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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

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeDec 6th 2022
    • (edited Dec 6th 2022)

    7: let me start understanding your picture. Lupercio and Uribe in 2001 paper Theorem 3.6.4 and Proposition (in fact Corollary) 3.6.6 (pdf) say that their loop groupoid is Morita equivalent to their inertia groupoid. Now the inertia orbifold entry says that in cohesive picture there is a comparison morphism which is in good cases faithful inclusion. So Lupercio-Uribe 3.6.4/3.6.6 does not survive in your cohesive version but is weakened in some sense ? By groupoid presentation you mean choosing in Morita class ?

    I spent some time in 1999-2003 understanding works of Lupercio and Uribe (including listening their seminars in Wisconsin) and somehow like comparison to their picture.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2022

    We are finally writing this up, preliminary but readable notes are at cyclic loop spaces 2022 (schreiber)

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeDec 6th 2022

    Great, do you plan to include remarks on various results around crepant resolution conjecture ?

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeDec 12th 2022

    11: You do not use cyclic nerve anywhere in your constructions planned for the writeup ?

    A couple of typoses.

    Page 5 formula 11: cyclified cocyle (green in the diagram at the top)

    Page 19 formula 19, also cocyle in blue

    Why do you say “the Cyc-adjunction” rather than the adjunction Cyc, it is not a modifier but a name.

    • CommentRowNumber14.
    • CommentAuthorzskoda
    • CommentTimeMar 14th 2023

    diff, v29, current

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 30th 2024

    Presumably the name of this concept owes its origins to ’inertia group’ and ’inertia field’ in number theory, e.g. as here. I’d add something to that effect if I knew anything about it.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 30th 2024

    So it’s coming from the idea of prime ideals remaining prime through field extensions, Inertial prime number.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2024

    I don’t know the historical origin of the terminology “inertia orbifold”. But, yes, it probably refers to the “inert actions” in the sense of “stabilizer subgroups” which make up the points of such an orbifold.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2024
    • (edited Apr 30th 2024)

    See also the first paragraph in section 3.4 of Noohi’s Foundations of Topological Stacks (p. 15).

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2024

    have expanded the Idea section (here) a bit: added little diagrams illustrating the objects and morphisms of an inertia groupoid, and added a paragraph on the terminology “inertia”.

    diff, v30, current

    • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 30th 2024

    Thanks!

    Re #18, four names for the same concept!

    Other synonyms for inertia group are stabilizer group, isotropy group and ramification group.

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