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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 25th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexthe equivariant generalization of the _[[Quillen adjunction between simplicial sets and connective dgc-algebras]]_ <a href="https://ncatlab.org/nlab/revision/Quillen+adjunction+between+equivariant+simplicial+sets+and+equivariant+connective+dgc-algebras/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Quillen+adjunction+between+equivariant+simplicial+sets+and+equivariant+connective+dgc-algebras">current</a>

   the equivariant generalization of the Quillen adjunction between simplicial sets and connective dgc-algebras

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2022
   • (edited Jul 21st 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI'd like to generalize the [[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]] by replacing the orbit category $Orb(G)$ by [tom Dieck's "fundamental category"](https://ncatlab.org/nlab/show/orbit%20category#FundamentalCategoryOfAGSpace) $\Pi_G(X)$ for any $G$-space $X$ (i.e. by the slice of the orbit category over the 1-truncation of the equivariant homotopy type of $X$). Such a generalization would be quite impactful: It would allow to handle equivariant rational homotopy theory by dgc-algebraic means for general equivariant rational homotopy types, not necessarily equivariantly simply connected (which is a much more serious constraint than plain simply connectedness). Half a year back I had checked on MathOverflow ([MO:q/412833](https://mathoverflow.net/q/412833/381)) if anyone would safe me the trouble, but not so far. Today I started playing with it more seriously. It looks like it's easy to transfer the Quillen equivalence under two assumptions: (i) that $X$ is equivariantly connected and (ii) that $G/H \mapsto \pi_1^H(X)$ is projectively cofibrant (a rough writeup of the simple argument in this case is currently in the *[[Sandbox]]*). Now assumption (i) is not too bad and it seems clear how to relax it with just a tad more effort. But assumption (ii) is way too strong to be of much use in practice. There are some evident things to try to work around this, such as slicing over a simplicial cofibrant replacement of $\pi_1^{(-)}(X)$. But while it is clear how to do this and similar variations on the side of the equivariant simplicial sets, it is not so clear (to me, at the moment) what to do on the side of the equivariant dgc-algebras. Alternatively, one could go step-by-step through Scull's original article * [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2)) and see if any of the proof steps break if one replaces $Orb(G)$ by $\Pi_G(X)$. My vague intuition is that the only relevant property of $Orb(G)$ which Scull's proof really uses is that it is an [[EI-category]], and that property is shared by $\Pi_G(X)$. But I haven't checked in detail, and I am shying away from the tedious task. But if all fails, maybe I'll dive into it.

   I’d like to generalize the Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras by replacing the orbit category $Orb(G)$ by tom Dieck’s “fundamental category” $\Pi_G(X)$ for any $G$-space $X$ (i.e. by the slice of the orbit category over the 1-truncation of the equivariant homotopy type of $X$).

   Such a generalization would be quite impactful: It would allow to handle equivariant rational homotopy theory by dgc-algebraic means for general equivariant rational homotopy types, not necessarily equivariantly simply connected (which is a much more serious constraint than plain simply connectedness).

   Half a year back I had checked on MathOverflow (MO:q/412833) if anyone would safe me the trouble, but not so far.

   Today I started playing with it more seriously. It looks like it’s easy to transfer the Quillen equivalence under two assumptions: (i) that $X$ is equivariantly connected and (ii) that $G/H \mapsto \pi_1^H(X)$ is projectively cofibrant (a rough writeup of the simple argument in this case is currently in the Sandbox).

   Now assumption (i) is not too bad and it seems clear how to relax it with just a tad more effort. But assumption (ii) is way too strong to be of much use in practice.

   There are some evident things to try to work around this, such as slicing over a simplicial cofibrant replacement of $\pi_1^{(-)}(X)$. But while it is clear how to do this and similar variations on the side of the equivariant simplicial sets, it is not so clear (to me, at the moment) what to do on the side of the equivariant dgc-algebras.

   Alternatively, one could go step-by-step through Scull’s original article

   • Laura Scull, A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)

   and see if any of the proof steps break if one replaces $Orb(G)$ by $\Pi_G(X)$. My vague intuition is that the only relevant property of $Orb(G)$ which Scull’s proof really uses is that it is an EI-category, and that property is shared by $\Pi_G(X)$. But I haven’t checked in detail, and I am shying away from the tedious task.

   But if all fails, maybe I’ll dive into it.

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 21st 2022
   • (edited Jul 21st 2022)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexWhat exactly is tom Dieck’s fundamental category? The hyperlink in the Sandbox links to [[fundamental category]], which does not seem to have any relevant information. Added: I see, it's just the [[equivariant fundamental groupoid]].

   What exactly is tom Dieck’s fundamental category? The hyperlink in the Sandbox links to fundamental category, which does not seem to have any relevant information.

   Added: I see, it’s just the equivariant fundamental groupoid.

4. • CommentRowNumber4.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 21st 2022
   • (edited Jul 21st 2022)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexBy the way, have looked in the book _Rational Homotopy Theory II_ (2015), which develops the theory of Sullivan algebras and minimal models in the non-simply-connected case? It looks like some techniques could be relevant.

   By the way, have looked in the book Rational Homotopy Theory II (2015), which develops the theory of Sullivan algebras and minimal models in the non-simply-connected case? It looks like some techniques could be relevant.

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 21st 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAnd concerning Scull's paper, she writes in the introduction: > to prove that under this model structure, there is a Quillen equivalence between the category of equivariant rational spaces which are 1-connected and of finite Q-type and the category of the diagrams of CDGAs with analogous restrictions (Section 5). So it appears that her proof is only for the simply connected case? And you want the non-simply-connected case?

   And concerning Scull’s paper, she writes in the introduction:

   to prove that under this model structure, there is a Quillen equivalence between the category of equivariant rational spaces which are 1-connected and of finite Q-type and the category of the diagrams of CDGAs with analogous restrictions (Section 5).

   So it appears that her proof is only for the simply connected case? And you want the non-simply-connected case?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2022
   • (edited Jul 21st 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexYes, so here is the bigger story: Without any conditions on connectivity, rationalization is "$\pi_1$-rationalization" as indicated [here](https://ncatlab.org/nlab/show/rationalization#PiOneRationalizationOfConnectedSpaces), namely equivalently localization at * maps which are isos on the 1-truncation and on all rationalized $\pi_{n \geq 2}$ * maps which are rational isos fiberwise over the 1-truncation unit $X \xrightarrow{ \eta^{[-]_1}_X } [X]_1$. So for general connected spaces $X$, rationalization is the operation that takes the universal cover fiber sequence $$ \widehat{X} \longrightarrow X \overset{ \eta^{[-]_1}_X }{\longrightarrow} B \pi_1(X) $$ and rationalizes the simply connected fiber space $\widehat{X}$ while leaving the 1-truncated base $B \pi_1(X)$ intact. This fiber sequence and its rationalization then automatically exhibit the residual $\pi_1(X)$-action on the (rationalized) fiber, whence "$\pi_1$-rationalization". By functoriality of $\pi_1$-rationalization, it immediately extends to $G$-equivariant homotopy types $\mathcal{X}$, given by $\infty$-presheaves over $Orb(G)$. By the equivalence/definition $\Pi_G(\mathcal{X}) \simeq Orb(G)_{/[\mathcal{X}]_1}$, it follows that the $\pi_1$-rationalization of any $\mathcal{X}$ is the fiberwise rationalization of $$ \big( \mathcal{X} ,\, \eta^{[-]_1}_{\mathcal{X}} \big) \;\in\; \big( G Grpd_\infty \big)_{\big/[\mathcal{X}]_1} \;\simeq\; \Big( PSh_\infty\big( Orb(G) \big) \Big)_{\big/[\mathcal{X}]_1} \;\simeq\; PSh_\infty\big( \Pi_G(X) \big) \,. $$ Chasing through this equivalence, the original equivariant homotopy type $\mathcal{X}$ is incarnated as a simply connected object of $PSh_\infty\big( \Pi_G(X) \big)$ and hence the generalization of Scull's construction to the site $\Pi_G(X)$ would be the desired equivariant $\pi_1$-rationalization in its dgc-algebra incarnation.

   Yes, so here is the bigger story:

   Without any conditions on connectivity, rationalization is “$\pi_1$-rationalization” as indicated here, namely equivalently localization at

   • maps which are isos on the 1-truncation and on all rationalized $\pi_{n \geq 2}$

   • maps which are rational isos fiberwise over the 1-truncation unit $X \xrightarrow{ \eta^{[-]_1}_X } [X]_1$.

   So for general connected spaces $X$, rationalization is the operation that takes the universal cover fiber sequence

   $$\widehat{X} \longrightarrow X \overset{ \eta^{[-]_1}_X }{\longrightarrow} B \pi_1(X)$$

   and rationalizes the simply connected fiber space $\widehat{X}$ while leaving the 1-truncated base $B \pi_1(X)$ intact. This fiber sequence and its rationalization then automatically exhibit the residual $\pi_1(X)$-action on the (rationalized) fiber, whence “$\pi_1$-rationalization”.

   By functoriality of $\pi_1$-rationalization, it immediately extends to $G$-equivariant homotopy types $\mathcal{X}$, given by $\infty$-presheaves over $Orb(G)$. By the equivalence/definition $\Pi_G(\mathcal{X}) \simeq Orb(G)_{/[\mathcal{X}]_1}$, it follows that the $\pi_1$-rationalization of any $\mathcal{X}$ is the fiberwise rationalization of

   $$\big( \mathcal{X} ,\, \eta^{[-]_1}_{\mathcal{X}} \big) \;\in\; \big( G Grpd_\infty \big)_{\big/[\mathcal{X}]_1} \;\simeq\; \Big( PSh_\infty\big( Orb(G) \big) \Big)_{\big/[\mathcal{X}]_1} \;\simeq\; PSh_\infty\big( \Pi_G(X) \big) \,.$$

   Chasing through this equivalence, the original equivariant homotopy type $\mathcal{X}$ is incarnated as a simply connected object of $PSh_\infty\big( \Pi_G(X) \big)$ and hence the generalization of Scull’s construction to the site $\Pi_G(X)$ would be the desired equivariant $\pi_1$-rationalization in its dgc-algebra incarnation.

7. • CommentRowNumber7.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 22nd 2022
   • (edited Jul 22nd 2022)
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexThis question is probably too naive, but the functor $i^*$ (in the [[Sandbox]]) also has a left adjoint $i_!$. So why not transfer along $i^*$ as a right adjoint? The right transfer has the same weak equivalences as the left transfer…

   This question is probably too naive, but the functor $i^*$ (in the Sandbox) also has a left adjoint $i_!$.

   So why not transfer along $i^*$ as a right adjoint? The right transfer has the same weak equivalences as the left transfer…

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 22nd 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI was thinking about this but didn't type up a note: When I right-transfer along $i^\ast$ with the same but dual argument, then I seem to dually need the condition that $Maps\big(\pi_1(X), Fib\big) \subset Fib$, which again leads to the overly strong requirement that $\pi_1(X) \in sPSh\big( Orb(G) \big)_{proj}$ be cofibrant. So nothing seems to be gained by switching between left and right transfer. Of course, in both cases the problem arises because I am not just checking that the (co-)anodyne maps are weak equivalences, but actually that their inages under $i^\ast$ are acyclic (co-)fibrations. This is much stronger than necessary, but more readily checked. So to salvage the argument, one should probably find another proof strategy altogether which just checks that the (co-)anodyne maps are weak equivalences. But I couldn't see yet how to do this.

   I was thinking about this but didn’t type up a note:

   When I right-transfer along $i^\ast$ with the same but dual argument, then I seem to dually need the condition that $Maps\big(\pi_1(X), Fib\big) \subset Fib$, which again leads to the overly strong requirement that $\pi_1(X) \in sPSh\big( Orb(G) \big)_{proj}$ be cofibrant. So nothing seems to be gained by switching between left and right transfer.

   Of course, in both cases the problem arises because I am not just checking that the (co-)anodyne maps are weak equivalences, but actually that their inages under $i^\ast$ are acyclic (co-)fibrations. This is much stronger than necessary, but more readily checked.

   So to salvage the argument, one should probably find another proof strategy altogether which just checks that the (co-)anodyne maps are weak equivalences. But I couldn’t see yet how to do this.

9. • CommentRowNumber9.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 22nd 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRe #8: Again, may be a bit too naive, but it does seem to me that showing [[anodyne maps]] to be weak equivalence is quite easy here. What's more, the argument is not specific to this setting, and works for transferring [[projective model structures]] along any right adjoint functor of the form $$i^*: sPSh(D) \to sPSh(C)$$ induced by an arbitrary functor of categories $$i: C\to D.$$ To this end, I am going to show that applying the left adjoint functor $i_!$ to generating projective acyclic cofibrations of $sPSh(C)$ produces projective acyclic cofibrations in $sPSh(D)$. (In fact, producing _injective_ acyclic cofibrations would already be perfectly sufficient.) The existence of the transferred model structure on $sPSh(C)$ then follows formally from the [[Kan transfer theorem]]. The generating projective acyclic cofibrations of $sPSh(C)$ are given by tensoring a horn inclusion with a representable presheaf of some object $X\in C$. The left adjoint functor $i_!$ sends these to the tensoring of the horn inclusion with the representable presheaf of $i(X)\in D$. But the latter is clearly a projective acyclic cofibration in $sPSh(D)$.

   Re #8: Again, may be a bit too naive, but it does seem to me that showing anodyne maps to be weak equivalence is quite easy here.

   What’s more, the argument is not specific to this setting, and works for transferring projective model structures along any right adjoint functor of the form

   $$i^*: sPSh(D) \to sPSh(C)$$

   induced by an arbitrary functor of categories

   $$i: C\to D.$$

   To this end, I am going to show that applying the left adjoint functor $i_!$ to generating projective acyclic cofibrations of $sPSh(C)$ produces projective acyclic cofibrations in $sPSh(D)$. (In fact, producing injective acyclic cofibrations would already be perfectly sufficient.)

   The existence of the transferred model structure on $sPSh(C)$ then follows formally from the Kan transfer theorem.

   The generating projective acyclic cofibrations of $sPSh(C)$ are given by tensoring a horn inclusion with a representable presheaf of some object $X\in C$.

   The left adjoint functor $i_!$ sends these to the tensoring of the horn inclusion with the representable presheaf of $i(X)\in D$.

   But the latter is clearly a projective acyclic cofibration in $sPSh(D)$.

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for insisting. Sorry if I am being slow, there must be something I keep misunderstanding in what you say. Don't we need to show (for right transfer, okay) that $$ f \;\;⧄\;\; (i^\ast)^{-1}(Fib) \;\;\;\;\; \Rightarrow \;\;\;\;\; i^\ast(f) \;\in\; W \,. $$ ? How does that follow from what you just said? Could you maybe point me to the number of the theorem in the entry whose assumptions you are checking? Sorry if I am missing the obvious. I understand that $i_!$ is left Quillen between projective model structures of simplicial presheaves, which is what you are emphasizing now, but here we are considering one projective and one transferred model structure, and the transferred one is not itself generally going to be the respective projective structure. Ah, wait, that's maybe the trick: In my example, with $i \,\colon\, Orb(G) \to \Pi_G(X)$, the right-transferred model structure *is* the projective one, sice $i$ is essentially surjective. So as soon as that is the case, we may forget about the transfer theorem and just consider the induced Quillen adjunction along a morphism of sites. (I am writing this somewhat on the run, have to go pick up my girls. Will be back later. )

    Thanks for insisting. Sorry if I am being slow, there must be something I keep misunderstanding in what you say.

    Don’t we need to show (for right transfer, okay) that

    $$f \;\;⧄\;\; (i^\ast)^{-1}(Fib) \;\;\;\;\; \Rightarrow \;\;\;\;\; i^\ast(f) \;\in\; W \,.$$

    ?

    How does that follow from what you just said? Could you maybe point me to the number of the theorem in the entry whose assumptions you are checking?

    Sorry if I am missing the obvious. I understand that $i_!$ is left Quillen between projective model structures of simplicial presheaves, which is what you are emphasizing now, but here we are considering one projective and one transferred model structure, and the transferred one is not itself generally going to be the respective projective structure.

    Ah, wait, that’s maybe the trick: In my example, with $i \,\colon\, Orb(G) \to \Pi_G(X)$, the right-transferred model structure is the projective one, sice $i$ is essentially surjective. So as soon as that is the case, we may forget about the transfer theorem and just consider the induced Quillen adjunction along a morphism of sites.

    (I am writing this somewhat on the run, have to go pick up my girls. Will be back later. )

11. • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 22nd 2022
    • (edited Jul 22nd 2022)
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexI am using the Kan transfer theorem, somewhat implicitly stated here: <https://ncatlab.org/nlab/show/transferred+model+structure#ConstructingFactorizationsForRightTransfer> The condition in your displayed formula follows from the fact that anodyne maps are weak equivalences, so there is no need to prove it separately. You are correct in pointing out that the transferred model structure has strictly fewer cofibrations in general. The generating cofibrations are given by tensoring representable presheaves of objects in the essential image of $i$ with simplicial boundary inclusions. But this does not prevent the existence of the transferred model structure: all what is needed is to verify that anodyne maps, more precisely, transfinite compositions of cobase changes of generating acyclic cofibrations, are weak equivalences. And the generating acyclic cofibrations are given by tensoring representable presheaves of objects in the essential image of $i$ with simplicial horn inclusions. All such maps are projective acyclic cofibrations, and the latter class is closed under cobase changes and transfinite compositions, which means that all anodyne maps are projective acyclic cofibrations. The converse is false in general: not every projective acyclic cofibration is an anodyne map for the transferred model structure. This property is not needed anyway.

    I am using the Kan transfer theorem, somewhat implicitly stated here: https://ncatlab.org/nlab/show/transferred+model+structure#ConstructingFactorizationsForRightTransfer

    The condition in your displayed formula follows from the fact that anodyne maps are weak equivalences, so there is no need to prove it separately.

    You are correct in pointing out that the transferred model structure has strictly fewer cofibrations in general. The generating cofibrations are given by tensoring representable presheaves of objects in the essential image of $i$ with simplicial boundary inclusions.

    But this does not prevent the existence of the transferred model structure: all what is needed is to verify that anodyne maps, more precisely, transfinite compositions of cobase changes of generating acyclic cofibrations, are weak equivalences.

    And the generating acyclic cofibrations are given by tensoring representable presheaves of objects in the essential image of $i$ with simplicial horn inclusions.

    All such maps are projective acyclic cofibrations, and the latter class is closed under cobase changes and transfinite compositions, which means that all anodyne maps are projective acyclic cofibrations.

    The converse is false in general: not every projective acyclic cofibration is an anodyne map for the transferred model structure. This property is not needed anyway.

12. • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for elaborating. I admit that I still feel unsure what you mean by saying > The condition in your displayed formula follows from the fact that anodyne maps are weak equivalences because, to my mind, that only holds for anodyne maps in a model structure, which is at this point what is yet to be established. (I apologize if I am being stupid here, it's also past my bedtime :-) $\,$ But in any case, we agree that the $i^\ast$ that I am wondering about gives right transfer from $Orb(G)$-simplicial sets to $\Pi_G(X)$-simplicial sets. Now with that in hand, let's look at this commuting diagram of right adjoint functors: $$ \array{ PSh \Big( \Pi_G(X) ,\, \big(dgcAlg_{\mathbb{Q}}^{\geq 0} \big)^{op} \Big)_{traScu} & \overset {\;\;\; exp \;\;\;} {\longrightarrow} & PSh \big( \Pi_G(X) \,, sSet \big)_{proj} \\ \mathllap{ {}^{i^\ast} } \big\downarrow && \big\downarrow \mathrlap{ {}^{i^\ast} } \\ PSh \Big( Orb(G) ,\, \big(dgcAlg_{\mathbb{Q}}^{\geq 0} \big)^{op} \Big)_{Scu} & \overset {\;\;\; exp \;\;\;} {\longrightarrow} & PSh \big( Orb(G) \,, sSet \big)_{proj} } \,, $$ Here at the bottom is the [[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]] due to Scull. My aim is to see that also the top functor is right Quillen, for the model structure right-transferred along $i^\ast$ in the top left. Now it looks like this follows as soon as the transferred model structure "traScu" actually exists: Because, by the transfer on the right, the top functor preserves fibrations iff its composite with the right functor does, which by commutativity of the diagram is the case iff the left+bottom composite is right Quillen, which is the case by Scull's result and if the $i^\ast$-transferred structure exists on the left. So that's good. But I still need to see that "traScu" actually exists...

    Thanks for elaborating. I admit that I still feel unsure what you mean by saying

    The condition in your displayed formula follows from the fact that anodyne maps are weak equivalences

    because, to my mind, that only holds for anodyne maps in a model structure, which is at this point what is yet to be established. (I apologize if I am being stupid here, it’s also past my bedtime :-)

    $\,$

    But in any case, we agree that the $i^\ast$ that I am wondering about gives right transfer from $Orb(G)$-simplicial sets to $\Pi_G(X)$-simplicial sets.

    Now with that in hand, let’s look at this commuting diagram of right adjoint functors:

    $$\array{ PSh \Big( \Pi_G(X) ,\, \big(dgcAlg_{\mathbb{Q}}^{\geq 0} \big)^{op} \Big)_{traScu} & \overset {\;\;\; exp \;\;\;} {\longrightarrow} & PSh \big( \Pi_G(X) \,, sSet \big)_{proj} \\ \mathllap{ {}^{i^\ast} } \big\downarrow && \big\downarrow \mathrlap{ {}^{i^\ast} } \\ PSh \Big( Orb(G) ,\, \big(dgcAlg_{\mathbb{Q}}^{\geq 0} \big)^{op} \Big)_{Scu} & \overset {\;\;\; exp \;\;\;} {\longrightarrow} & PSh \big( Orb(G) \,, sSet \big)_{proj} } \,,$$

    Here at the bottom is the Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras due to Scull.

    My aim is to see that also the top functor is right Quillen, for the model structure right-transferred along $i^\ast$ in the top left.

    Now it looks like this follows as soon as the transferred model structure “traScu” actually exists: Because, by the transfer on the right, the top functor preserves fibrations iff its composite with the right functor does, which by commutativity of the diagram is the case iff the left+bottom composite is right Quillen, which is the case by Scull’s result and if the $i^\ast$-transferred structure exists on the left.

    So that’s good. But I still need to see that “traScu” actually exists…

13. • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 22nd 2022
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItex> because, to my mind, that only holds for anodyne maps in a model structure, which is at this point what is yet to be established. (I apologize if I am being stupid here, it’s also past my bedtime :-) Yes, so the point is that first you show that anodyne maps (defined as retracts of transfinite compositions of cobase changes of candidates for generating acyclic cofibrations) are actually weak equivalences, and then you conclude by invoking the small object argument, which tells you that the class of anodyne maps coincides with the class of maps with a left lifting property with respect to the transferred fibrations. Concerning the model structure traScu: its existence is proved in the same way. Specifically, we apply the Kan recognition theorem again, so the whole claim boils down to showing that transfinite compositions of cobase changes of (candidates for) generating acyclic cofibrations are contained in the transferred weak equivalences. The candidates for generating acyclic cofibrations are obtained by tensoring the representable presheaf of an object in the essential image of $i$ with a generating acyclic cofibration for rational dgcas. The latter is a projective acyclic cofibration on the projective model structure on presheaves on Π_G(X) valued in rational dgcas. Therefore, candidates for generating acyclic cofibrations are contained within the weakly saturated class of projective acyclic cofibrations, which themselves are contained within the class of transferred weak equivalences. Therefore, the condition for the Kan recognition theorem is satisfied and the model structure traScu exists.

    because, to my mind, that only holds for anodyne maps in a model structure, which is at this point what is yet to be established. (I apologize if I am being stupid here, it’s also past my bedtime :-)

    Yes, so the point is that first you show that anodyne maps (defined as retracts of transfinite compositions of cobase changes of candidates for generating acyclic cofibrations) are actually weak equivalences, and then you conclude by invoking the small object argument, which tells you that the class of anodyne maps coincides with the class of maps with a left lifting property with respect to the transferred fibrations.

    Concerning the model structure traScu: its existence is proved in the same way.

    Specifically, we apply the Kan recognition theorem again, so the whole claim boils down to showing that transfinite compositions of cobase changes of (candidates for) generating acyclic cofibrations are contained in the transferred weak equivalences.

    The candidates for generating acyclic cofibrations are obtained by tensoring the representable presheaf of an object in the essential image of $i$ with a generating acyclic cofibration for rational dgcas. The latter is a projective acyclic cofibration on the projective model structure on presheaves on Π_G(X) valued in rational dgcas. Therefore, candidates for generating acyclic cofibrations are contained within the weakly saturated class of projective acyclic cofibrations, which themselves are contained within the class of transferred weak equivalences.

    Therefore, the condition for the Kan recognition theorem is satisfied and the model structure traScu exists.

14. • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJul 23rd 2022
    • (edited Jul 23rd 2022)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks Dmitri!! This would be great if this holds. I'll think about it when I find the time (I am not doubting it, but still need to absorb the argument for myself and have no leisure right now. Right now we are about to hit the road to leave for a vacation. I'll have no Wifi for the next two weeks, but maybe I'll be able to hack myself into the Matrix via phone from time to time.) By the way, once this "fundamental theorem of dgc-algebraic $\pi_1$-rational equivariant homotopy theory" is established, the only remaining item I need for happiness (namely to have a fully satisfactory general TED cohomology theory) is the existence (and uniqueness) of *relative* equivariant minimal models (that's another open MO question of mine: [MO:q/373819](https://mathoverflow.net/q/373819/381)). Proving this should be a straightforward if tedious variation of the Postnikov-decomposition argument which proves the non-equivariant version. If anyone knows or can make appear such a proof in citable form, we'll happily cite it.

    Thanks Dmitri!!

    This would be great if this holds. I’ll think about it when I find the time

    (I am not doubting it, but still need to absorb the argument for myself and have no leisure right now. Right now we are about to hit the road to leave for a vacation. I’ll have no Wifi for the next two weeks, but maybe I’ll be able to hack myself into the Matrix via phone from time to time.)

    By the way, once this “fundamental theorem of dgc-algebraic $\pi_1$-rational equivariant homotopy theory” is established, the only remaining item I need for happiness (namely to have a fully satisfactory general TED cohomology theory) is the existence (and uniqueness) of relative equivariant minimal models (that’s another open MO question of mine: MO:q/373819).

    Proving this should be a straightforward if tedious variation of the Postnikov-decomposition argument which proves the non-equivariant version. If anyone knows or can make appear such a proof in citable form, we’ll happily cite it.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJul 28th 2022
    • (edited Jul 28th 2022)
    • PermaLink
    Author: Urs
    Format: MarkdownItexFinally I have found a quiet moment to think about this. I understand now what you have been saying above: To make use of the projective model structure over the codomain of $i$ as an auxiliary tool for getting control over the saturation of $(i_!)\big( Cof \cap W\big)$, using that we know $i_!(J)$ and that $i^*$ preserves projective weak equivalences.

    Finally I have found a quiet moment to think about this. I understand now what you have been saying above: To make use of the projective model structure over the codomain of $i$ as an auxiliary tool for getting control over the saturation of $(i_!)\big( Cof \cap W\big)$, using that we know $i_!(J)$ and that $i^*$ preserves projective weak equivalences.

16. • CommentRowNumber16.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJul 29th 2022
    • PermaLink
    Author: Dmitri Pavlov
    Format: MarkdownItexRe #15: Yes, exactly. (Did you resolve your question about the existence of the projective structure on presheaves valued in the opposite category of cdgas?) Re #14: Concerning the existence (and uniqueness) of relative equivariant minimal models: it seems to me that the proof can be obtained by amalgamating the proof for relative minimal models and the proof for equivariant minimal models. Is there a specific step in the proof for which you are anticipating difficulties?

    Re #15: Yes, exactly. (Did you resolve your question about the existence of the projective structure on presheaves valued in the opposite category of cdgas?)

    Re #14: Concerning the existence (and uniqueness) of relative equivariant minimal models: it seems to me that the proof can be obtained by amalgamating the proof for relative minimal models and the proof for equivariant minimal models. Is there a specific step in the proof for which you are anticipating difficulties?

17. • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJul 29th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexOn the first point: Yes, I think so: For a moment I thought we need the projective model structure on functors with values in $dgcAlg^{op}$, and I didn't see how to apply the usual existence results. But we can instead use the opposite of the projective model structure on functors (on the opposite site) with values in $dcgAlg$ itself (in fact, that makes manifest the desired acyclic cofibrancy of elements in $i_!(J)$), and since $dgcAlg$ is cofibrantly generated and (I suppose) locally presentable, it follows that this exists. On the second point: As I said, I expect it to be straightforward, but somewhat tedious. (It's "bureaucratically" tedious, I expect, to write this out cleanly. But maybe I am just procrastinating... :-)

    On the first point: Yes, I think so:

    For a moment I thought we need the projective model structure on functors with values in $dgcAlg^{op}$, and I didn’t see how to apply the usual existence results. But we can instead use the opposite of the projective model structure on functors (on the opposite site) with values in $dcgAlg$ itself (in fact, that makes manifest the desired acyclic cofibrancy of elements in $i_!(J)$), and since $dgcAlg$ is cofibrantly generated and (I suppose) locally presentable, it follows that this exists.

    On the second point:

    As I said, I expect it to be straightforward, but somewhat tedious. (It’s “bureaucratically” tedious, I expect, to write this out cleanly. But maybe I am just procrastinating… :-)

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexFinally back from vacation and with more than a phone in hand. Just thinking out loud: For the fact that $dgcAlg_{proj}$ (rational, in non-negative degrees) is locally presentable I would point to Prop. 3.6 in Shipley's [arXiv:math/0209215](https://arxiv.org/abs/math/0209215) and for cofibrant generation to p. 6 of Hess's [arXiv:math/0604626](https://arxiv.org/abs/math/0604626). Years ago I had written a remark [here](https://ncatlab.org/nlab/show/model+structure+on+dg-algebras#ForUnboundedDGAlgebrasDefinition) that an argument for $dgcAlg_{proj}$ (rational and unbounded) being combinatorial is in Toën-Vezzosi's [arXiv:math/0404373](https://arxiv.org/abs/math/0404373), but it's not made very explicit in there.

    Finally back from vacation and with more than a phone in hand.

    Just thinking out loud:

    For the fact that $dgcAlg_{proj}$ (rational, in non-negative degrees) is locally presentable I would point to Prop. 3.6 in Shipley’s arXiv:math/0209215

    and for cofibrant generation to p. 6 of Hess’s arXiv:math/0604626.

    Years ago I had written a remark here that an argument for $dgcAlg_{proj}$ (rational and unbounded) being combinatorial is in Toën-Vezzosi’s arXiv:math/0404373, but it’s not made very explicit in there.

19. • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeAug 13th 2022
    • (edited Aug 13th 2022)
    • PermaLink
    Author: Urs
    Format: MarkdownItexJust to say that I have finally typed up a proof of the transfer that I was after. It follows your (Dmitri's) strategy above, but I ended up needing the left transfer after all, and the fine detail is slightly different. This is now in a draft file currently in my Dropbox [here](https://www.dropbox.com/s/x7usdp24g2fti8d/Twisted_Equivariant_Character.pdf?dl=0) -- the transfer is Prop. 3.42. Comments are welcome. Please just note that it's really not more than a draft with the hope of summing up the above discussion. The file won't stay online for very long.

    Just to say that I have finally typed up a proof of the transfer that I was after.

    It follows your (Dmitri’s) strategy above, but I ended up needing the left transfer after all, and the fine detail is slightly different.

    This is now in a draft file currently in my Dropbox here – the transfer is Prop. 3.42.

    Comments are welcome. Please just note that it’s really not more than a draft with the hope of summing up the above discussion. The file won’t stay online for very long.