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It’s time to become serious about “higher order” aspects of applications of the the “sharp-modality” ♯ in a cohesive (infinity,1)-topos H – I am thinking of the construction of moduli ∞-stacks for differential cocycles.
Consider, as usual, the running example H=Sh∞(CartSp)= Smooth∞Grpd.
Here is the baby example, which below I discuss how to refine:
there is an object called Ω1∈H, which is just the good old sheaf of differential 1-forms. Consider also a smooth manifold X∈H. On first thought one might want to say that the internal hom object [X,Ω1] is the “moduli 0-stack of differential 1-forms on X”. But that’s not quite right. For U∈ 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,Ω1] contain a bit more stuff. They are of course 1-forms on U×X and the actual families that we want to see are only those 1-forms on U×X which have “no leg along U”. But one sees easily that the correct moduli stack of 1-forms on X is
Ω1(X):=♯1[X,Ω1]↪♯[X,Ω1],where ♯1[X,Ω1]:=image([X,Ω1]→♯[X,Ω1]) is the concretification of [X,Ω1].
This above kind of issue persists as we refine differential 1-forms to circle-principal connections: write BU(1)conn∈H for the stack of circle-principal connections. Then for X a manifold, one might be inclined to say that the mapping stack [X,BU(1)conn] is the moduli stack of circle-principal connections on X. But again it is not quite right: a U-plot of [X,BU(1)conn] is a circle-principal connection on U×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 ♯1[X,BU(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] 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 Γ(U)-parameterized sets of gauge transformations between these, where Γ(U) is the underlying discrete set of the test manifold U – and that’s of course not how it should be. The reflection ♯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) of circle principal connections on some X is the homotopy pullback in
U(1)Conn(X)→[X,BU(1)]↓↓♯1[X,BU(1)conn]→♯1[X,BU(1)]where the bottom morphism is induced from the canonical map BU(1)conn→BU(1) from circle-principal connections to their underlying circle-principal bundles.
Here the ♯1 in the bottom takes care of making the 0-cells come out right, whereas the pullback restricts among those dustified Γ(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,BU(1)]≃♯2[X,BU(1)]where ♯2 is the 2-image of id→♯, hence the factorization by a 0-connected morphism followed by a 0-truncated one. For the 1-truncated object [X,BU(1)] the 2-image doesn’t change anything. Generally we have a tower
id=♯∞→⋯→♯2→♯1→♯0=♯.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
BU(1)conn=DK(U(1)→Ω1)BU(1)=DK(U(1)→0).If we pass to circle-principal 2-connections, this becomes
B2U(1)conn1=B2U(1)conn=DK(U(1)→Ω1→Ω2)BU(1)conn2=DK(U(1)→Ω1→0)BU(1)conn3=B2U(1)=DK(U(1)→0→0)and so on.
And a little reflection show that the correct moduli 2-stack (BU(1))Conn(X) of circle-principal 2-connections on some X is the homotopy limit in
(BU(1))Conn(X)→→[X,B2U(1)]↓♯2[X,B2U(1)conn2]→♯2[X,B2U(1)]↓↓♯1[X,B2U(1)conn]→♯1[X,B2U(1)conn2].This is a “3-stage ♯-reflection” of sorts, which fixes the naive moduli 2-stack [X,B2U(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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