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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 20th 2012
   • (edited Nov 20th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt's time to become serious about "higher order" aspects of applications of the the "sharp-modality" $\sharp$ in a [[cohesive (infinity,1)-topos]] $\mathbf{H}$ -- I am thinking of the construction of moduli $\infty$-stacks for differential cocycles. Consider, as usual, the running example $\mathbf{H} = Sh_\infty(CartSp) = $ [[Smooth∞Grpd]]. ## Simple motivating example: moduli of differential forms Here is the baby example, which below I discuss how to refine: there is an object called $\Omega^1 \in \mathbf{H}$, which is just the good old sheaf of differential 1-forms. Consider also a smooth manifold $X \in \mathbf{H}$. On first thought one might want to say that the internal hom object $[X, \Omega^1]$ is the "moduli 0-stack of differential 1-forms on $X$". But that's not quite right. For $U \in $ [[CartSp]], the $U$-plots of the latter should be smoothly $U$-parameterized sets of differential 1-forms on $X$, but the $U$-plots of $[X,\Omega^1]$ contain a bit more stuff. They are of course 1-forms on $U \times X$ and the actual families that we want to see are only those 1-forms on $U \times X$ which have "no leg along $U$". But one sees easily that the correct moduli stack of 1-forms on $X$ is $$ \mathbf{\Omega}^1(X) := \sharp_1 [X,\Omega^1] \hookrightarrow \sharp [X, \Omega^1] \,, $$ where $\sharp_1 [X,\Omega^1] := image( [X, \Omega^1] \to \sharp [X, \Omega^1] )$ is the _[[concrete object|concretification]]_ of $[X,\Omega^1]$. ## Next easy example: moduli of connections This above kind of issue persists as we refine differential 1-forms to circle-principal connections: write $\mathbf{B}U(1)_{conn} \in \mathbf{H}$ for the stack of circle-principal connections. Then for $X$ a manifold, one might be inclined to say that the mapping stack $[X, \mathbf{B}U(1)_{conn}]$ is the moduli stack of circle-principal connections on $X$. But again it is not quite right: a $U$-plot of $[X,\mathbf{B}U(1)_{conn}]$ is a circle-principal connection on $U \times X$, but it should be one with no form components along $U$, so that we can interpret it as a smoothly $U$-parameterized set of connections on $X$. The previous example might make one think that this is again fixed by considering $\sharp_1 [X, \mathbf{B}U(1)_{conn}]$. But now that we have a genuine 1-stack and not a 0-stack anymore, this is not good enough: the stack $\sharp_1 [X, \mathbf{B}U(1)_{conn}]$ has as $U$-plots the groupoid whose objects are smoothly $U$-parameterized sets of connections on $X$ -- that's as it should be -- , but whose morphisms are $\Gamma(U)$-parameterized sets of gauge transformations between these, where $\Gamma(U)$ is the underlying discrete set of the test manifold $U$ -- and that's of course not how it should be. The reflection $\sharp_1$ fixes the moduli in degree 0 correctly, but it "dustifies" their automorphisms in degree 1. We can correct this as follows: the correct moduli stack $U(1)\mathbf{Conn}(X)$ of circle principal connections on some $X$ is the homotopy pullback in $$ \array{ U(1)\mathbf{Conn}(X) &\to& [X, \mathbf{B} U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}U(1)_{conn}] &\to& \sharp_1 [X, \mathbf{B} U(1)] } $$ where the bottom morphism is induced from the canonical map $\mathbf{B}U(1)_{conn} \to \mathbf{B}U(1)$ from circle-principal connections to their underlying circle-principal bundles. Here the $\sharp_1$ in the bottom takes care of making the 0-cells come out right, whereas the pullback restricts among those dustified $\Gamma(U)$-parameterized sets of gauge transformations to those that actually do have a smooth parameterization. ## More serious example: moduli of 2-connections The previous example is controlled by a hidden pattern, which we can bring out by noticing that $$ [X, \mathbf{B}U(1)] \simeq \sharp_2 [X, \mathbf{B}U(1)] $$ where $\sharp_2$ is the 2-image of $id \to \sharp$, hence the factorization by a 0-connected morphism followed by a 0-truncated one. For the 1-truncated object $[X, \mathbf{B}U(1)]$ the 2-image doesn't change anything. Generally we have a tower $$ id = \sharp_\infty \to \cdots \to \sharp_2 \to \sharp_1 \to \sharp_0 = \sharp \,. $$ Moreover, if we write $DK$ for the Dold-Kan map from sheaves of chain complexes to sheaves of groupoids (and let stackification be implicit), then $$ \begin{aligned} \mathbf{B}U(1)_{conn} &= DK( U(1) \to \Omega^1 ) \\ \mathbf{B}U(1) &= DK( U(1) \to 0 ) \end{aligned} \,. $$ If we pass to circle-principal 2-connections, this becomes $$ \begin{aligned} \mathbf{B}^2 U(1)_{conn^1} = \mathbf{B}^2U(1)_{conn} &= DK( U(1) \to \Omega^1 \to \Omega^2 ) \\ \mathbf{B}U(1)_{conn^2} & = DK( U(1) \to \Omega^1 \to 0 ) \\ \mathbf{B}U(1)_{conn^3} = \mathbf{B}^2 U(1) & = DK( U(1) \to 0 \to 0 ) \end{aligned} $$ and so on. And a little reflection show that the correct moduli 2-stack $(\mathbf{B}U(1))\mathbf{Conn}(X)$ of circle-principal 2-connections on some $X$ is the homotopy limit in $$ \array{ (\mathbf{B}U(1))\mathbf{Conn}(X) &\to& &\to& [X, \mathbf{B}^2 U(1)] \\ && && \downarrow \\ && \sharp_2 [X, \mathbf{B}^2 U(1)_{conn^2}] &\to& \sharp_2 [X, \mathbf{B}^2 U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}^2 U(1)_{conn}] &\to& \sharp_1 [X, \mathbf{B}^2 U(1)_{conn^2}] } \,. $$ This is a "3-stage $\sharp$-reflection" of sorts, which fixes the naive moduli 2-stack $[X, \mathbf{B}^2 U(1)]$ first in degree 0 (thereby first completely messing it up in the higher degrees), then fixes it in degree 1, then in degree 2. Then we are done.

   It’s time to become serious about “higher order” aspects of applications of the the “sharp-modality” $\sharp$ in a cohesive (infinity,1)-topos $\mathbf{H}$ – I am thinking of the construction of moduli $\infty$-stacks for differential cocycles.

   Consider, as usual, the running example $\mathbf{H} = Sh_\infty(CartSp) =$ Smooth∞Grpd.

   Simple motivating example: moduli of differential forms

   Here is the baby example, which below I discuss how to refine:

   there is an object called $\Omega^1 \in \mathbf{H}$, which is just the good old sheaf of differential 1-forms. Consider also a smooth manifold $X \in \mathbf{H}$. On first thought one might want to say that the internal hom object $[X, \Omega^1]$ is the “moduli 0-stack of differential 1-forms on $X$”. But that’s not quite right. For $U \in$ CartSp, the $U$-plots of the latter should be smoothly $U$-parameterized sets of differential 1-forms on $X$, but the $U$-plots of $[X,\Omega^1]$ contain a bit more stuff. They are of course 1-forms on $U \times X$ and the actual families that we want to see are only those 1-forms on $U \times X$ which have “no leg along $U$”. But one sees easily that the correct moduli stack of 1-forms on $X$ is

   $$\mathbf{\Omega}^1(X) := \sharp_1 [X,\Omega^1] \hookrightarrow \sharp [X, \Omega^1] \,,$$

   where $\sharp_1 [X,\Omega^1] := image( [X, \Omega^1] \to \sharp [X, \Omega^1] )$ is the concretification of $[X,\Omega^1]$.

   Next easy example: moduli of connections

   This above kind of issue persists as we refine differential 1-forms to circle-principal connections: write $\mathbf{B}U(1)_{conn} \in \mathbf{H}$ for the stack of circle-principal connections. Then for $X$ a manifold, one might be inclined to say that the mapping stack $[X, \mathbf{B}U(1)_{conn}]$ is the moduli stack of circle-principal connections on $X$. But again it is not quite right: a $U$-plot of $[X,\mathbf{B}U(1)_{conn}]$ is a circle-principal connection on $U \times X$, but it should be one with no form components along $U$, so that we can interpret it as a smoothly $U$-parameterized set of connections on $X$.

   The previous example might make one think that this is again fixed by considering $\sharp_1 [X, \mathbf{B}U(1)_{conn}]$. But now that we have a genuine 1-stack and not a 0-stack anymore, this is not good enough: the stack $\sharp_1 [X, \mathbf{B}U(1)_{conn}]$ has as $U$-plots the groupoid whose objects are smoothly $U$-parameterized sets of connections on $X$ – that’s as it should be – , but whose morphisms are $\Gamma(U)$-parameterized sets of gauge transformations between these, where $\Gamma(U)$ is the underlying discrete set of the test manifold $U$ – and that’s of course not how it should be. The reflection $\sharp_1$ fixes the moduli in degree 0 correctly, but it “dustifies” their automorphisms in degree 1.

   We can correct this as follows: the correct moduli stack $U(1)\mathbf{Conn}(X)$ of circle principal connections on some $X$ is the homotopy pullback in

   $$\array{ U(1)\mathbf{Conn}(X) &\to& [X, \mathbf{B} U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}U(1)_{conn}] &\to& \sharp_1 [X, \mathbf{B} U(1)] }$$

   where the bottom morphism is induced from the canonical map $\mathbf{B}U(1)_{conn} \to \mathbf{B}U(1)$ from circle-principal connections to their underlying circle-principal bundles.

   Here the $\sharp_1$ in the bottom takes care of making the 0-cells come out right, whereas the pullback restricts among those dustified $\Gamma(U)$-parameterized sets of gauge transformations to those that actually do have a smooth parameterization.

   More serious example: moduli of 2-connections

   The previous example is controlled by a hidden pattern, which we can bring out by noticing that

   $$[X, \mathbf{B}U(1)] \simeq \sharp_2 [X, \mathbf{B}U(1)]$$

   where $\sharp_2$ is the 2-image of $id \to \sharp$, hence the factorization by a 0-connected morphism followed by a 0-truncated one. For the 1-truncated object $[X, \mathbf{B}U(1)]$ the 2-image doesn’t change anything. Generally we have a tower

   $$id = \sharp_\infty \to \cdots \to \sharp_2 \to \sharp_1 \to \sharp_0 = \sharp \,.$$

   Moreover, if we write $DK$ for the Dold-Kan map from sheaves of chain complexes to sheaves of groupoids (and let stackification be implicit), then

   $$\begin{aligned} \mathbf{B}U(1)_{conn} &= DK( U(1) \to \Omega^1 ) \\ \mathbf{B}U(1) &= DK( U(1) \to 0 ) \end{aligned} \,.$$

   If we pass to circle-principal 2-connections, this becomes

   $$\begin{aligned} \mathbf{B}^2 U(1)_{conn^1} = \mathbf{B}^2U(1)_{conn} &= DK( U(1) \to \Omega^1 \to \Omega^2 ) \\ \mathbf{B}U(1)_{conn^2} & = DK( U(1) \to \Omega^1 \to 0 ) \\ \mathbf{B}U(1)_{conn^3} = \mathbf{B}^2 U(1) & = DK( U(1) \to 0 \to 0 ) \end{aligned}$$

   and so on.

   And a little reflection show that the correct moduli 2-stack $(\mathbf{B}U(1))\mathbf{Conn}(X)$ of circle-principal 2-connections on some $X$ is the homotopy limit in

   $$\array{ (\mathbf{B}U(1))\mathbf{Conn}(X) &\to& &\to& [X, \mathbf{B}^2 U(1)] \\ && && \downarrow \\ && \sharp_2 [X, \mathbf{B}^2 U(1)_{conn^2}] &\to& \sharp_2 [X, \mathbf{B}^2 U(1)] \\ \downarrow && \downarrow \\ \sharp_1 [X, \mathbf{B}^2 U(1)_{conn}] &\to& \sharp_1 [X, \mathbf{B}^2 U(1)_{conn^2}] } \,.$$

   This is a “3-stage $\sharp$-reflection” of sorts, which fixes the naive moduli 2-stack $[X, \mathbf{B}^2 U(1)]$ first in degree 0 (thereby first completely messing it up in the higher degrees), then fixes it in degree 1, then in degree 2. Then we are done.