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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.
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 $[X,\Omega^1]$.
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.
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.
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