} Show that a presheaf \( F \) is a sheaf on \( {\mathsf{Sch}}_\text{Ét} \) iff - \( F \) is a sheaf on \( {\mathsf{Sch}}_{\mathrm{Zar}} \) and - For all etale surjections \( U' \twoheadrightarrow_{\text{ét}} U \) of affines, the equalizer diagram is exact. ::: ::: {.proposition title="Yoneda"} For \( X\in {\mathsf{Sch}} \), the presheaf \[ h_X \coloneqq\operatorname{Mor}({-}, X): {\mathsf{Sch}}\to {\mathsf{Set}} \] is a sheaf on \( {\mathsf{Sch}}_{\text{Ét}} \). ::: ::: {.remark} We'll often consider *moduli functors*: functors \( F: {\mathsf{Sch}}\to {\mathsf{Set}} \) where \( F(S) \) is a family of objects over \( S \). Then \( F \) will be a sheaf iff families glue uniquely in the étale topology, and representability of such functors will imply they are sheaves. ::: ::: {.example title="A non-sheaf"} Consider the following moduli functor: ```{=tex} \begin{tikzpicture} \node {% \(\begin{aligned} F_{{\mathsf{Alg}}}: {\mathsf{Sch}}&\to {\mathsf{Set}}\\ S &\mapsto \left\{ \begin{tikzcd} \mathcal{C} \ar[d] \\ S \end{tikzcd} \right. \begin{aligned} \text{Smooth families of}\\ \text{genus $g$ curves.} \end{aligned} \end{aligned}\) }; \end{tikzpicture} ``` This is *not* representable by a scheme and not a sheaf. ::: ::: {.remark} Why care about representability? Suppose there were a scheme \( M \), so \[ F_{{\mathsf{Alg}}}(S) \simeq \operatorname{Mor}(S, M) .\] Then taking \( \operatorname{id}_M \in \operatorname{Mor}(M, M) \) should yield a universal family \( {\mathcal{U}}\to M \): ![](figures/2021-09-06_14-50-50.png) Then the points of \( M \) would correspond to isomorphism classes of curves, and every family of curves would be a pullback of this. For any \( S\in{\mathsf{Sch}} \) and a family \( {\mathcal{C}}\xrightarrow{f} S \), the fiber \( f^{-1}(s)\in{\mathcal{C}} \) is a curve for any \( s\in S \), so one could define a map \[ g: S &\to M \\ s &\mapsto [s] ,\] where we send a curve to its isomorphism class. Then \( {\mathcal{C}} \) would fit into a pullback diagram: ```{=tex} \begin{tikzcd} {\mathcal{C}}&& {\mathcal{U}}\\ \\ S && M \arrow[from=1-3, to=3-3] \arrow[from=3-1, to=3-3] \arrow[from=1-1, to=3-1] \arrow[dashed, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jYyJdLFsyLDAsIlxcbWN1Il0sWzAsMiwiUyJdLFsyLDIsIk0iXSxbMSwzXSxbMiwzXSxbMCwyXSxbMCwxLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=) If \( S \) was itself a curve, then \( g: S\to M \) would be a path in \( M \) deforming a base curve. ::: ## Groupoids ::: {.remark} Recall that a **groupoid** is a category where every morphism is an isomorphism. Morphisms of groupoids are functors, and isomorphisms of groupoids are equivalences of categories. ::: ::: {.example title="Groupoid of a set"} A basic example is the category of sets where \[ \operatorname{Mor}(A, B) \coloneqq \begin{cases} \operatorname{id}_A & A=B \\ \emptyset & \text{else}. \end{cases} \] A similar construction: for any set \( \Sigma \), one can form a groupoid \( {\mathcal{C}}_\Sigma \): - Object: Elements \( x\in \Sigma \). - Morphisms: \( \operatorname{id}_x \) ::: ::: {.example title="Moduli of curves"} Define a category \( { \mathcal{M}_{g} }({\mathbb{C}}) \): - Objects: smooth projective curves over \( {\mathbb{C}} \) of genus \( g \). - Morphisms: \[ \operatorname{Mor}(C, C') = \mathop{\mathrm{Isom}}_{{\mathsf{Sch}}_{/ {{\mathbb{C}}}} }(C, C') \subseteq \operatorname{Mor}_{{\mathsf{Sch}}_{/ {{\mathbb{C}}}} }(C, C') .\] ::: ::: {.example title="Equivalence of groupoids"} Groupoids are equivalent iff they are equivalent as categories. The following is an example of mapping the quotient groupoid \( [C_2/C_4] \) to \( {\mathsf{B}}C_2 \): ![](figures/2021-09-06_19-13-21.png) ::: ::: {.example title="Groupoids equivalent to sets"} If a groupoid \( {\mathfrak{X}} \) is equivalent to \( \mathsf{C}_{\Sigma} \) for any \( \Sigma \in {\mathsf{Set}} \), we say \( {\mathfrak{X}} \) is **equivalent to a set**. For example, the following groupoid is equivalent to a 2-element set: ![](figures/2021-09-06_19-15-23.png) ::: ::: {.example title="Quotient groupoids"} For \( G\curvearrowright\Sigma \) a group acting on any set, define the **quotient groupoid** \( [\Sigma/G] \) in the following way: - Objects: \( x\in \Sigma \), i.e. one object for each element of the set \( \Sigma \). - Morphisms: \( \operatorname{Mor}(x, x') = \left\{{g\in G {~\mathrel{\Big|}~}gx' = x}\right\} \). ::: ::: {.exercise title="Groupoids equivalent to sets"} Show that \( [\Sigma/G] \) is equivalent to a set iff \( G\curvearrowright\Sigma \) is a free action. ::: ::: {.example title="Classifying stacks"} For \( \Sigma = \left\{{{\operatorname{pt}}}\right\} \), we obtain \[ {\mathsf{B}}G \coloneqq[{\operatorname{pt}}/ G] ,\] where there is one object \( {\operatorname{pt}} \) and \( \operatorname{Mor}({\operatorname{pt}}, {\operatorname{pt}}) = G \). ::: ::: {.example title="from representation stability"} Define \( {\mathsf{FinSet}} \) to be the category of finite sets where the morphisms are set bijections. Then \( {\mathsf{FinSet}}= \displaystyle\coprod_{n\in {\mathbb{Z}}_{\geq 0}} {\mathsf{B}}S_n \) for \( S_n \) the symmetric group. ::: ::: {.definition title="Fiber products of groupoids"} For \( C, D' \to D \) morphisms of groupoids, we can construct their **fiber product** as the cartesian diagram: ```{=tex} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{C{ \underset{\scriptscriptstyle {D} }{\times} }D'} && {D'} \\ \\ C && D \arrow["f"', from=3-1, to=3-3] \arrow["{g}", from=1-3, to=3-3] \arrow["{{\operatorname{pr}}_1}"', from=1-1, to=3-1] \arrow["{{\operatorname{pr}}_2}", from=1-1, to=1-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJDIl0sWzIsMiwiRCJdLFsyLDAsIkQnIl0sWzAsMCwiQ1xcZmliZXJwcm9ke0R9RCciLFsyNDAsNjAsNjAsMV1dLFswLDEsImYiLDJdLFsyLDEsImYnIl0sWzMsMCwiXFxwcl8xIiwyXSxbMywyLCJcXHByXzIiXV0=) It can be constructed as the following category: \[ {\operatorname{Ob}}(C{ \underset{\scriptscriptstyle {D} }{\times} } D') \coloneqq \left\{ \begin{array}{l} (c, d', \alpha) \end{array} \middle\vert \begin{array}{l} c\in C, d'\in D', \\ \\ \alpha: f(c) \xrightarrow{\sim} g(d') \end{array} \right\} \] `\includegraphics{figures/BigDiagram1.pdf}`{=tex} ```{=html} ``` ::: ::: {.exercise title="Universal property of pullbacks in Groupoids"} Describe the universal property of the pullback in the 2-category of groupoids. ::: ::: {.example title="$G$ is a pullback of $\\B G$"} \( G \) regarded as a groupoid is the pullback over inclusions of points into \( {\mathsf{B}}G \): ```{=tex} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{G} && {\operatorname{pt}}\\ \\ {\operatorname{pt}}&& {{\mathsf{B}}G} \arrow[from=3-1, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=3-1] \arrow[from=1-3, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJHIixbMjQwLDYwLDYwLDFdXSxbMiwwLCJcXHB0Il0sWzAsMiwiXFxwdCJdLFsyLDIsIlxcQiBHIl0sWzIsM10sWzAsMiwiIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXX1dLFsxLDNdLFswLDEsIiIsMix7ImNvbG91ciI6WzI0MCw2MCw2MF19XSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=) ::: ::: {.example title="Orbit/Stabilizer"} Let \( G\curvearrowright\Sigma \) and \( x\in \Sigma \), and let \( Gx \) be the orbit and \( G_x \) be the stabilizer. Then there is a morphism of groupoids \( f \in \operatorname{Mor}({\mathsf{B}}G_x, [\Sigma/G]) \) inducing a pullback: ```{=tex} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{G_x} & {} & \Sigma \\ \\ {{\mathsf{B}}G_x} && {[\Sigma/G]} \\ {\operatorname{pt}}&& x \arrow["{\exists f}", from=3-1, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=3-1] \arrow[from=1-3, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, color={rgb,255:red,92;green,92;blue,214}, draw=none, from=1-1, to=3-3] \arrow[maps to, from=4-1, to=4-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNyxbMCwwLCJHX3giLFsyNDAsNjAsNjAsMV1dLFsxLDBdLFsyLDAsIlxcU2lnbWEiXSxbMCwyLCJcXEIgR194Il0sWzIsMiwiW1xcU2lnbWEvR10iXSxbMCwzLCJcXHB0Il0sWzIsMywieCJdLFszLDQsIlxcZXhpc3RzIGYiXSxbMCwzLCIiLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdfV0sWzIsNF0sWzAsMiwiIiwyLHsiY29sb3VyIjpbMjQwLDYwLDYwXX1dLFswLDQsIiIsMSx7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFs1LDYsIiIsmfx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=) ::: ## Prestacks ::: {.remark} Motivation: to specify a moduli functor, we'll need the data of - Families over \( S \), - How to pull back families under morphisms, and - *How* objects are isomorphic. As a first attempt, we might try to define a 2-functor \( F: {\mathsf{Sch}}\to {\mathsf{Grpd}} \) between 2-categories, where the latter is the category of groupoids. For this, we need the following data: - For all \( S\in {\mathsf{Sch}} \), an assignment of a groupoid \( F(S) \), - For all morphisms \( f\in \operatorname{Mor}_{{\mathsf{Sch}}}(S, T) \), an assignment of morphisms of groupoids \[ f^* \in \operatorname{Mor}_{{\mathsf{Grpd}}}(F(T), F(S)) .\] - For compositions of morphisms of schemes \( S \xrightarrow{f} T \xrightarrow{g} U \), an isomorphism of functors \[ \psi_{fg}: g^* \circ f^* \xrightarrow{\sim} (g \circ f)^* .\] - Compatibility of these isomorphisms on chains of compositions \( S \to T \to U \to V \to \cdots \). [^1] This is a lot of data to track, so instead we'll construct a large category \( {\mathfrak{X}} \) that encodes all of this, along with a fibration ```{=tex} \begin{tikzcd} {\mathfrak{X}}\coloneqq\displaystyle\coprod_{S\in {\mathsf{Sch}}} F(S) \ar[d, "p"] & (S, \alpha \in F(S)) \ar[d, maps to] \\ {\mathsf{Sch}}& S \end{tikzcd} ``` Here \( S \in {\mathsf{Sch}} \) and \( F(S) \in {\mathsf{Grpd}} \), so the "fibers" above \( S \) are groupoids. ::: ::: {.definition title="Prestack"} Let \( p:{\mathfrak{X}}\to \mathsf{C} \) be a functor between two 1-categories, so we have the following data: ```{=tex} \begin{tikzcd} {\mathfrak{X}}&& a && b & {\in {\operatorname{Ob}}({\mathfrak{X}})} \\ \\ \mathsf{C} && S && T & {\in {\operatorname{Ob}}(\mathsf{C})} \arrow["f", from=3-3, to=3-5] \arrow["p"', from=1-1, to=3-1] \arrow[maps to, from=1-3, to=3-3] \arrow[maps to, from=1-5, to=3-5] \arrow["\alpha", from=1-3, to=1-5] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCJcXG1meCJdLFswLDIsIlMiXSxbMiwwLCJhIl0sWzQsMCwiYiJdLFsyLDIsIlMiXSxbNCwyLCJUIl0sWzUsMCwiXFxpbiBcXE9iKFxcbWZ4KSJdLFs1LDIsIlxcaW4gXFxPYihTKSJdLFs0LDUsImYiXSxbMCwxLCJwIiwyXSxbMiw0LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzMsNSwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFsyLDNdXQ==) Then \( {\mathfrak{X}}, p \) define a **prestack** over \( \mathsf{C} \) iff - Pullbacks exist: for \( S \xrightarrow{f} T \), there exists a (not necessarily unique) map \( f^*b \), sometimes denoted \( { \left.{{b}} \right|_{{f}} } \), yielding a cartesian square: ```{=tex} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{\exists a} && b \\ \\ S && T \arrow[from=3-1, to=3-3] \arrow[from=1-3, to=3-3] \arrow["{f^* b = { \left.{{b}} \right|_{{f}} }}", color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=1-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=3-1] \arrow["\lrcorner"{anchor=center, pos=0.125}, color={rgb,255:red,92;green,92;blue,214}, draw=none, from=1-1, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXGV4aXN0cyBhIixbMjQwLDYwLDYwLDFdXSxbMiwwLCJiIl0sWzAsMiwiUyJdLFsyLDIsIlQiXSxbMiwzXSxbMSwzXSxbMCwxLCJmXiogYiA9IFxccm97Yn17Zn0iLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19LFsyNDAsNjAsNjAsMV1dLFswLDIsIiIsMCx7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDMsIiIsMSx7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==) - A universal property making \( {\mathfrak{X}} \) a *fibered category*: every arrow in \( {\mathfrak{X}} \) is a pullback, so there are always lifts of the following form: ```{=tex} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{a} && b && c \\ \\ R && S && R \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[maps to, from=1-3, to=3-3] \arrow[maps to, from=1-5, to=3-5] \arrow[from=1-3, to=1-5] \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, maps to, from=1-1, to=3-1] \arrow["{\exists !}", color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=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) ::: ::: {.slogan} An alternative definition: a prestack is a category *fibered in groupoids*. ::: ::: {.warnings} We often conflate \( {\mathfrak{X}} \) and the functor \( {\mathfrak{X}}\xrightarrow{p} S \), and don't spell out the composition law in \( {\mathfrak{X}} \). Moreover, we write \( f^*b \) or \( { \left.{{b}} \right|_{{f}} } \) for a *choice* of a pullback. ::: ::: {.definition title="Fiber Categories"} For \( p: {\mathfrak{X}}\to \mathsf{C} \) a functor and \( S\in {\operatorname{Ob}}(\mathsf{C}) \) any fixed object, the associated **fiber category over \( S \)**, denoted \( {\mathfrak{X}}(S) \), is the subcategory of \( {\mathfrak{X}} \) defined by: - Objects: \( a\in {\operatorname{Ob}}({\mathfrak{X}}) \) such that \( a \xrightarrow{p} S \), - Morphisms: \( \operatorname{Mor}(a, a') \) are morphisms \( f\in \operatorname{Mor}_{{\mathfrak{X}}}(a, a') \) over \( \operatorname{id}_S \): ```{=tex} \begin{tikzcd} a \ar[rd, ""] \ar[rr, "f"] & & a' \ar[ld, ""] \\ & S & \end{tikzcd} ``` ::: ::: {.remark} We can now equivalently define presheaves as categories fibered in sets. ::: ::: {.exercise title="Justifying 'category fibered in groupoids'"} Show that if \( {\mathfrak{X}}\to \mathsf{C} \) is a prestack, then for all \( S\in \mathsf{C} \), all maps in \( {\mathfrak{X}}(S) \) are invertible. Conclude that the fiber categories \( {\mathfrak{X}}(S) \) are all groupoids. ::: ::: {.example title="Presheaves"} Every presheaf forms a prestack. Let \( F \in \underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{Sch}}, {\mathsf{Set}}) \) be a presheaf of sets, and define \( {\mathfrak{X}}_F \) as the following category: - Objects: Pairs \( (S, a \in F(S)) \) where \( S\in {\mathsf{Sch}} \) and \( F(s) \in {\mathsf{Set}} \). - Morphisms: \[ \operatorname{Mor}( (S, a), (T, b) ) \coloneqq\left\{{ S \xrightarrow{f} T {~\mathrel{\Big|}~}a = f^* b}\right\} .\] Note that we'll often conflate \( F \) and \( {\mathfrak{X}}_F \). This yields the fibration ```{=tex} \begin{tikzcd} {{\mathfrak{X}}_F} && {(S, a)} \\ \\ {\mathsf{Sch}}&& S \arrow[from=1-1, to=3-1, "p"] \arrow[maps to, from=1-3, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jeF9GIl0sWzAsMiwiXFxTY2giXSxbMiwwLCIoUywgYSkiXSxbMiwyLCJTIl0sWzAsMV0sWzIsMywiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==) ::: ::: {.example title="Schemes"} For \( X\in {\mathsf{Sch}} \), take its Yoneda functor \( h_X: {\mathsf{Sch}}\to {\mathsf{Set}} \). Then define the category \( {\mathfrak{X}}_X \): - Objects: Morphisms \( S\to X \) of schemes. - Morphisms: \( \operatorname{Mor}(S\to X, T\to X) \) are morphisms over \( X \): ```{=tex} \begin{tikzcd} S \ar[rd, ""] \ar[rr, ""] & & T \ar[ld, ""] \\ & X & \end{tikzcd} ``` This yields the fibration ```{=tex} \begin{tikzcd} {{\mathfrak{X}}_X} && {(S\to X)} \\ \\ {\mathsf{Sch}}&& S \arrow[from=1-1, to=3-1, "p"] \arrow[maps to, from=1-3, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jeF9GIl0sWzAsMiwiXFxTY2giXSxbMiwwLCIoUywgYSkiXSxbMiwyLCJTIl0sWzAsMV0sWzIsMywiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==) ::: ::: {.example title="Moduli of curves"} Define \( { \mathcal{M}_{g} } \) as the following category: - Objects: families \( {\mathcal{C}}\to S \) of smooth genus \( g \) curves, - Morphisms: \( \operatorname{Mor}({\mathcal{C}}\to S, {\mathcal{C}}'\to S') \): cartesian squares ```{=tex} \begin{tikzcd} {\mathcal{C}}&& {{\mathcal{C}}'} \\ \\ S && S' \arrow[from=3-1, to=3-3] \arrow[from=1-1, to=3-1] \arrow[from=1-3, to=3-3] \arrow[from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJDIl0sWzAsMiwiUyJdLFsyLDIsIlMiXSxbMiwwLCJDJyJdLFsxLDIsIlxcaWRfUyIsMl0sWzAsMV0sWzMsMl0sWzAsM10sWzAsMiwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d) This yields a fibration ```{=tex} \begin{tikzcd} {{ \mathcal{M}_{g} }} && {({\mathcal{C}}\to S)} \\ \\ {\mathsf{Sch}}&& S \arrow[from=1-1, to=3-1] \arrow[maps to, from=1-3, to=3-3] \end{tikzcd} ``` ::: ::: {.example title="Bundles"} For \( C \) a smooth connected projective curve over \( k \) a field, define \( {\mathsf{Bun}}(C) \) as the following category: - Objects: pairs \( (S, F) \) where \( F \) is a vector bundle over \( C\times S \). - Morphisms: \[ \operatorname{Mor}((S, F), (S', F')) = \left\{ \begin{array}{l} f\in \operatorname{Mor}_{{\mathsf{Sch}}}(S, S') \\ \text{and a chosen isomorphism} \\ \alpha: (f\times \operatorname{id})^* \circ F' \xrightarrow{\sim} F \end{array} \right\} .\] ::: {.remark} A technical point: the choice of pushforward here is not necessarily canonical. However, as part of the data, one can take morphisms \( F' \to (f\times\operatorname{id})_* \circ F \) such that the adjunction yields an isomorphism. ::: ::: ::: {.example title="Quotient prestack"} Let \( X_{/ {S}} \in {\mathsf{Grp}}{\mathsf{Sch}} \) where \( G\curvearrowright X \). Then define a category \( [X/G]^{\mathsf{pre}} \): - Objects: Morphisms over \( \operatorname{id}_S \): ```{=tex} \begin{tikzcd} T \ar[rd, ""] \ar[rr, ""] & & X \ar[ld, ""] \\ & S & \end{tikzcd} ``` - Morphisms: \[ \operatorname{Mor}(T\to X, T'\to X) \coloneqq \left\{ \begin{array}{l} T\to T' \end{array} \,\, \middle\vert \begin{array}{l} (T \to T' \to X ) = g(T \to X) \\ g\in G(T) \\ G(T) \curvearrowright X(T) \end{array} \right\} .\] ::: ::: {.remark} A group scheme can alternatively be thought of as a functor with a factorization through \( {\mathsf{Grp}} \). ::: ::: {.exercise title="Quotient prestacks and quotient groupoids"} Show that for \( T\in {\mathsf{Sch}} \), there is an equivalence \[ [X/G]^{\mathsf{pre}}(T) \xrightarrow{\sim} [X(T) / G(T)] ,\] where the left-hand side is a fibered category over \( T \) and the right-hand side is a quotient groupoid. ::: ### Morphisms of Prestacks ::: {.definition title="Morphisms of prestacks"} A **morphism of prestacks** is a functor \( {\mathfrak{X}}\xrightarrow{f} {\mathfrak{X}}' \) such that there is a (strictly) commutative triangle ```{=tex} \begin{tikzcd} {\mathfrak{X}}&& {\mathfrak{X}}' \\ \\ & \mathsf{C} \arrow["f", from=1-1, to=1-3] \arrow["{p_X}"', from=1-1, to=3-2] \arrow["{p_Y}", from=1-3, to=3-2] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXG1jeCJdLFsyLDAsIlxcbWN5Il0sWzEsMiwiXFxTY2giXSxbMCwxLCJmIl0sWzAsMiwicF9YIiwyXSxbMSwyLCJwX1kiXV0=) Here we require a strict equality \( p_X(a) = p_Y(f(a)) \) for any \( a\in {\mathfrak{X}} \) A **2-morphism** \( \alpha \) between morphisms \( f, g \) is a natural transformation: ```{=tex} \begin{tikzcd} {\mathfrak{X}}&&& {\mathfrak{X}}' \arrow[""{name=0, anchor=center, inner sep=0}, "f", curve={height=-30pt}, from=1-1, to=1-4] \arrow[""{name=1, anchor=center, inner sep=0}, "g"', curve={height=30pt}, from=1-1, to=1-4] \arrow["\alpha", shorten <=8pt, shorten >=8pt, Rightarrow, from=0, to=1] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsMixbMCwwLCJcXG1meCJdLFszLDAsIlxcbWZ5Il0sWzAsMSwiZiIsMCx7ImN1cnZlIjotNX1dLFswLDEsImciLDIseyJjdXJ2ZSI6NX1dLFsyLDMsIlxcYWxwaGEiLDAseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9fV1d) such that for all \( a\in {\mathfrak{X}} \), the following triangle \( \alpha_a\in \operatorname{Mor}_{{\mathfrak{X}}'}(f(a), g(a)) \) is a morphisms over \( \operatorname{id}_S \) for any \( S\in \mathsf{C} \): ```{=tex} \begin{tikzcd} f(a) \ar[rd, ""] \ar[rr, ""] & & g(a) \ar[ld, ""] \\ & S & \end{tikzcd} ``` We define a category \( \operatorname{Mor}({\mathfrak{X}}, {\mathfrak{X}}') \) by: - Objects: morphisms of prestacks. - Morphisms: 2-morphisms of prestacks. ::: ::: {.exercise title="?"} Show that \( \operatorname{Mor}({\mathfrak{X}}, {\mathfrak{X}}') \) is a groupoid. ::: ::: {.definition title="2-commutativity"} A diagram is **2-commutative** iff there exists a 2-morphism \( \alpha: g \circ f' \xrightarrow{\sim} f\circ g' \) which is an isomorphism: ```{=tex} \begin{tikzcd} {{\mathfrak{X}}{ \underset{\scriptscriptstyle {{\mathfrak{X}}'} }{\times} } {\mathfrak{X}}''} && {{\mathfrak{X}}''} \\ \\ {\mathfrak{X}}&& {\mathfrak{X}}' \arrow["g", from=1-3, to=3-3] \arrow["f"', from=3-1, to=3-3] \arrow["{g'}"', from=1-1, to=3-1] \arrow["{f'}", from=1-1, to=1-3] \arrow["\alpha", Rightarrow, from=3-1, to=1-3] \end{tikzcd} ``` > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1meCBcXGZpYmVycHJvZHtcXG1meX0gXFxtZnknIl0sWzIsMCwiXFxtZnknIl0sWzIsMiwiXFxtZnkiXSxbMCwyLCJcXG1meCJdLFsxLDIsImciXSxbMywyLCJmIiwyXSxbMCwzLCJnJyIsMl0sWzAsMSwiZiciXSxbMywxLCJcXGFscGhhIiwwLHsibGV2ZWwiOjJ9XV0=) ::: ::: {.definition title="Isomorphisms of prestacks"} An **isomorphism** of prestacks is a 1-isomorphism of prestacks \( f: {\mathfrak{X}}\to {\mathfrak{X}}' \) along with 2-isomorphisms \( g\circ f \xrightarrow{\sim} \operatorname{id}_{{\mathfrak{X}}} \) and \( f\circ g \xrightarrow{\sim} \operatorname{id}_{{\mathfrak{X}}'} \). ::: ::: {.exercise title="Isomorphisms of prestacks can be checked on fibers"} Show that \( {\mathfrak{X}}\to {\mathfrak{X}}' \) is an isomorphism iff \( {\mathfrak{X}}(S) \xrightarrow{\sim} {\mathfrak{X}}'(S) \) is an isomorphism on all fibers. ::: ::: {.proposition title="2-Yoneda"} If \( {\mathfrak{X}}\in {\underset{ \mathsf{pre} } {\mathsf{St} } } {}_{/ {\mathsf{C}}} \) is a prestack over \( \mathsf{C} \), then for any \( S\in {\operatorname{Ob}}(\mathsf{C}) \), there is an equivalence of categories induced by the following functor: \[ \operatorname{Mor}(S, {\mathfrak{X}}) & \xrightarrow{\sim} {\mathfrak{X}}(S) \\ f &\mapsto f_S(\operatorname{id}_S ) .\] ::: ::: {.remark} For \( S\in {\mathsf{Sch}} \), view \( S \) as a prestack and consider a morphism \( f:S\to {\mathfrak{X}} \). How is this specified? For all \( T\in {\mathsf{Sch}} \), the objects of \( S_{/ {T}} \) are morphisms \[ f_T: \operatorname{Mor}(T, S) \to {\mathfrak{X}}(T) \] and if \( T=S \) this sends \( \operatorname{id}_S \) to \( f_S(\operatorname{id}_S)\in {\mathfrak{X}}(S) \). What is the inverse? For \( a\in {\mathfrak{X}}(S) \) and for each \( T \xrightarrow{g} S \), **choose** a pullback \( g^* a \). Then define \( f: S \to {\mathfrak{X}} \) by \[ f_T: \operatorname{Mor}(T, S) &\to {\mathfrak{X}}(T) \\ g &\mapsto g^* a .\] ::: ::: {.exercise title="?"} Define what this equivalence should do on morphisms. ::: ::: {.remark} Next time: fiber products of prestacks. ::: [^1]: This leads to the notion of **lax** or **pseudofunctors**.