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nLab > Latest Changes: inertia orbifold

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2021

    have brushed-up the Definition-section

    diff, v16, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2021

    added a remark (here) on the two different notions of free loop space objects for orbifolds, related by pullback along the shape-unit of the circle:

    Λ𝒳=[ʃS 1,𝒳]→[η S 1,𝒳][S 1,𝒳]=ℒ𝒳. \Lambda \mathcal{X} \;=\; \big[ ʃS^1, \, \mathcal{X} \big] \xrightarrow{ \; [\eta_{S^1},\mathcal{X}]\; } \big[ S^1, \, \mathcal{X} \big] \;=\; \mathcal{L} \mathcal{X} \,.

    diff, v16, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2021

    added statement (here) of the skeleton of the inertia orboifold of proper good orbifolds

    diff, v16, current

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 26th 2021

    Added reference:

    A Fréchet–Lie groupoid presenting the cohesive free loop space-orbifold is given in

    diff, v17, current

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 26th 2021

    We also claim in the announcement

    that the Lie groupoid presenting the resulting infinite-dimensional differentiable stack is even proper étale, hence an orbifold groupoid, though the full write-up of this is still on the back-burner.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2021

    How about making an entry loop orbifold or loop stack where this would fit?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 29th 2021

    It has become tradition to say that the inertia orbifold is the S 1S^1-fixed locus in a groupoid presentation of the free loop stack (e.g. Lupercio & Uribe 2001, Thm 3.6.4). However, it seems to me that this statement is not homotopy-meaningful:

    The identification of the smooth free loop stack [S 1,𝒳][S^1, \mathcal{X}] with the co-free S 1S^1-action induced by 𝒳\mathcal{X} (as seen in the diagram here) shows at once that its homotopy-fixed locus is just 𝒳\mathcal{X}:

    ([𝒮,𝒳]) 𝒮≃(p B𝒮) *∘(pt B𝒮) *(𝒳)≃(id) *(𝒳)≃𝒳, \Big( \big[ \mathcal{S}, \, \mathcal{X} \big] \Big)^{\mathcal{S}} \;\simeq\; (p_{\mathbf{B}\mathcal{S}})_\ast \circ (\mathrm{pt}_{\mathbf{B}\mathcal{S}})_\ast \big( \mathcal{X} \big) \;\simeq\; (\mathrm{id})_\ast \big( \mathcal{X} \big) \;\simeq\; \mathcal{X} \,,

    where

    id:*→pt B𝒮B𝒮→p B𝒮* id \;\colon\; \ast \xrightarrow{pt_{\mathbf{B}\mathcal{S}}} \mathbf{B}\mathcal{S} \xrightarrow{ p_{{}_{\mathbf{B}\mathcal{S}}} } \ast

    (here for 𝒮≔S 1\mathcal{S} \coloneqq S^1)

    It seems to me that in order to relate the inertia and the smooth loop stack in a homotopy-meaningful way, one needs to invoke cohesion.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 11th 2021

    I have made explicit (here) the following basic fact, which, for what it’s worth, becomes a triviality only through appeal to the path-notation for the Eilenberg-Zilber decomposition:

    The nerve of the inertia groupoid of a delooping groupoid of a finite group GG is isomorphic to the simplicial hom complex out of the minimal simplicial circle SS (Def. \ref{MinimalSimplicialCircle}) into the simplicial classifying space W¯G\overline{W}G:

    N(ΛBG) •≃[S,W¯G] •. N\big( \Lambda \mathbf{B}G\big)_\bullet \;\simeq\; [S,\overline{W}G]_\bullet \,.

    Under this isomorphism, an n-simplex (γ,g n−1,⋯,g 1,g 0)(\gamma, g_{n-1}, \cdots, g_1, g_0) of N(Func(Bℤ,BG))N\big( Func( \mathbf{B}\mathbb{Z}, \, \mathbf{B}G )\big), being a sequence of natural transformations of the form

    • →g n−1 • →g n−2 ⋯ →g n−j−1 • →g n−j−2 • →g n−j−3 ⋯ →g 0 • ↓ γ ↓ g n−1 −1⋅γ⋅g n−1 ↓ ↓ ↓ (g n−1⋯g 0) −1⋅γ⋅(g n−1⋯g 0) • →g n−1 • →g n−2 ⋯ →g n−j−1 • →g n−j−2 • →g n−j−3 ⋯ →g 0 • \array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet \\ \big\downarrow {}^{\mathrlap{ \gamma }} && \big\downarrow {}^{\mathrlap{ g_{n-1}^{-1} \cdot \gamma \cdot g_{n-1} }} && && \big\downarrow && \big\downarrow && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{0} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{0} ) }} \\ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet }

    is sent to the homomorphism

    Δ[n]×S→W¯G \Delta[n] \times S \xrightarrow{\;\;} \overline{W}G

    which, in turn, sends a non-degenerate (n+1)(n+1)-simplex in Δ[n]×S\Delta[n] \times S of the form (in the path notation discussed at product of simplices)

    (0,[0]) → (1,[0]) → ⋯ → (j,[0]) ↓ (j,[1]) → (j+1,[1]) → ⋯ → (n,[1]) \array{ (0,[0]) &\to& (1,[0]) &\to& \cdots &\to& (j,[0]) \\ && && && \big\downarrow \\ && && && (j,[1]) &\to& (j+1,[1]) &\to& \cdots &\to& (n,[1]) }

    to the n+1n+1-simplex in W¯G\overline{W}G of the form

    • →g n−1 • →g n−2 ⋯ →g n−j−1 • ↓ (g n−1⋯g n−j−1) −1⋅γ⋅(g n−1⋯g n−j−1) • →g n−j−2 • →g n−j−3 ⋯ →g 0 • \array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{n-j-1} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{n-j-1} ) }} \\ && && && \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet }

    diff, v23, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 12th 2021
    • (edited Jul 12th 2021)

    added (here) the consequence for the form of the simplicial evaluation map:

    The evaluation map

    [S,W¯G]×S→W¯G [S,\overline{W}G] \times S \xrightarrow{\;\;} \overline{W}G

    (out of the product of the simplicial hom complex out of SS with SS) takes any non-degenerate n+1n+1-simplex of [S,W¯G]×S[S,\overline{W}G] \times S of the form (still in the path notation discussed at product of simplices)

    (γ,[0]) →(g n−1,id) (Ad n−1(γ),[0]) →(g n−2,id) ⋯ →(g j,id) (Ad j(γ),[0]) ↓ (e,[[0],[1]]) (Ad j(γ),[1]) →(g n−j,id) ⋯ →(g 0,id) (Ad 0(γ),[1]), \array{ (\gamma, [0]) &\xrightarrow{ (g_{n-1}, id) }& ( Ad_{n-1}(\gamma), [0] ) &\xrightarrow{ (g_{n-2}, id) }& \cdots &\xrightarrow{ (g_{j}, id) }& \big( Ad_{j}(\gamma), [0] \big) \\ && && && \big\downarrow {}^{\mathrlap{ (e, [ [0],[1] ]) }} \\ && && && (Ad_j(\gamma), [1]) &\xrightarrow{ (g_{n-j},id) } & \cdots &\xrightarrow{ (g_0,id) }& (Ad_0(\gamma), [1]) \mathrlap{\,,} }

    where we are abbreviating

    Ad j(γ) ≔Ad (g n−1⋯g j)(γ) ≔(g n−1⋯g j) −1⋅γ⋅(g n−1⋯g j), \begin{aligned} Ad_j(\gamma) & \;\coloneqq\; Ad_{(g_{n-1} \cdots g_j)}(\gamma) \\ & \;\coloneqq\; (g_{n-1} \cdots g_j)^{-1} \cdot \gamma \cdot (g_{n-1} \cdots g_j) \,, \end{aligned}

    to the following n+1n+1 simplex of W¯G\overline{W}G:

    • →g n−1 • →g n−2 ⋯ →g j • ↓ Ad j(γ) • →g n−j ⋯ →g 0 •. \array{ \bullet &\xrightarrow{ g_{n-1} }& \bullet &\xrightarrow{ g_{n-2} }& \cdots &\xrightarrow{ g_{j} }& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ Ad_j(\gamma) }} \\ && && && \bullet &\xrightarrow{ g_{n-j} } & \cdots & \xrightarrow{ g_0 } & \bullet \mathrlap{\,.} }

    diff, v24, current

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