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to go alongside cyclic loop space and free loop stack. Not quite done yet, but need to save
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 $\ast \to \mathbf{B} S^1 \to \ast \to \mathbf{B} S^1 \to \ast$.
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.
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?
True, if Sasha really means what he writes there, then it’s different.
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