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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 27th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexto go alongside *[[cyclic loop space]]* and *[[free loop stack]]*. Not quite done yet, but need to save <a href="https://ncatlab.org/nlab/revision/cyclic+loop+stack/1">v1</a>, <a href="https://ncatlab.org/nlab/show/cyclic+loop+stack">current</a>

   to go alongside cyclic loop space and free loop stack. Not quite done yet, but need to save

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 1st 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexCarrying on the conversation from [here](https://nforum.ncatlab.org/discussion/7938/mysterious-duality/?Focus=93475#Comment_93475), I guess with cyclification here as a right adjoint followed by a left adjoint, there's no structure around to help with iteration. Double cyclical iteration isn't cyclification by the torus, etc. Presumably one could factor toroidal cyclification as two right base changes and two left base changes along $\ast \to \mathbf{B} S^1 \to \mathbf{B} (S^1 \times S^1) \to \mathbf{B} S^1 \to \ast$. But this isn't comparable to right-left base change twice along $\ast \to \mathbf{B} S^1 \to \ast \to \mathbf{B} S^1 \to \ast$.

   Carrying on the conversation from here, I guess with cyclification here as a right adjoint followed by a left adjoint, there’s no structure around to help with iteration. Double cyclical iteration isn’t cyclification by the torus, etc.

   Presumably one could factor toroidal cyclification as two right base changes and two left base changes along $\ast \to \mathbf{B} S^1 \to \mathbf{B} (S^1 \times S^1) \to \mathbf{B} S^1 \to \ast$. But this isn’t comparable to right-left base change twice along $\ast \to \mathbf{B} S^1 \to \ast \to \mathbf{B} S^1 \to \ast$.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 1st 2021
   • (edited Jul 1st 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes. This is related to the fact that the Extension$\dashv$Cyclification-adjunction keeps the cyclification in the slice. Iterating it looks as $$ \array{ \mathbf{H} & \underoverset {\underset{Cyc_{\mathcal{S}_2}}{\longrightarrow}} {\overset{Ext_{\mathcal{S}_2}}{\longleftarrow}} {\phantom{AAA}\bot\phantom{AAA}} & \mathbf{H}_{/\mathbf{B}\mathcal{S}_2} & \underoverset {\underset{Cyc_{\mathcal{S}_1}}{\longrightarrow}} {\overset{Ext_{\mathcal{S}_1}}{\longleftarrow}} {\phantom{AAA}\bot\phantom{AAA}} & \mathbf{H}_{/\mathbf{B}(\mathcal{S}_1 \times \mathcal{S}_2)} \\ \ast &\xrightarrow{\phantom{AAAAAA}}& \mathbf{B}\mathcal{S}_2 &\xrightarrow{\phantom{AAAAAA}}& \mathbf{B}\mathcal{S}_1\times \mathbf{B}\mathcal{S}_2 } $$ This is maybe most transparent for the left adjoint: $Ext_{\mathcal{S}_1}$ reads in a spacetime that is a KK-compactification on a fiber $\mathcal{S}_1 \times \mathcal{S}_2$ and it first "de-compactifies" the $\mathcal{S}_1$-factor. Then $Ext_{\mathcal{S}_2}$ de-compactifies the remaining $\mathcal{S}_2$-factor.

   Yes. This is related to the fact that the Extension$\dashv$Cyclification-adjunction keeps the cyclification in the slice. Iterating it looks as

   $$\array{ \mathbf{H} & \underoverset {\underset{Cyc_{\mathcal{S}_2}}{\longrightarrow}} {\overset{Ext_{\mathcal{S}_2}}{\longleftarrow}} {\phantom{AAA}\bot\phantom{AAA}} & \mathbf{H}_{/\mathbf{B}\mathcal{S}_2} & \underoverset {\underset{Cyc_{\mathcal{S}_1}}{\longrightarrow}} {\overset{Ext_{\mathcal{S}_1}}{\longleftarrow}} {\phantom{AAA}\bot\phantom{AAA}} & \mathbf{H}_{/\mathbf{B}(\mathcal{S}_1 \times \mathcal{S}_2)} \\ \ast &\xrightarrow{\phantom{AAAAAA}}& \mathbf{B}\mathcal{S}_2 &\xrightarrow{\phantom{AAAAAA}}& \mathbf{B}\mathcal{S}_1\times \mathbf{B}\mathcal{S}_2 }$$

   This is maybe most transparent for the left adjoint: $Ext_{\mathcal{S}_1}$ reads in a spacetime that is a KK-compactification on a fiber $\mathcal{S}_1 \times \mathcal{S}_2$ and it first “de-compactifies” the $\mathcal{S}_1$-factor. Then $Ext_{\mathcal{S}_2}$ de-compactifies the remaining $\mathcal{S}_2$-factor.

4. • CommentRowNumber4.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 1st 2021
   • (edited Jul 1st 2021)
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexBut then if toroidal cyclification (your diagram in #3 + left base change for the homotopy fibre) is not double $S^1$-cyclification, why is Voronov looking at iterated $S^1$-cyclication (slide 9) with a view to its relevance for toroidal-compactification (slide 5)? Isn't that to connect the two processes mentioned at the end of #2?

   But then if toroidal cyclification (your diagram in #3 + left base change for the homotopy fibre) is not double $S^1$-cyclification, why is Voronov looking at iterated $S^1$-cyclication (slide 9) with a view to its relevance for toroidal-compactification (slide 5)? Isn’t that to connect the two processes mentioned at the end of #2?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJul 1st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexTrue, if Sasha really means what he writes there, then it's different.

   True, if Sasha really means what he writes there, then it’s different.