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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeOct 29th 2009
   • PermaLink
   Author: Urs
   Format: MarkdownI created at [[equivariant cohomology]] separate subsections for, so far, Borel equivariant and Bredon equivariant cohomology. At [[Bredon cohomology]] I added a sentence about the coefficient objects.

   I created at equivariant cohomology separate subsections for, so far, Borel equivariant and Bredon equivariant cohomology.

   At Bredon cohomology I added a sentence about the coefficient objects.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 23rd 2009
   • (edited Dec 23rd 2009)
   • PermaLink
   Author: Urs
   Format: Markdownadded a reference (lecture notes on basics) to [[equivariant cohomology]]

   added a reference (lecture notes on basics) to equivariant cohomology

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 23rd 2010
   • (edited Jan 23rd 2010)
   • PermaLink
   Author: Urs
   Format: Markdownworked on the entry [[equivariant cohomology]] * mainly I pasted in Mike's latest blog comment <a href="http://golem.ph.utexas.edu/category/2010/01/the_sacred_and_the_profane.html#c031280">here</a> on how to think of Bredon equivariant cohomology as being [[cohomology]] of an (oo,1)-topos * in the course of this I gave the entry three main sections 1. Borel equivariant cohomology 2. Multiplicative equivarian cohomology (not sure what a better name is, this is the refinement of Borel EC used when E-oo ring spectra and formal group laws etc are around) 3. Bredon equivariant cohomology To go with this, I expanded the Idea-section , trying to give an overview of the material to follow.

   worked on the entry equivariant cohomology

   • mainly I pasted in Mike's latest blog comment here on how to think of Bredon equivariant cohomology as being cohomology of an (oo,1)-topos

   • in the course of this I gave the entry three main sections

     1. Borel equivariant cohomology

     2. Multiplicative equivarian cohomology (not sure what a better name is, this is the refinement of Borel EC used when E-oo ring spectra and formal group laws etc are around)

     3. Bredon equivariant cohomology

   To go with this, I expanded the Idea-section , trying to give an overview of the material to follow.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 26th 2010
   • PermaLink
   Author: Urs
   Format: Markdowntwo more references at [[equivariant stable homotopy theory]] from the blog discussion (hopefully more complete list later...)

   two more references at equivariant stable homotopy theory from the blog discussion

   (hopefully more complete list later...)

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 14th 2014
   • (edited Apr 14th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have completely rewritten the Idea-section at _[[equivariant cohomology]]_. The previous version left it a conceptual mystery of why Bredon cohomology appears, apart from the after-the-fact observation that somehow it "behaves better". I think to understand the situation conceptually, two ingredients are crucial: 1. [[Elmendorf's theorem]], which clarifies why the orbit category appears when talking about [[G-spaces]]; 1. the observation that G-equivariant cohomology for $G$-acting on both domain and coeffcients is just the $G$-homotopy fixed points of the non-equivariant cocycles spaces $\mathbf{H}(X,A)$ equipped with their inherited $G$-conjugation action. The new version of the Idea-section now reads as follows. I think it contains everything that was good about the previous Idea-section and hence is an actual improvement; but of course if anyone feels that some of the previous version which I removed needs to be retained, let me know. *** [ text extracted from entry, see [[equivariant cohomology|there]] for current version ] What is called _equivariant cohomology_ is [[cohomology]] in the presence of and taking into account [[group]]-[[actions]] (and [[∞-group]] [[∞-actions]]) both on the domain space and on the [[coefficients]]. In the simplest situation the group action on the [[coefficients]] is trivial and one is dealing with cohomology of [[spaces]] $X$ that are equipped with a $G$-action ([[G-spaces]]). Here a class in equivariant cohomology is an ordinary cohomology class $c \in \mathbf{H}(X,A)$ on $X$, together with ([[coherence|coherently]]) an [[equivalence]] $c \simeq g^\ast c$ for each [[generalized element]] $g$ of $G$, hence is a cocycle which is $G$- _invariant_ , but up to coherent equivalences. Diagrammatically this means that where a non-equivariant [[cocycle]] on $X$ with [[coefficients]] in $A$ is just a map $c \colon X \to A$ (see at _[[cohomology]]_) an equivariant cocycle is a natural system of [[diagrams]] of the form $$ \array{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{\mathllap{\rho_X(g)}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{=}} \\ X &\underset{c}{\longrightarrow}& A } $$ Standard examples of this kind of equivariant cocycles are traditional [[equivariant bundles]]. This kind of equivariant cocycle is the same as just a single cocycle on the [[homotopy quotient]] $X//G$. Since a standard model for homotopy quotients is the [[Borel construction]], this kind of equivariant cohomology with trivial $G$-action on the coefficients is also called **Borel equivariant cohomology**. In general the group $G$ also acts on the [[coefficients]] $A$, and then an equivariant cocycle is a map $c \;\colon\; X \to A$ which is invariant, up to equivalence, under the _joint_ action of $G$ on base space and coefficients. Diagrammatically this is a natural system of diagrams of the form $$ \array{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{\mathllap{\rho_X(g)}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\rho_A(G)}} \\ X &\underset{c}{\longrightarrow}& A } \,. $$ More concisely this means that an equivariant cocycle is a [[homotopy fixed point]] of the non-equivariant [[cocycle]] [[∞-groupoid]] $\mathbf{H}(X,A)$: $$ H^G(X,A) \simeq \pi_0(\mathbf{H}(X,G)^G) \,. $$ By the discussion at _[[∞-action]]_ one may phrase this abstractly as follows: spaces and coefficients with $G$-[[∞-action]] are objects in the [[slice (∞,1)-topos]] of the ambient [[(∞,1)-topos]] $\mathbf{H}$, and $G$-equivariant cohomology is the [[dependent product]] [[base change]] of [[internal homs]] in the slice over $\mathbf{B}G$: $$ H^G(X,A) \simeq \pi_0 \left( \underset{\mathbf{B}G}{\prod} [X,A] \right) \,. $$ Hence equivariant cohomology is a natural generalization of [[group cohomology]], to which it reduces when the base space is a point. If here the cohomology is to be $\mathbb{Z}$-graded this means that the coefficients $A$ are the stages in a [[spectrum object]] in the $\mathbf{H}_{/\mathbf{B}G}$, which is a [[spectrum with G-action]]. These are hence the [[coefficients]] for equivariant [[generalized (Eilenberg-Steenrod) cohomology]]. (More generally one considers [[genuine G-spectra]] in [[equivariant stable homotopy theory]], but as coefficients in cohomology they appear only through their underlying [[spectra with G-action]] as above.) Among the simplest non-trivial example of this $G$-equivariance with joint action on domain and coefficients is [[real-oriented cohomology|real oriented]] [[generalized cohomology theory]] such as notably [[KR-theory]], which is equivariance with respect to a $\mathbb{Z}_2$-action. This appears notably in [[type II string theory]] on [[orientifold]] backgrounds, where the extra group action on the coefficients is exhibited by what is called the [[worldsheet parity operator]]. The word "[[orientifold]]" is modeled on that of "[[orbifold]]" to reflect precisely this extra action (on coefficients) of non-Borel $\mathbb{Z}_2$-equivariant cohomology. Various explicit models for this equivariance are known and discussed in the literature. Notably by [[Elmendorf's theorem]], the [[homotopy theory]] of [[G-spaces]] is equivalently the [[(∞,1)-category of (∞,1)-presheaves]] over the [[orbit category]] $Orb_G$. This way equivariant cohomology may be expressed in terms of coefficient systems which are presheaves of abelian groups over the orbit category. This perspective is called **[[Bredon cohomology]]**.

   I have completely rewritten the Idea-section at equivariant cohomology.

   The previous version left it a conceptual mystery of why Bredon cohomology appears, apart from the after-the-fact observation that somehow it “behaves better”.

   I think to understand the situation conceptually, two ingredients are crucial:

   1. Elmendorf’s theorem, which clarifies why the orbit category appears when talking about G-spaces;

   2. the observation that G-equivariant cohomology for $G$-acting on both domain and coeffcients is just the $G$-homotopy fixed points of the non-equivariant cocycles spaces $\mathbf{H}(X,A)$ equipped with their inherited $G$-conjugation action.

   The new version of the Idea-section now reads as follows. I think it contains everything that was good about the previous Idea-section and hence is an actual improvement; but of course if anyone feels that some of the previous version which I removed needs to be retained, let me know.

   ---

   [ text extracted from entry, see there for current version ]

   What is called equivariant cohomology is cohomology in the presence of and taking into account group-actions (and ∞-group ∞-actions) both on the domain space and on the coefficients.

   In the simplest situation the group action on the coefficients is trivial and one is dealing with cohomology of spaces $X$ that are equipped with a $G$-action (G-spaces). Here a class in equivariant cohomology is an ordinary cohomology class $c \in \mathbf{H}(X,A)$ on $X$, together with (coherently) an equivalence $c \simeq g^\ast c$ for each generalized element $g$ of $G$, hence is a cocycle which is $G$- invariant , but up to coherent equivalences. Diagrammatically this means that where a non-equivariant cocycle on $X$ with coefficients in $A$ is just a map $c \colon X \to A$ (see at cohomology) an equivariant cocycle is a natural system of diagrams of the form

   $$\array{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{\mathllap{\rho_X(g)}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{=}} \\ X &\underset{c}{\longrightarrow}& A }$$

   Standard examples of this kind of equivariant cocycles are traditional equivariant bundles. This kind of equivariant cocycle is the same as just a single cocycle on the homotopy quotient $X//G$. Since a standard model for homotopy quotients is the Borel construction, this kind of equivariant cohomology with trivial $G$-action on the coefficients is also called Borel equivariant cohomology.

   In general the group $G$ also acts on the coefficients $A$, and then an equivariant cocycle is a map $c \;\colon\; X \to A$ which is invariant, up to equivalence, under the joint action of $G$ on base space and coefficients. Diagrammatically this is a natural system of diagrams of the form

   $$\array{ X &\stackrel{c}{\longrightarrow}& A \\ {}^{\mathllap{\rho_X(g)}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\rho_A(G)}} \\ X &\underset{c}{\longrightarrow}& A } \,.$$

   More concisely this means that an equivariant cocycle is a homotopy fixed point of the non-equivariant cocycle ∞-groupoid $\mathbf{H}(X,A)$:

   $$H^G(X,A) \simeq \pi_0(\mathbf{H}(X,G)^G) \,.$$

   By the discussion at ∞-action one may phrase this abstractly as follows: spaces and coefficients with $G$-∞-action are objects in the slice (∞,1)-topos of the ambient (∞,1)-topos $\mathbf{H}$, and $G$-equivariant cohomology is the dependent product base change of internal homs in the slice over $\mathbf{B}G$:

   $$H^G(X,A) \simeq \pi_0 \left( \underset{\mathbf{B}G}{\prod} [X,A] \right) \,.$$

   Hence equivariant cohomology is a natural generalization of group cohomology, to which it reduces when the base space is a point.

   If here the cohomology is to be $\mathbb{Z}$-graded this means that the coefficients $A$ are the stages in a spectrum object in the $\mathbf{H}_{/\mathbf{B}G}$, which is a spectrum with G-action. These are hence the coefficients for equivariant generalized (Eilenberg-Steenrod) cohomology. (More generally one considers genuine G-spectra in equivariant stable homotopy theory, but as coefficients in cohomology they appear only through their underlying spectra with G-action as above.)

   Among the simplest non-trivial example of this $G$-equivariance with joint action on domain and coefficients is real oriented generalized cohomology theory such as notably KR-theory, which is equivariance with respect to a $\mathbb{Z}_2$-action. This appears notably in type II string theory on orientifold backgrounds, where the extra group action on the coefficients is exhibited by what is called the worldsheet parity operator. The word “orientifold” is modeled on that of “orbifold” to reflect precisely this extra action (on coefficients) of non-Borel $\mathbb{Z}_2$-equivariant cohomology.

   Various explicit models for this equivariance are known and discussed in the literature. Notably by Elmendorf’s theorem, the homotopy theory of G-spaces is equivalently the (∞,1)-category of (∞,1)-presheaves over the orbit category $Orb_G$. This way equivariant cohomology may be expressed in terms of coefficient systems which are presheaves of abelian groups over the orbit category. This perspective is called Bredon cohomology.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 14th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexmade to further additions to the Idea-section 1. a remark on how the fact that general equivariant reduces to Borel equivariance if the coefficients have trivial action is a special case of the [[projection formula]] in [[Wirthmüller contexts]]; 1. a paragraph on the "third case" where the action on the domain space is trivial but on the coefficients it's not (leading to ordinary cohomology with fixed-point coefficients) Summed this up with a little table at the end: | Borel equivariant cohomology | $\leftarrow$ | general equivariant (Bredon) cohomology | $\rightarrow$ | non-equivariant [[cohomology]] with [[homotopy fixed point]] [[coefficients]] | |---|---|---|---|---| | $\mathbf{H}(X_G, A)$ | trivial action on coefficients | $[X,A]^G$ | trivial action on domain space | $\mathbf{H}(X, A^G)$ |

   made to further additions to the Idea-section

   1. a remark on how the fact that general equivariant reduces to Borel equivariance if the coefficients have trivial action is a special case of the projection formula in Wirthmüller contexts;

   2. a paragraph on the “third case” where the action on the domain space is trivial but on the coefficients it’s not (leading to ordinary cohomology with fixed-point coefficients)

   Summed this up with a little table at the end:

   Borel equivariant cohomology	$\leftarrow$	general equivariant (Bredon) cohomology	$\rightarrow$	non-equivariant cohomology with homotopy fixed point coefficients
   $\mathbf{H}(X_G, A)$	trivial action on coefficients	$[X,A]^G$	trivial action on domain space	$\mathbf{H}(X, A^G)$

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeApr 14th 2014
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexIt's showing >assigns to a coset space G/H is (since $)theH−homotopyfixedpointsoftheG−actiononE$: because of 'since $$'. I don't know what you wanted there.

   It’s showing

   assigns to a coset space G/H is (since $)theH−homotopyfixedpointsoftheG−actiononE$:

   because of ’since $$’. I don’t know what you wanted there.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 14th 2014
   • (edited Apr 14th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. I have fixed the typesetting for the moment by removing the incomplete parenthesis. I had intended to give more of a hint here of the formal argument for why this is true, but then I needed to think more...

   Thanks. I have fixed the typesetting for the moment by removing the incomplete parenthesis. I had intended to give more of a hint here of the formal argument for why this is true, but then I needed to think more…

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeApr 14th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexYeah, so there is some subtlety hidden there which my paragraphs are not accounting for properly yet. I'll come back to it...

   Yeah, so there is some subtlety hidden there which my paragraphs are not accounting for properly yet. I’ll come back to it…

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2014
    • (edited Apr 15th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexOkay, so I had some puzzlement as to whether the derived function spaces in equivariant homotopy theory would include the cofibrant replacement necessary to make equivariant cohomology involve _homotopy_ quotients/fixed points instead of plain quotients/fixed points. (Somehow standard literature here seems remarkably shy on addressing this point, or maybe I have just all been looking in the wrong places.) But from * Bert Guillou, "A short note on models for equivariant homotopy theory" ([pdf](http://www.math.uiuc.edu/~bertg/EquivModels.pdf)) I gather that by theorem 3.12+example 4.2+section5 in that article, there is a LEFT Quillen functor from the "coarse equivariant model structure"/[[Borel model structure]] to the genuine/fine one. Since cofibrant replacement in the coarse/Borel model structure is given by crossing with $W G$, this gives the desired relation between derived hom-spaces and _homotopy_ (co)invariants that one would hope to see in equivariant cohomology.

    Okay, so I had some puzzlement as to whether the derived function spaces in equivariant homotopy theory would include the cofibrant replacement necessary to make equivariant cohomology involve homotopy quotients/fixed points instead of plain quotients/fixed points.

    (Somehow standard literature here seems remarkably shy on addressing this point, or maybe I have just all been looking in the wrong places.)

    But from

    • Bert Guillou, “A short note on models for equivariant homotopy theory” (pdf)

    I gather that by theorem 3.12+example 4.2+section5 in that article, there is a LEFT Quillen functor from the “coarse equivariant model structure”/Borel model structure to the genuine/fine one.

    Since cofibrant replacement in the coarse/Borel model structure is given by crossing with $W G$, this gives the desired relation between derived hom-spaces and homotopy (co)invariants that one would hope to see in equivariant cohomology.

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2014
    • (edited Apr 16th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI have now added a chunk to the Idea section that spells out a bit more how the "coarse" and then the "fine"/Bredon cohomology come out from first principles. Since that made the Idea-section expand a good bit more, I have now split it into two subsections: 1. _[Equivariance](http://ncatlab.org/nlab/show/equivariant+cohomology#IdeaEquivariance)_ 1. _[Geometricity](http://ncatlab.org/nlab/show/equivariant+cohomology#IdeaGeometricity)_

    I have now added a chunk to the Idea section that spells out a bit more how the “coarse” and then the “fine”/Bredon cohomology come out from first principles.

    Since that made the Idea-section expand a good bit more, I have now split it into two subsections:

    1. Equivariance

    2. Geometricity

12. • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI don't understand the claim that genuine $G$-spectra only appear in cohomology through their underlying spectra-with-$G$-action. Why is that?

    I don’t understand the claim that genuine $G$-spectra only appear in cohomology through their underlying spectra-with-$G$-action. Why is that?

13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexHm, true. I forget what I was after at that point. For the moment I have removed the clause you cite, but I should get back to this.

    Hm, true. I forget what I was after at that point. For the moment I have removed the clause you cite, but I should get back to this.

14. • CommentRowNumber14.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexCoincidentally, I've been thinking about equivariant cohomology this past weekend, because Megan finally posted her [thesis](http://arxiv.org/abs/1405.1770) on the arxiv, and that started me wondering whether the spectral sequences we can construct in homotopy type theory might yield new and useful results when applied to equivariant homotopy theory. Right now, I feel like the point of "genuine spectra" is just that one may want to "stabilize" an $(\infty,1)$-category with respect to objects more general than the categorical spheres $S^n$.

    Coincidentally, I’ve been thinking about equivariant cohomology this past weekend, because Megan finally posted her thesis on the arxiv, and that started me wondering whether the spectral sequences we can construct in homotopy type theory might yield new and useful results when applied to equivariant homotopy theory. Right now, I feel like the point of “genuine spectra” is just that one may want to “stabilize” an $(\infty,1)$-category with respect to objects more general than the categorical spheres $S^n$.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexThat's how its mostly presented. On the other hand, the equivalence to Mackey functors gives genuine equivariant spectra a natural home in... linear homotopy type theory.

    That’s how its mostly presented. On the other hand, the equivalence to Mackey functors gives genuine equivariant spectra a natural home in… linear homotopy type theory.

16. • CommentRowNumber16.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexAlso, I would have been inclined to describe Bredon cohomology first in terms of the $(\infty,1)$-topos of presheaves on the "local" orbit category of $G$, and only then observe that one might also slice the global situation over $B G$; the former seems simpler.

    Also, I would have been inclined to describe Bredon cohomology first in terms of the $(\infty,1)$-topos of presheaves on the “local” orbit category of $G$, and only then observe that one might also slice the global situation over $B G$; the former seems simpler.

17. • CommentRowNumber17.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexWell, linear homotopy type theory should be the internal language of any notion of parametrized spectrum. The question is, *what* notion of parametrized spectrum are we interested in?

    Well, linear homotopy type theory should be the internal language of any notion of parametrized spectrum. The question is, what notion of parametrized spectrum are we interested in?

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexI certainly agree there is much room to improve the discussion in the entry. I won't have resources to work on it further at the moment, though.

    I certainly agree there is much room to improve the discussion in the entry. I won’t have resources to work on it further at the moment, though.

19. • CommentRowNumber19.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI guess by "the equivalence to Mackey functors" you mean the Guillou-May characterization as spectrally enriched presheaves on a Burnside category? I don't really find that illuminating, because (1) I don't see how to motivate the Burnside category in question unless you already know what stable category you're trying to get out, and (2) I don't see how to view it as a construction on the $(\infty,1)$-category of equivariant spaces.

    I guess by “the equivalence to Mackey functors” you mean the Guillou-May characterization as spectrally enriched presheaves on a Burnside category? I don’t really find that illuminating, because (1) I don’t see how to motivate the Burnside category in question unless you already know what stable category you’re trying to get out, and (2) I don’t see how to view it as a construction on the $(\infty,1)$-category of equivariant spaces.

20. • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014
    • (edited May 29th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI am thinking of the perspective on Guillou-May due to Barwick ([arXiv:1404.0108](http://arxiv.org/abs/1404.0108)). He uses the "effective Burnside category" which, for equivariance over a finite group, is just the category of spans in the orbit category (see his remarks on p. 3). That seems to address both your concerns, but let me know if I am missing something.

    I am thinking of the perspective on Guillou-May due to Barwick (arXiv:1404.0108). He uses the “effective Burnside category” which, for equivariance over a finite group, is just the category of spans in the orbit category (see his remarks on p. 3).

    That seems to address both your concerns, but let me know if I am missing something.

21. • CommentRowNumber21.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexCan you explain how that addresses my concerns?

    Can you explain how that addresses my concerns?

22. • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexStarting with spectra on the orbit category, which are stable objects in equivariant spaces, one generalizes to spectra on spans in the orbit category, hence one adds compatible push-forward to the given pullbacks. Seen this was this is a natural and well-motivated step. But maybe I a misunderstanding your concerns.

    Starting with spectra on the orbit category, which are stable objects in equivariant spaces, one generalizes to spectra on spans in the orbit category, hence one adds compatible push-forward to the given pullbacks. Seen this was this is a natural and well-motivated step.

    But maybe I a misunderstanding your concerns.

23. • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexHow is it motivated?

    How is it motivated?

24. • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI don't mean to sound snarky, by the way; I am really trying to understand. If there is an answer here I would love to know it.

    I don’t mean to sound snarky, by the way; I am really trying to understand. If there is an answer here I would love to know it.

25. • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexSorry, I am not sure what else you are looking for. We have naiveSpectra = stable objects in G-spaces = stable objects in functors over the orbit category and we pass to stable objects in functors on spans in the orbit category. That's an example of the usual construction of "adding transfer".

    Sorry, I am not sure what else you are looking for.

    We have

    naiveSpectra = stable objects in G-spaces = stable objects in functors over the orbit category

    and we pass to

    stable objects in functors on spans in the orbit category.

    That’s an example of the usual construction of “adding transfer”.

26. • CommentRowNumber26.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI'm not familiar with this construction of "adding transfer", can you give a link or an explanation? Why should we expect that to get a good notion of spectrum in a diagram category we should look at diagrams of spectra on spans? What are we trying to achieve? For example, here's a motivation for stabilizing with respect to representation spheres: the Thom collapse map of a $G$-manifold lands in a represenation sphere, and we want $G$-manifolds to be stably dualizable. Regarding (2), I expect a way to phrase the definition of spectrum in the internal homotopy type theory of the ambient $(\infty,1)$-topos, i.e. $G$-spaces, or better, diagrams of spaces on the orbit category. I think I see how to do that if what we're doing is stabilizing representation spheres (modulo just assuming that we're given the representation spheres as known objects), but passing to spans on the orbit category seems a purely external construction.

    I’m not familiar with this construction of “adding transfer”, can you give a link or an explanation? Why should we expect that to get a good notion of spectrum in a diagram category we should look at diagrams of spectra on spans? What are we trying to achieve?

    For example, here’s a motivation for stabilizing with respect to representation spheres: the Thom collapse map of a $G$-manifold lands in a represenation sphere, and we want $G$-manifolds to be stably dualizable.

    Regarding (2), I expect a way to phrase the definition of spectrum in the internal homotopy type theory of the ambient $(\infty,1)$-topos, i.e. $G$-spaces, or better, diagrams of spaces on the orbit category. I think I see how to do that if what we’re doing is stabilizing representation spheres (modulo just assuming that we’re given the representation spheres as known objects), but passing to spans on the orbit category seems a purely external construction.

27. • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexOkay, now I see what you are aiming for. Regarding "transfer": the definition of Mackey functor is to presheaves on the orbit category pretty much the concept of a _[[sheaf with transfer]]_ (please see there for further pointers) used in the definition of [[Voevodsky motives]].

    Okay, now I see what you are aiming for.

    Regarding “transfer”: the definition of Mackey functor is to presheaves on the orbit category pretty much the concept of a sheaf with transfer (please see there for further pointers) used in the definition of Voevodsky motives.

28. • CommentRowNumber28.
    • CommentAuthorMike Shulman
    • CommentTimeMay 29th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexHmm, okay. It seems a little backwards to me --- I think of transfer maps as coming from dualities, so the duality is the fundamental concept we should be motivating ourselves by, not the transfer. But I suppose I can see how someone might come to view the transfer maps as basic. This doesn't help with (2), though. Does it?

    Hmm, okay. It seems a little backwards to me — I think of transfer maps as coming from dualities, so the duality is the fundamental concept we should be motivating ourselves by, not the transfer. But I suppose I can see how someone might come to view the transfer maps as basic.

    This doesn’t help with (2), though. Does it?

29. • CommentRowNumber29.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 30th 2014
    • (edited May 30th 2014)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexIs there a sense in which there could be something HoTT-like, providing a more natural formal means to describe relations, spans, etc.? So filling the gaps >Set:Rel = topos:allegory/cartesian bicategory = $\infty$-topos:??? = HoTT:??? Hermida suggested modality had something to do with it [here](http://maggie.cs.queensu.ca/chermida/papers/sat-sim-IandC.pdf): >We set about exploring intuitionistic/constructive modalities from a categorical logic perspective, along the lines of Lawvere’s analysis [16] of logical connectives and quantifiers as adjoint functors. We are primarily interested in the use of modal formulae to analyse properties of transition systems and we thus consider modal logic as a logic of sets and relations, in the same spirit as first-order predicate logic is a logic of sets and functions.

    Is there a sense in which there could be something HoTT-like, providing a more natural formal means to describe relations, spans, etc.? So filling the gaps

    Set:Rel = topos:allegory/cartesian bicategory = $\infty$-topos:??? = HoTT:???

    Hermida suggested modality had something to do with it here:

    We set about exploring intuitionistic/constructive modalities from a categorical logic perspective, along the lines of Lawvere’s analysis [16] of logical connectives and quantifiers as adjoint functors. We are primarily interested in the use of modal formulae to analyse properties of transition systems and we thus consider modal logic as a logic of sets and relations, in the same spirit as first-order predicate logic is a logic of sets and functions.

30. • CommentRowNumber30.
    • CommentAuthorMike Shulman
    • CommentTimeMay 30th 2014
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexProbably your first ??? is some kind of "cartesian $(\infty,2)$-category". But I doubt that the second is anything different from HoTT, since an allegory doesn't have an internal logic that's distinguishable from that of its corresponding topos.

    Probably your first ??? is some kind of “cartesian $(\infty,2)$-category”. But I doubt that the second is anything different from HoTT, since an allegory doesn’t have an internal logic that’s distinguishable from that of its corresponding topos.