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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 25th 2021

have brushed-up the Definition-section

• 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:

$\Lambda \mathcal{X} \;=\; \big[ ʃS^1, \, \mathcal{X} \big] \xrightarrow{ \; [\eta_{S^1},\mathcal{X}]\; } \big[ S^1, \, \mathcal{X} \big] \;=\; \mathcal{L} \mathcal{X} \,.$
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJun 25th 2021

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

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeJun 26th 2021

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

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

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

where

$id \;\colon\; \ast \xrightarrow{pt_{\mathbf{B}\mathcal{S}}} \mathbf{B}\mathcal{S} \xrightarrow{ p_{{}_{\mathbf{B}\mathcal{S}}} } \ast$

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

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 $G$ is isomorphic to the simplicial hom complex out of the minimal simplicial circle $S$ (Def. \ref{MinimalSimplicialCircle}) into the simplicial classifying space $\overline{W}G$:

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

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

$\array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet \\ \big\downarrow {}^{\mathrlap{ \gamma }} && \big\downarrow {}^{\mathrlap{ g_{n-1}^{-1} \cdot \gamma \cdot g_{n-1} }} && && \big\downarrow && \big\downarrow && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{0} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{0} ) }} \\ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet }$

is sent to the homomorphism

$\Delta[n] \times S \xrightarrow{\;\;} \overline{W}G$

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

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

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

$\array{ \bullet &\xrightarrow{g_{n-1}}& \bullet &\xrightarrow{g_{n-2}}& \cdots &\xrightarrow{g_{n-j-1}}& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ ( g_{n-1} \cdots g_{n-j-1} )^{-1} \cdot \gamma \cdot ( g_{n-1} \cdots g_{n-j-1} ) }} \\ && && && \bullet &\xrightarrow{g_{n-j-2}}& \bullet &\xrightarrow{g_{n-j-3}}& \cdots &\xrightarrow{g_{0}}& \bullet }$
• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJul 12th 2021
• (edited Jul 12th 2021)

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

$[S,\overline{W}G] \times S \xrightarrow{\;\;} \overline{W}G$

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

$\array{ (\gamma, [0]) &\xrightarrow{ (g_{n-1}, id) }& ( Ad_{n-1}(\gamma), [0] ) &\xrightarrow{ (g_{n-2}, id) }& \cdots &\xrightarrow{ (g_{j}, id) }& \big( Ad_{j}(\gamma), [0] \big) \\ && && && \big\downarrow {}^{\mathrlap{ (e, [ [0],[1] ]) }} \\ && && && (Ad_j(\gamma), [1]) &\xrightarrow{ (g_{n-j},id) } & \cdots &\xrightarrow{ (g_0,id) }& (Ad_0(\gamma), [1]) \mathrlap{\,,} }$

where we are abbreviating

\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 $n+1$ simplex of $\overline{W}G$:

$\array{ \bullet &\xrightarrow{ g_{n-1} }& \bullet &\xrightarrow{ g_{n-2} }& \cdots &\xrightarrow{ g_{j} }& \bullet \\ && && && \big\downarrow {}^{\mathrlap{ Ad_j(\gamma) }} \\ && && && \bullet &\xrightarrow{ g_{n-j} } & \cdots & \xrightarrow{ g_0 } & \bullet \mathrlap{\,.} }$