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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeFeb 5th 2017
   • (edited Feb 5th 2017)
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   Author: Urs
   Format: MarkdownItexHas anyone developed models for the homotopy theory of $H \mathbb{Q}$.module spectra over rational topological spaces a bit? I expect there should be a model on the opposite category of dg-modules over rational dg-algebras. Restricted to the trivial modules it should reduce to the standard Sullivan/Quillen model of rational homotopy theory. Restricted to the dg-modules over $\mathbb{Q}$ it should reduce to the standard model for the homotopy theory of rational chain complexes, hence equivalently that of $H \mathbb{Q}$-module spectra. Is there any work on this?

   Has anyone developed models for the homotopy theory of $H \mathbb{Q}$.module spectra over rational topological spaces a bit?

   I expect there should be a model on the opposite category of dg-modules over rational dg-algebras. Restricted to the trivial modules it should reduce to the standard Sullivan/Quillen model of rational homotopy theory. Restricted to the dg-modules over $\mathbb{Q}$ it should reduce to the standard model for the homotopy theory of rational chain complexes, hence equivalently that of $H \mathbb{Q}$-module spectra.

   Is there any work on this?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeFeb 9th 2017
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   Author: Urs
   Format: MarkdownItexI have forwarded this question to MO, [here](http://mathoverflow.net/q/261747/381).

   I have forwarded this question to MO, here.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeFeb 9th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat the usual simply-connectedness assumption of rational homotopy theory is necessary to make this work even over a fixed base space was pointed out earlier on MO [here](http://mathoverflow.net/a/221836/381). So I am really re-asking the question ([here](http://mathoverflow.net/q/221541/381)) that this was in reply to, but restricted to simply connected base spaces and in the simplified case after rationalization.

   That the usual simply-connectedness assumption of rational homotopy theory is necessary to make this work even over a fixed base space was pointed out earlier on MO here. So I am really re-asking the question (here) that this was in reply to, but restricted to simply connected base spaces and in the simplified case after rationalization.

4. • CommentRowNumber4.
   • CommentAuthorCharles Rezk
   • CommentTimeFeb 9th 2017
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   Author: Charles Rezk
   Format: MarkdownItexThe thing that naturally corresponds to parameterized spectra over $X$ is modules over the group ring $\mathbb{S}[\Omega X]$. There is supposed to be a Koszul duality relationship between these and modules over the cochain algebra $\mathbb{S}^{X}$, and that's what requires some kind of connectivity/finite type hypotheses. Probably somebody already knows whether this works out the way you want.

   The thing that naturally corresponds to parameterized spectra over $X$ is modules over the group ring $\mathbb{S}[\Omega X]$. There is supposed to be a Koszul duality relationship between these and modules over the cochain algebra $\mathbb{S}^{X}$, and that’s what requires some kind of connectivity/finite type hypotheses. Probably somebody already knows whether this works out the way you want.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeFeb 10th 2017
   • (edited Feb 10th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItex> The thing that naturally corresponds to parameterized spectra over $X$ is modules over the group ring $\mathbb{S}[\Omega X]$. There is supposed to be a Koszul duality relationship between these and modules over the cochain algebra $\mathbb{S}^{X}$, and that's what requires some kind of connectivity/finite type hypotheses. Thanks! What's a reference for this? The $\infty$-group ring $\mathbb{S}[\Omega X]$ seems like it should directly connect to the Quillen model of rational homotopy theory, where one forms the rational simplicial group ring $\mathbb{Q}[G Sing_*(X)]$ of the Quillen model for the loop group of $X$. Maybe that would be a more direct road, then from there using the classical equivalence to the Sullivan model.

   The thing that naturally corresponds to parameterized spectra over $X$ is modules over the group ring $\mathbb{S}[\Omega X]$. There is supposed to be a Koszul duality relationship between these and modules over the cochain algebra $\mathbb{S}^{X}$, and that’s what requires some kind of connectivity/finite type hypotheses.

   Thanks! What’s a reference for this?

   The $\infty$-group ring $\mathbb{S}[\Omega X]$ seems like it should directly connect to the Quillen model of rational homotopy theory, where one forms the rational simplicial group ring $\mathbb{Q}[G Sing_*(X)]$ of the Quillen model for the loop group of $X$. Maybe that would be a more direct road, then from there using the classical equivalence to the Sullivan model.

6. • CommentRowNumber6.
   • CommentAuthorCharles Rezk
   • CommentTimeFeb 10th 2017
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   Author: Charles Rezk
   Format: MarkdownItexI never know references for these sorts of things. To spell out the idea: if $A$ is an augmented algebra over $k$, then you can form $C:= k\otimes_A k$, which is a $k$-coalgebra (everything is derived here). Then you get a functor $F$: ($A$-modules) $\to$ ($C$-comodules) by $F(M):= k\otimes_A M$. Koszul duality (or perhaps more accurately, bar-cobar duality), says that $F$ induces an equivalence on certain full subcategories. Identifying the appropriate full subcategories is where connectivity hypotheses can enter. If $A=\mathbb{S}[\Omega X]$ over $k=\mathbb{S}$, then $C=\mathbb{S}[B\Omega X]=\mathbb{S}[X]$, the chains on $X$, which is a coalgebra via the diagonal map of $X$. Passing from chains to cochains by dualizing should give an equivalence between finite type $C$-coalgebras and finite type $C^*=\mathbb{S}^{X}$-algebras.

   I never know references for these sorts of things.

   To spell out the idea: if $A$ is an augmented algebra over $k$, then you can form $C:= k\otimes_A k$, which is a $k$-coalgebra (everything is derived here). Then you get a functor $F$: ($A$-modules) $\to$ ($C$-comodules) by $F(M):= k\otimes_A M$. Koszul duality (or perhaps more accurately, bar-cobar duality), says that $F$ induces an equivalence on certain full subcategories. Identifying the appropriate full subcategories is where connectivity hypotheses can enter.

   If $A=\mathbb{S}[\Omega X]$ over $k=\mathbb{S}$, then $C=\mathbb{S}[B\Omega X]=\mathbb{S}[X]$, the chains on $X$, which is a coalgebra via the diagonal map of $X$. Passing from chains to cochains by dualizing should give an equivalence between finite type $C$-coalgebras and finite type $C^*=\mathbb{S}^{X}$-algebras.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 11th 2017
   • (edited Feb 11th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. I meant the statement that parameterized spectra over some connected space $X$ is equivalently $\mathbb{S}[\Omega X]$-module spectra. I suppose I see how to prove it (we are after the category of functors $X \to Spectra$ hence $B \Omega X \to Spectra$ and that it is equivalent to $\mathbb{S} Mod$-enriched functors $B \mathbb{S}[\Omega X] \to Spectra$ which is $\mathbb{S}[\Omega X] Mod$) but I'd just like to know where this and maybe more related to it has been discussed before. If you happen to know off the top of your head. Thanks!

   Thanks. I meant the statement that parameterized spectra over some connected space $X$ is equivalently $\mathbb{S}[\Omega X]$-module spectra.

   I suppose I see how to prove it (we are after the category of functors $X \to Spectra$ hence $B \Omega X \to Spectra$ and that it is equivalent to $\mathbb{S} Mod$-enriched functors $B \mathbb{S}[\Omega X] \to Spectra$ which is $\mathbb{S}[\Omega X] Mod$) but I’d just like to know where this and maybe more related to it has been discussed before. If you happen to know off the top of your head. Thanks!

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeFeb 13th 2017
   • (edited Feb 13th 2017)
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   Author: Urs
   Format: MarkdownItexI'll chat a bit about why I am interested in this. It is related to my ongoing quest to understand which mathematics formalizes the physics folklore of "gauge enhancement at fixed points". It seems to me that the minimal dg-module model for $S^4 /S^1 \to S^3$ that Roig-Saralegi produce [here](https://ncatlab.org/nlab/show/4-sphere#CircleAction) (kindly pointed out to me by Hisham Sati) provides the missing link, if only we correctly put read it into context. In the following I'll write $\mathfrak{l}$ for the Sullivan-Quillen map from topological spaces to $L_\infty$-algebras. So recall that the mathematical incarnation of the physics story that there is the M2-brane and the M5-brane in 11d is that there is a non-trivial super $L_\infy$-homomorphism $$ \mathbb{R}^{10,1\vert \mathbf{32}} \longrightarrow \mathfrak{l}(S^4) \,. $$ Moreover, the mathematical incarnation of the physics story that under [[double dimensional reduction]] along the central extension $\mathbb{R}^{10,1\vert \mathbf{32}} \to \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}} }$ this gives rise to the F1 (the string), the NS5 and the D0, D2 and D4 is that under the adjunction $$ L_\infty Alg \underoverset {\underset{\mathfrak{L}(-)/(\mathfrak{l}(S^1))}{\longrightarrow}} {\overset{hoib}{\leftarrow}} {\bot} L_\infty Alg_{/B \mathbb{R}} $$ this turns into a map of the form $$ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} \longrightarrow \mathfrak{L}(S^4)/{\mathfrak{l}(S^1)} $$ and the coefficient on the right turns out to be the rational image of the 6-truncated twisted K-theory space $ku\langle 0\cdots 6\rangle/BU(1)$, capturing the cocycles for the F1, the D0, D2 and D4, as well as an NS5-brane contribution ([here](https://ncatlab.org/nlab/show/4-sphere#FreeLoopSpace)). (continued in next comment)

   I’ll chat a bit about why I am interested in this. It is related to my ongoing quest to understand which mathematics formalizes the physics folklore of “gauge enhancement at fixed points”. It seems to me that the minimal dg-module model for $S^4 /S^1 \to S^3$ that Roig-Saralegi produce here (kindly pointed out to me by Hisham Sati) provides the missing link, if only we correctly put read it into context.

   In the following I’ll write $\mathfrak{l}$ for the Sullivan-Quillen map from topological spaces to $L_\infty$-algebras.

   So recall that the mathematical incarnation of the physics story that there is the M2-brane and the M5-brane in 11d is that there is a non-trivial super $L_\infy$-homomorphism

   $$\mathbb{R}^{10,1\vert \mathbf{32}} \longrightarrow \mathfrak{l}(S^4) \,.$$

   Moreover, the mathematical incarnation of the physics story that under double dimensional reduction along the central extension $\mathbb{R}^{10,1\vert \mathbf{32}} \to \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}} }$ this gives rise to the F1 (the string), the NS5 and the D0, D2 and D4 is that under the adjunction

   $$L_\infty Alg \underoverset {\underset{\mathfrak{L}(-)/(\mathfrak{l}(S^1))}{\longrightarrow}} {\overset{hoib}{\leftarrow}} {\bot} L_\infty Alg_{/B \mathbb{R}}$$

   this turns into a map of the form

   $$\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} \longrightarrow \mathfrak{L}(S^4)/{\mathfrak{l}(S^1)}$$

   and the coefficient on the right turns out to be the rational image of the 6-truncated twisted K-theory space $ku\langle 0\cdots 6\rangle/BU(1)$, capturing the cocycles for the F1, the D0, D2 and D4, as well as an NS5-brane contribution (here).

   (continued in next comment)

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeFeb 13th 2017
   • (edited Feb 13th 2017)
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   Author: Urs
   Format: MarkdownItex(continued from previous comment) Next, physics folklore says that the missing [[D6-brane]] appears by a different mechanism, namely it appears as the [[KK-monopole]] at the locus where the 11d circle fibration over 10d has an $S^1$-fixed point of codimension 4. Now one may check by hand that there are coycles on $https://nforum.ncatlab.org/post/61307/\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ which correspond to the D6 (and D8 and D10) that did not appear from the double dimensional reduction above, but which consitute a non-trivial map $$ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} \longrightarrow \mathfrak{l}(ku/BU(1)) $$ but the question is if apart from observing that this map may be written down, we find a principle akin to the "cyclification" $\mathcal{L}(-)/S^1$ above which puts this into a bigger perspective. Now since we are looking at that cyclificatin adjunction in the first place, and since we built the double dimensional reduction using the adjunction unit $$ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} \longrightarrow \mathfrak{L}(\mathbb{R}^{10,1\vert \mathbf{32}})/{\mathfrak{l}(S^1)} $$ it is natural to consider the same adjunction unit also on the coefficients. That _requires_ us to think of $S^4$ as equipped with an $S^1$-action, and the evident non-trivial such is the canonical one induced by the identification $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$, using the left action of $S^1 \hookrightarrow SU(2)$. Writing $S^4 / S^1$ for the corresponding homotopy quotient, then the adjunction unit has the form $$ S^4/S^1 \longrightarrow \mathcal{L}(S^4)/S^1 $$ and so it is natural to ask for a lift of the above cocycle through this map. This is where the result by Roig-Saralegi comes in: if we think of this adjunction unit as a map over $S^3$ and approximate $S^4 / S^1 \longrightarrow S^3$ by its fiberwise suspension spectrum $\Sigma^\infty_{S^3} (S^4/S^1)$, then they find that rationally, this is just the parameterized spectrum for un-truncated twisted K-theory ([here](https://ncatlab.org/nlab/show/4-sphere#CircleAction))... (continued in next comment)

   (continued from previous comment)

   Next, physics folklore says that the missing D6-brane appears by a different mechanism, namely it appears as the KK-monopole at the locus where the 11d circle fibration over 10d has an $S^1$-fixed point of codimension 4.

   Now one may check by hand that there are coycles on $https://nforum.ncatlab.org/post/61307/\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ which correspond to the D6 (and D8 and D10) that did not appear from the double dimensional reduction above, but which consitute a non-trivial map

   $$\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} \longrightarrow \mathfrak{l}(ku/BU(1))$$

   but the question is if apart from observing that this map may be written down, we find a principle akin to the “cyclification” $\mathcal{L}(-)/S^1$ above which puts this into a bigger perspective.

   Now since we are looking at that cyclificatin adjunction in the first place, and since we built the double dimensional reduction using the adjunction unit

   $$\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} \longrightarrow \mathfrak{L}(\mathbb{R}^{10,1\vert \mathbf{32}})/{\mathfrak{l}(S^1)}$$

   it is natural to consider the same adjunction unit also on the coefficients. That requires us to think of $S^4$ as equipped with an $S^1$-action, and the evident non-trivial such is the canonical one induced by the identification $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$, using the left action of $S^1 \hookrightarrow SU(2)$.

   Writing $S^4 / S^1$ for the corresponding homotopy quotient, then the adjunction unit has the form

   $$S^4/S^1 \longrightarrow \mathcal{L}(S^4)/S^1$$

   and so it is natural to ask for a lift of the above cocycle through this map.

   This is where the result by Roig-Saralegi comes in: if we think of this adjunction unit as a map over $S^3$ and approximate $S^4 / S^1 \longrightarrow S^3$ by its fiberwise suspension spectrum $\Sigma^\infty_{S^3} (S^4/S^1)$, then they find that rationally, this is just the parameterized spectrum for un-truncated twisted K-theory (here)…

   (continued in next comment)

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 13th 2017
    • (edited Feb 13th 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItex(continuation from previous comment) ... or rather, it is the parameterized spectrum for twisted K-theory, rationally, plus another sumand which asks for trivializations of the twist. To see what this means, notice that at the locus of $S^1$ fixed point, then the homotopy quotient of the "11d" spacetime is of the form $X_{10} \times B S^1$, hence is locally an $S^1$-gerbe over the 10d spacetime. The infinitesimal model for this is the super Lie 2-algebra $\mathfrak{string}_{IIA}$, which is the homotopy fiber of $\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}} \overset{\mu{F1}^{IIA}}{\longrightarrow} b^2 \mathbb{R}$. On _that_ "fixed point locus" we do have a canonical map of dg-modules $$ \array{ \mathfrak{string}_{IIA} && \longrightarrow && \mathfrak{l }\Sigma^{\infty}_{S^3} (S^4 / S^1) \\ & \searrow && \swarrow \\ && \mathfrak{l}(S^3)\simeq b^2 \mathbb{R} } $$ namely the one which, dually, sends each generator to the generator of the same name, in the naming conventions as [here](https://ncatlab.org/nlab/show/4-sphere#CircleAction). (continued in next comment)

    (continuation from previous comment)

    … or rather, it is the parameterized spectrum for twisted K-theory, rationally, plus another sumand which asks for trivializations of the twist.

    To see what this means, notice that at the locus of $S^1$ fixed point, then the homotopy quotient of the “11d” spacetime is of the form $X_{10} \times B S^1$, hence is locally an $S^1$-gerbe over the 10d spacetime. The infinitesimal model for this is the super Lie 2-algebra $\mathfrak{string}_{IIA}$, which is the homotopy fiber of $\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}} \overset{\mu{F1}^{IIA}}{\longrightarrow} b^2 \mathbb{R}$. On that “fixed point locus” we do have a canonical map of dg-modules

    $$\array{ \mathfrak{string}_{IIA} && \longrightarrow && \mathfrak{l }\Sigma^{\infty}_{S^3} (S^4 / S^1) \\ & \searrow && \swarrow \\ && \mathfrak{l}(S^3)\simeq b^2 \mathbb{R} }$$

    namely the one which, dually, sends each generator to the generator of the same name, in the naming conventions as here.

    (continued in next comment)

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeFeb 13th 2017
    • (edited Feb 13th 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItex(continued from previous comment) So it seems that things are falling into place nicely. But to solidify this, I need to check some things about those would-be dg-module models for the parameterized spectra. Foremost, I should check that we have a homotopy-commuting triangle $$ \array{ S^4/S^1 && \longrightarrow && \mathcal{L}(S^4)/S^1 \\ & \searrow && \swarrow \\ && S^3 } $$ where the vertical maps are the canonical ones -- rationally at least, and for the vertical maps linearized to parameterized spectra, at least. (am forced offline now. More later)

    (continued from previous comment)

    So it seems that things are falling into place nicely. But to solidify this, I need to check some things about those would-be dg-module models for the parameterized spectra. Foremost, I should check that we have a homotopy-commuting triangle

    $$\array{ S^4/S^1 && \longrightarrow && \mathcal{L}(S^4)/S^1 \\ & \searrow && \swarrow \\ && S^3 }$$

    where the vertical maps are the canonical ones – rationally at least, and for the vertical maps linearized to parameterized spectra, at least.

    (am forced offline now. More later)

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeFeb 20th 2017
    • (edited Feb 20th 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexBack to the question on how to model rational spectra parameterized over a rational space in terms of dg-modules over a Quillen/Sullivan-model for that space: If any of the model categories used in classical rational homotopy theory -- i.e. dgc-algebras or dgc-coalgebra or dg-Lie algebras, all in suitably truncated degrees -- were proper model categories, then the analogous argument as in section 3 of [Schwede 97](https://ncatlab.org/nlab/show/parametrized+spectrum#Schwede97) should go through. Are any of these three model categories proper? It seems that the projective model structure on cochain dgc-algebras in non-negative degrees is not proper (as opposed to its untruncated version). The problem is the generating cofibration from the graded algebra free on a single closed generator in degree 0 and mapping to the ground field: pushout along this map does not preserve quasi-isos. Not sure about the other two.

    Back to the question on how to model rational spectra parameterized over a rational space in terms of dg-modules over a Quillen/Sullivan-model for that space:

    If any of the model categories used in classical rational homotopy theory – i.e. dgc-algebras or dgc-coalgebra or dg-Lie algebras, all in suitably truncated degrees – were proper model categories, then the analogous argument as in section 3 of Schwede 97 should go through.

    Are any of these three model categories proper?

    It seems that the projective model structure on cochain dgc-algebras in non-negative degrees is not proper (as opposed to its untruncated version). The problem is the generating cofibration from the graded algebra free on a single closed generator in degree 0 and mapping to the ground field: pushout along this map does not preserve quasi-isos.

    Not sure about the other two.

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2017
    • (edited Feb 21st 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexAh, maybe we don't need left properness for the argument in [Schwede 97, section 3](https://ncatlab.org/nlab/show/parametrized+spectrum#Schwede97) to go through. There left and right properness is assumed on the one hand in order to appeal to the [[Bousfield-Friedlander theorem]]. But the proof given in that entry (from Goerss-Jardine), only right properness is needed to establish the model structure (and [Stanculescu 08](https://ncatlab.org/nlab/show/Bousfield-Friedlander+theorem#Stanculescu08) claims that not even right properness is necessary here.) On the other hand a scan through [Schwede 97, section 2](https://ncatlab.org/nlab/show/parametrized+spectrum#Schwede97) shows (I think) that (only) right properness is used to establish the assumptions for the BF-theorem on the spectrification functor $Q$, namely in the proof of lemma 2.1.3 (only) right properness is used on the bottom of p. 92 (16 of 28) in the guise of "the dual of the gluing lemma (Lemma 1.1.9)". Now the projective model structure on cochain dgc-algebras in non-negative degrees is right proper (being transferred from the projective model structure on chain complexes in non-negative degree) and it readily admits the analogous $(Sym \dashv U \circ AugmentationIdeal)$ adjunction as in [Schwede 97, first lines of section 3.2](https://ncatlab.org/nlab/show/parametrized+spectrum#Schwede97). Therefore, it seems to me, the direct analogue of the proof of [Schwede 97, lemma 3.2.2](https://ncatlab.org/nlab/show/parametrized+spectrum#Schwede97) should go through for the projective model structure on dgc-algebras in non-negative degrees (in place of simplicial commutative algebras as considered in the article) and hence the main result theorem 3.2.3 should be obtained, which would be the desired statement.

    Ah, maybe we don’t need left properness for the argument in Schwede 97, section 3 to go through.

    There left and right properness is assumed on the one hand in order to appeal to the Bousfield-Friedlander theorem. But the proof given in that entry (from Goerss-Jardine), only right properness is needed to establish the model structure (and Stanculescu 08 claims that not even right properness is necessary here.)

    On the other hand a scan through Schwede 97, section 2 shows (I think) that (only) right properness is used to establish the assumptions for the BF-theorem on the spectrification functor $Q$, namely in the proof of lemma 2.1.3 (only) right properness is used on the bottom of p. 92 (16 of 28) in the guise of “the dual of the gluing lemma (Lemma 1.1.9)”.

    Now the projective model structure on cochain dgc-algebras in non-negative degrees is right proper (being transferred from the projective model structure on chain complexes in non-negative degree) and it readily admits the analogous $(Sym \dashv U \circ AugmentationIdeal)$ adjunction as in Schwede 97, first lines of section 3.2.

    Therefore, it seems to me, the direct analogue of the proof of Schwede 97, lemma 3.2.2 should go through for the projective model structure on dgc-algebras in non-negative degrees (in place of simplicial commutative algebras as considered in the article) and hence the main result theorem 3.2.3 should be obtained, which would be the desired statement.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2017
    • PermaLink
    Author: Urs
    Format: MarkdownItexHm, maybe I am wrong that $(dgcAlg^{\geq 0}_\mathbb{Q})_{proj}$ verifies enough properties for the proof in Schwede 97 to still go through. I suppose what is missing is that forming the dg-algebra anolog of based loop spaces is Quillen adjoint to tensor product with the dg-algebra of polynomial forms on the simplicial circle. I suspect for dg-coalgebras all this works better, but I haven't collected the relevant bits yet.

    Hm, maybe I am wrong that $(dgcAlg^{\geq 0}_\mathbb{Q})_{proj}$ verifies enough properties for the proof in Schwede 97 to still go through. I suppose what is missing is that forming the dg-algebra anolog of based loop spaces is Quillen adjoint to tensor product with the dg-algebra of polynomial forms on the simplicial circle.

    I suspect for dg-coalgebras all this works better, but I haven’t collected the relevant bits yet.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2017
    • (edited Feb 22nd 2017)
    • PermaLink
    Author: Urs
    Format: MarkdownItexOkay, so I need to switch model. Of all the algebraic models of rational homotopy theory, the one (?) that seems to be both a simplicial model category as well as right proper is the [[model structure on simplicial Lie algebras]]. To that now [Schwede 97, theorem 3.2.3](https://ncatlab.org/nlab/show/parametrized+spectrum#Schwede97) applies directly without any ado. And so that should be it.

    Okay, so I need to switch model. Of all the algebraic models of rational homotopy theory, the one (?) that seems to be both a simplicial model category as well as right proper is the model structure on simplicial Lie algebras.

    To that now Schwede 97, theorem 3.2.3 applies directly without any ado. And so that should be it.

16. • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 1st 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexIs the use of rational homotopy theory in this physics that you're developing a temporary measure, an approximation to some full homotopy theory with torsion?

    Is the use of rational homotopy theory in this physics that you’re developing a temporary measure, an approximation to some full homotopy theory with torsion?

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2017
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    Author: Urs
    Format: MarkdownItexYes, it's a stepping stone. The point is that by the systematic analysis of the brane bouquet, even though it's just rational, one already sees some things that were missed, or not properly appreciated before. So it's a blueprint for a systematic derivation of the full story.

    Yes, it’s a stepping stone. The point is that by the systematic analysis of the brane bouquet, even though it’s just rational, one already sees some things that were missed, or not properly appreciated before. So it’s a blueprint for a systematic derivation of the full story.

18. • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 1st 2017
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    Author: David_Corfield
    Format: MarkdownItexThanks. And what happens to the modelling of rational homotopy spaces by dg-Lie algebras with the passage to the full story? As I mentioned elsewhere, people (Behrens, Rezk, Heuts) are looking for Lie algebra models for $v_n$-periodic unstable homotopy theory (Heuts promising this in a forthcoming 'Periodicity in unstable homotopy theory'). Will this be an approximation to what is already known to exist?

    Thanks. And what happens to the modelling of rational homotopy spaces by dg-Lie algebras with the passage to the full story? As I mentioned elsewhere, people (Behrens, Rezk, Heuts) are looking for Lie algebra models for $v_n$-periodic unstable homotopy theory (Heuts promising this in a forthcoming ’Periodicity in unstable homotopy theory’). Will this be an approximation to what is already known to exist?

19. • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 1st 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexThanks. And what happens to the modelling of rational homotopy spaces by dg-Lie algebras with the passage to the full story? As I mentioned elsewhere, people (Behrens, Rezk, Heuts) are looking for Lie algebra models for $v_n$-periodic unstable homotopy theory (Heuts promising this in a forthcoming 'Periodicity in unstable homotopy theory'). Will this be an approximation to what is already known to exist?

    Thanks. And what happens to the modelling of rational homotopy spaces by dg-Lie algebras with the passage to the full story? As I mentioned elsewhere, people (Behrens, Rezk, Heuts) are looking for Lie algebra models for $v_n$-periodic unstable homotopy theory (Heuts promising this in a forthcoming ’Periodicity in unstable homotopy theory’). Will this be an approximation to what is already known to exist?

20. • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 25th 2017
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexI see Lurie is giving a seminar on [Unstable Chromatic Homotopy Theory](http://www.math.harvard.edu/~lurie/Thursday2017.html) which looks to extend beyond Quillen and Sullivan to the non-rational setting chromatically.

    I see Lurie is giving a seminar on Unstable Chromatic Homotopy Theory which looks to extend beyond Quillen and Sullivan to the non-rational setting chromatically.

21. • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2018
    • (edited Mar 23rd 2018)
    • PermaLink
    Author: Urs
    Format: MarkdownItexMy initial efforts of proving, with [[Vincent Braunack-Mayer]] (formerly Vincent Schlegel), that/how the homotopy theory of dg-(co-)modules models [[rational parameterized stable homotopy theory]] have meanwhile been reworked and completed by Vincent, who has now finished a PhD thesis, establishing the foundations of rational parameterized stable homotopy theory as well as its Lie integration to smooth parameterized spectra (twisted differential cohomology theories): * Vincent Braunack-Mayer, _[[schreiber:thesis Braunack-Mayer|Rational parameterized stable homotopy theory]]_, Zurich 2018 ([pdf](https://ncatlab.org/schreiber/show/thesis+Braunack-Mayer#pdf)) This will be defended in April.

    My initial efforts of proving, with Vincent Braunack-Mayer (formerly Vincent Schlegel), that/how the homotopy theory of dg-(co-)modules models rational parameterized stable homotopy theory have meanwhile been reworked and completed by Vincent, who has now finished a PhD thesis, establishing the foundations of rational parameterized stable homotopy theory as well as its Lie integration to smooth parameterized spectra (twisted differential cohomology theories):

    • Vincent Braunack-Mayer, Rational parameterized stable homotopy theory, Zurich 2018 (pdf)

    This will be defended in April.

22. • CommentRowNumber22.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 23rd 2018
    • PermaLink
    Author: DavidRoberts
    Format: MarkdownItexHurrah! (I saw Vincent the week before he was married, didn't know his plans to re-name himself :-)

    Hurrah! (I saw Vincent the week before he was married, didn’t know his plans to re-name himself :-)

23. • CommentRowNumber23.
    • CommentAuthorcrogers
    • CommentTimeMar 23rd 2018
    • PermaLink
    Author: crogers
    Format: TextThis is great. I've been looking forward to reading this. One minor remark: Vincent may want to cite, in the "vista's section" on integration/homotopy Lie theory, Severa-Siran's paper https://arxiv.org/abs/1506.04898 on integrating Lie n-algebroids. Also, in my work with Chenchang Zhu, we exhibit Henriques' integration, just for n-algebras, as an exact functor (in the sense of a functor between categories of fibrant objects). The latter work is https://arxiv.org/abs/1609.01394 I'm finishing up revisions to a third version that includes more details. Anyway, congrats to Vincent!

    This is great. I've been looking forward to reading this.
    
    One minor remark: Vincent may want to cite, in the "vista's section" on integration/homotopy Lie theory, Severa-Siran's paper
    
    https://arxiv.org/abs/1506.04898
    
    on integrating Lie n-algebroids.
    
    Also, in my work with Chenchang Zhu, we exhibit Henriques' integration, just for n-algebras, as an exact functor (in the sense of a functor between categories of fibrant objects).
    
    The latter work is https://arxiv.org/abs/1609.01394
    
    I'm finishing up revisions to a third version that includes more details.
    
    Anyway, congrats to Vincent!
24. • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2018
    • (edited Mar 25th 2018)
    • PermaLink
    Author: Urs
    Format: MarkdownItexHi Chris, thanks for your comments! I have forwarded them to Vincent. Vincent's Lie integration follows that of [arXiv:1011.4735](https://arxiv.org/abs/1011.4735), which disregards the question of whether the result is representable by a finite-dimensional Kan-simplicial manifold, in favor of first understanding its abstract behaviour as a Lie integration adjunction into the $\infty$-topos of smooth $\infty$-stacks. It seemed to me at times that the insistence on Kan simplicial manifold representatives is driven more by habit than by mathematical motivation. I expect that it will eventually play a role for constructing push-forward maps of (differential) generalized cohomology classes, because Umkehr maps via Pontryagin-Thom collapse is, curiously, the key point where abstract homotopy theory insists on manifold structure in favor of more general homotopy types. But until one begins to see how that speculation wants to pan out in detail, it seems unreasonable to me to insist on a constraint which is not being put to use. I have raised that remark a few times before, for instance when Joost got told to seek on Kan simplicial manifolds the adjunction that Vincent now established on general $\infty$-stacks, and I remain interested in hearing what people think.

    Hi Chris,

    thanks for your comments! I have forwarded them to Vincent.

    Vincent’s Lie integration follows that of arXiv:1011.4735, which disregards the question of whether the result is representable by a finite-dimensional Kan-simplicial manifold, in favor of first understanding its abstract behaviour as a Lie integration adjunction into the $\infty$-topos of smooth $\infty$-stacks.

    It seemed to me at times that the insistence on Kan simplicial manifold representatives is driven more by habit than by mathematical motivation. I expect that it will eventually play a role for constructing push-forward maps of (differential) generalized cohomology classes, because Umkehr maps via Pontryagin-Thom collapse is, curiously, the key point where abstract homotopy theory insists on manifold structure in favor of more general homotopy types. But until one begins to see how that speculation wants to pan out in detail, it seems unreasonable to me to insist on a constraint which is not being put to use.

    I have raised that remark a few times before, for instance when Joost got told to seek on Kan simplicial manifolds the adjunction that Vincent now established on general $\infty$-stacks, and I remain interested in hearing what people think.