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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 20th 2012
    • (edited Nov 20th 2012)

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

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

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

    where 1[X,Ω 1]:=image([X,Ω 1][X,Ω 1])\sharp_1 [X,\Omega^1] := image( [X, \Omega^1] \to \sharp [X, \Omega^1] ) is the concretification of [X,Ω 1][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 BU(1) connH\mathbf{B}U(1)_{conn} \in \mathbf{H} for the stack of circle-principal connections. Then for XX a manifold, one might be inclined to say that the mapping stack [X,BU(1) conn][X, \mathbf{B}U(1)_{conn}] is the moduli stack of circle-principal connections on XX. But again it is not quite right: a UU-plot of [X,BU(1) conn][X,\mathbf{B}U(1)_{conn}] is a circle-principal connection on U×XU \times X, but it should be one with no form components along UU, so that we can interpret it as a smoothly UU-parameterized set of connections on XX.

    The previous example might make one think that this is again fixed by considering 1[X,BU(1) conn]\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 1[X,BU(1) conn]\sharp_1 [X, \mathbf{B}U(1)_{conn}] has as UU-plots the groupoid whose objects are smoothly UU-parameterized sets of connections on XX – that’s as it should be – , but whose morphisms are Γ(U)\Gamma(U)-parameterized sets of gauge transformations between these, where Γ(U)\Gamma(U) is the underlying discrete set of the test manifold UU – and that’s of course not how it should be. The reflection 1\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)Conn(X)U(1)\mathbf{Conn}(X) of circle principal connections on some XX is the homotopy pullback in

    U(1)Conn(X) [X,BU(1)] 1[X,BU(1) conn] 1[X,BU(1)] \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 BU(1) connBU(1)\mathbf{B}U(1)_{conn} \to \mathbf{B}U(1) from circle-principal connections to their underlying circle-principal bundles.

    Here the 1\sharp_1 in the bottom takes care of making the 0-cells come out right, whereas the pullback restricts among those dustified Γ(U)\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,BU(1)] 2[X,BU(1)] [X, \mathbf{B}U(1)] \simeq \sharp_2 [X, \mathbf{B}U(1)]

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

    id= 2 1 0=. id = \sharp_\infty \to \cdots \to \sharp_2 \to \sharp_1 \to \sharp_0 = \sharp \,.

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

    BU(1) conn =DK(U(1)Ω 1) BU(1) =DK(U(1)0). \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

    B 2U(1) conn 1=B 2U(1) conn =DK(U(1)Ω 1Ω 2) BU(1) conn 2 =DK(U(1)Ω 10) BU(1) conn 3=B 2U(1) =DK(U(1)00) \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 (BU(1))Conn(X)(\mathbf{B}U(1))\mathbf{Conn}(X) of circle-principal 2-connections on some XX is the homotopy limit in

    (BU(1))Conn(X) [X,B 2U(1)] 2[X,B 2U(1) conn 2] 2[X,B 2U(1)] 1[X,B 2U(1) conn] 1[X,B 2U(1) conn 2]. \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,B 2U(1)][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.