Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

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

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2017
    • (edited Feb 21st 2017)

    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)ModSpectraCh () (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 2) /[0]Uker(ε ())SymcnCh () (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 (Σ Ω )(\Sigma^\infty \dashv \Omega^\infty). But I haven’t type the proof into the entry yet.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2017

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

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 23rd 2017
    • (edited Feb 23rd 2017)

    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

    Ho(Spectra ,fin) Ho(Ch ,,fin) cn 2𝕃i 2 Ho(Ch ,>1,fin) () * Ho(Ch ,fin >1) op (𝕃Sym /[0]) op((Uker(ε ()))) op Ho(dgcAlg ,fin >0) /[0]) op Ho(Top ,>1,fin) \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

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

    to simply connected rational homotopy types of finite type.

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)