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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 21st 2017
   • (edited Feb 21st 2017)
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   Author: Urs
   Format: MarkdownItexfor the purposes of having direct links to it, I gave a side-remark at _[[stable Dold-Kan correspondence]]_ its own page: [[rational stable homotopy theory]], recording the equivalence $$ (H \mathbb{Q}) ModSpectra \;\simeq\; Ch_\bullet(\mathbb{Q}) $$ I also added the claim that under this identification and that of classical rational homotopy theory then the derived version of the free-forgetful adjunction $$ (dgcAlg^{\geq 2}_{\mathbb{Q}})_{/\mathbb{Q}[0]} \underoverset {\underset{U \circ ker(\epsilon_{(-)})}{\longrightarrow}} {\overset{Sym \circ cn}{\longleftarrow}} {\bot} Ch^{\bullet}(\mathbb{Q}) $$ models the [[stabilization]] adjunction $(\Sigma^\infty \dashv \Omega^\infty)$. But I haven't type the proof into the entry yet.

   for the purposes of having direct links to it, I gave a side-remark at stable Dold-Kan correspondence its own page: rational stable homotopy theory, recording the equivalence

   $$(H \mathbb{Q}) ModSpectra \;\simeq\; Ch_\bullet(\mathbb{Q})$$

   I also added the claim that under this identification and that of classical rational homotopy theory then the derived version of the free-forgetful adjunction

   $$(dgcAlg^{\geq 2}_{\mathbb{Q}})_{/\mathbb{Q}[0]} \underoverset {\underset{U \circ ker(\epsilon_{(-)})}{\longrightarrow}} {\overset{Sym \circ cn}{\longleftarrow}} {\bot} Ch^{\bullet}(\mathbb{Q})$$

   models the stabilization adjunction $(\Sigma^\infty \dashv \Omega^\infty)$. But I haven’t type the proof into the entry yet.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 22nd 2017
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   Author: Urs
   Format: MarkdownItexI have added to the beginning of the entry ([[rational stable homotopy theory]]) a remark that rational spectra _are_ $H\mathbb{Q}$-module spectra. Deserves to be further expanded.

   I have added to the beginning of the entry (rational stable homotopy theory) a remark that rational spectra are $H\mathbb{Q}$-module spectra. Deserves to be further expanded.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 23rd 2017
   • (edited Feb 23rd 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIn #1 I wrote: > I also added the claim that $[...]$ But I haven't type the proof into the entry yet. Done now, [here](https://ncatlab.org/nlab/show/rational+stable+homotopy+theory#RationalStabilization): The following composite total derived functors $$ \array{ \mathrm{Ho}( \mathrm{Spectra}_{\mathbb{Q}, \mathrm{fin}} ) \\ \downarrow \simeq \uparrow \\ \mathrm{Ho}( \mathrm{Ch}_{\mathbb{Q},\bullet,\mathrm{fin}} ) & \underoverset \underset{\mathbb{R}\mathrm{cn}_2}{\longrightarrow} \overset{\mathbb{L} i_2}{\longleftarrow} {\bot} & \mathrm{Ho}( \mathrm{Ch}_{\mathbb{Q}, \gt 1, \mathrm{fin}} ) \\ && \downarrow \simeq \uparrow^{(-)^\ast} \\ && \mathrm{Ho}( \mathrm{Ch}^{\gt 1}_{\mathbb{Q}, \mathrm{fin}} )^{\mathrm{op}} & \underoverset \underset{ (\mathbb{L}\mathrm{Sym}_{/\mathbb{Q}[0]})^{\mathrm{op}} }{\longrightarrow} \overset{ (\mathbb{R}( U \circ \mathrm{ker}(\epsilon_{(-)}) ))^{\mathrm{op}} }{\longleftarrow} {\bot} & \mathrm{Ho}( \mathrm{dgcAlg}^{\gt 0}_{\mathbb{Q}, \mathrm{fin}})_{/\mathbb{Q}[0]} )^{\mathrm{op}} \\ && && \updownarrow \simeq \\ && && \mathrm{Ho}(\mathrm{Top}_{\mathbb{Q}, \gt 1 , \mathrm{fin}}) } $$ agree with the restriction of the [[stabilization]] infinity-adjunction $$ Spectra \underoverset {\underset{\Omega^\infty}{\longrightarrow}} {\overset{\Sigma^\infty}{\longleftarrow}} {\bot} \infty Grpd^{\ast/} $$ to simply connected rational homotopy types of finite type.

   In #1 I wrote:

   I also added the claim that $[...]$ But I haven’t type the proof into the entry yet.

   Done now, here:

   The following composite total derived functors

   $$\array{ \mathrm{Ho}( \mathrm{Spectra}_{\mathbb{Q}, \mathrm{fin}} ) \\ \downarrow \simeq \uparrow \\ \mathrm{Ho}( \mathrm{Ch}_{\mathbb{Q},\bullet,\mathrm{fin}} ) & \underoverset \underset{\mathbb{R}\mathrm{cn}_2}{\longrightarrow} \overset{\mathbb{L} i_2}{\longleftarrow} {\bot} & \mathrm{Ho}( \mathrm{Ch}_{\mathbb{Q}, \gt 1, \mathrm{fin}} ) \\ && \downarrow \simeq \uparrow^{(-)^\ast} \\ && \mathrm{Ho}( \mathrm{Ch}^{\gt 1}_{\mathbb{Q}, \mathrm{fin}} )^{\mathrm{op}} & \underoverset \underset{ (\mathbb{L}\mathrm{Sym}_{/\mathbb{Q}[0]})^{\mathrm{op}} }{\longrightarrow} \overset{ (\mathbb{R}( U \circ \mathrm{ker}(\epsilon_{(-)}) ))^{\mathrm{op}} }{\longleftarrow} {\bot} & \mathrm{Ho}( \mathrm{dgcAlg}^{\gt 0}_{\mathbb{Q}, \mathrm{fin}})_{/\mathbb{Q}[0]} )^{\mathrm{op}} \\ && && \updownarrow \simeq \\ && && \mathrm{Ho}(\mathrm{Top}_{\mathbb{Q}, \gt 1 , \mathrm{fin}}) }$$

   agree with the restriction of the stabilization infinity-adjunction

   $$Spectra \underoverset {\underset{\Omega^\infty}{\longrightarrow}} {\overset{\Sigma^\infty}{\longleftarrow}} {\bot} \infty Grpd^{\ast/}$$

   to simply connected rational homotopy types of finite type.