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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 15th 2013
    • (edited May 3rd 2014)

    Some comments on the implementation of Atiyah- and Courant Lie n-algebroids in cohesive homotopy type theory, and some general thoughts on “0-strict” \infty-groupoids (\infty-groupoids equipped with a 1-epimorphism out of a 0-truncated object).

    Let H\mathbf{H} be some cohesive (infinity,1)-topos.

    The homotopy Lie-Rinehart pair-perspective on an \infty-groupoid 𝒳H\mathcal{X} \in \mathbf{H} which is equipped with a 1-epimorphism ϕ:X𝒳\phi \colon X \to \mathcal{X} is to consider the pair consisting of XX and of the infinity-group of bisections

    BiSect(𝒳,X)Aut 𝒳(X)𝒳Equiv /𝒳(X) . BiSect(\mathcal{X},X) \coloneqq \mathbf{Aut}_{\mathcal{X}}(X) \coloneqq \underset{\mathcal{X}}{\prod} Equiv_{/\mathcal{X}}(X)_ \,.

    (Traditionally in a (homtopy-)Lie-Rinehart algebra of course one only remembers the L-infinity algebra Lie(Aut 𝒳(X))Lie( \mathbf{Aut}_{\mathcal{X}}(X) ) obtained from this under Lie differentiation, but here I will stick to the complete Lie integrated picture.)

    Now, it turns out that famous homotopy Lie-Rinehart algebras out there are constructed (secretly, but one can see that this is what happens) by starting with a map

    χ:XF \chi \colon X \to \mathbf{F}

    to some moduli infinity-stack F\mathbf{F} and then taking the group of bisections to be the automorphism group of this χ\chi over F\mathbf{F}.

    For instance

    • the Atiyah Lie algebroid assigned to a circle principal bundle modulated by 0:XBU(1)\nabla^0 \colon X \to \mathbf{B}U(1) is the Lie differentiaton of (X,Aut BU(1)( 0))(X, \mathbf{Aut}_{\mathbf{B}U(1)}(\nabla^0));

    • the Courant Lie 2-algebroid assigned to a map 1:XB 2U(1) conn 1\nabla^1 \colon X \to \mathbf{B}^2 U(1)_{conn^1} modulating a “bundle gerbe with connective data but no curving” is the Lie differentiation of (X,concAut B 2U(1) conn 1( 1))(X, conc \mathbf{Aut}_{\mathbf{B}^2 U(1)_{conn^1}}(\nabla^1)).

      (here concconc stands for “differential concretification”, a technical subtlety related to the right cohesive structure on these objects, which for the purpose of the present discussion one should ignore)

    Now given such an “integrated homotopy Lie-Rinehart pair” consisting of an object XX and an automorphism \infty-group of a map χ:XF\chi : X \to \mathbf{F} over F\mathbf{F}, can we canonically find for it the corresponding \infty-groupoid, hence the 𝒳H\mathcal{X} \in \mathbf{H} such that

    1. X𝒳X \to \mathcal{X} is a 1-epimorphism;

    2. Aut F(χ)Aut 𝒳(X)\mathbf{Aut}_{\mathbf{F}}(\chi) \simeq \mathbf{Aut}_{\mathcal{X}}(X)


    Well, that’s just the 1-image of χ\chi:

    𝒳im 1(χ) \mathcal{X} \simeq im_1(\chi)

    isn’t it?

    Let’s look at the first example in the above series:

    Let H=\mathbf{H} = SmoothGrpd, let XSmthMfdHX \in SmthMfd \hookrightarrow \mathbf{H} be a smooth manifold, and let 0:XBU(1)\nabla^0 \colon X \to \mathbf{B}U(1) be the map modulating a circle principal bundle PXP \to X.

    Then what is

    X𝒳im 1( 0). X \to \mathcal{X}\coloneqq im_1(\nabla^0) \,.

    To check this, remember that the 1-image of a map may be computed as the homotopy colimit over its homotopy Cech nerve. Doing so here, we find that

    im 1( 0)(P× U(1)PX). im_1(\nabla^0) \simeq ( P \times_{U(1)} P \stackrel{\to}{\to} X) \,.

    This is the Lie integration of the Atiyah Lie algebroid of the circle bundle PXP \to X modulated by 0\nabla^0.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2013
    • (edited Feb 20th 2013)

    I have now started to write out a bunch of details along the above lines in a new entry

    (Not proof-read yet, as it is getting too late for me now. Will fix typos and furher refine tomorrow…)