added pointer to:

- Graeme Segal,
*Some results in equivariant homotopy theory*(1978) [scan: web, pdf]

This document seems to have existed only as a scan hosted as 6 separate html pages at homepages.warwick.ac.uk/~maaac/segal.html, /segal2.html, /segal3.html, /segal4.html, /segal5.html, /segal6.html.

I have merged these six html pages into a single pdf file (here). The result has overly large margins and bad page breaks, but otherwise is readable.

]]>re #30: ugh, yes; thanks!

]]>[ mysterious duplicate removed ]

]]>Thanks!

I have re-instantiated (here) the cross-link with *Arf-Kervaire invariant problem*, assuming that you deleted it by accident.

I’ve updated this Hill, Hopkins, Ravenel reference to point to their 2021 book, since it’s a more thorough treatment of the material. As far as I can tell, nothing on the page points directly to the previously listed article, but I can put the article back anyway if someone wishes.

]]>added pointer to:

- Bert Guillou,
*Equivariant Homotopy and Cohomology*, lecture notes, 2020 (pdf)

added pointer to today’s

- Mehmet Akif Erdal, Aslı Güçlükan İlhan,
*A model structure via orbit spaces for equivariant homotopy*(arXiv:1903.03152)

(just for completeness)

]]>added some links

]]>added the following diagram, for illustration purposes

$\array{ Ho(G Top_{cof}) &\underset{}{\longrightarrow}& Ho(Top_{cof}) \\ {\mathllap{\text{equivariant} \atop \text{Whitehead}}}\big\downarrow{\mathrlap{\simeq}} && {\mathllap{\simeq}}\big\downarrow{\mathrlap{\text{Whitehead}}} \\ Ho(G Top_{loc}) &\overset{}{\longrightarrow}& Ho(Top_{loc}) \\ {\mathllap{Elmendorf}}\big\downarrow{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} \\ Ho( PSh( Orb_G, Top_{loc} ) )_{proj} &\longrightarrow& Ho( \ast, Top_{loc} )_{proj} }$wanted to assign a name to the top left arrow (Bredon?) but maybe there is no particular name associated with it besides Whitehead

]]>Made explicit the pointer to the *equivariant Whitehead theorem* (notably in the statement of this theorem)

Re #20, what if anything stands in the way of global equivariance for all homotopy types with finite homotopy groups? I see Charles Rezk thought things should work out with 2-groups here.

Is there a suitable sense of compactness for Lie n-groups?

]]>BTW, this is different from the naive/genuine distinction for $G$-*spectra*; that’s a further bifurcation within the world of fixed-point-wise equivariant homotopy theory. So perhaps “naive” is a poor choice of word here…

Note that there’s actually a whole range of possible “$G$-equivariant homotopy theories” parametrized by a family of subgroups of $G$; the “most naive” one corresponds to choosing only the trivial subgroup.

]]>Sounds interesting. What already exists as equivariant Goodwillie theory? I see there is

- Emanuele Dotto, Higher Equivariant Excision, arXiv:1507.01909

which refers to another couple of his own papers.

]]>That quote gives natural language for speaking about the homotopy theory that is presented by the orbit category. But I would think that what you were after is something like an intrinsic internal characterization of this homotopy theory.

The following vague thought had occured to me:

if we consider global equiviance not under Lie groups but under finite groups, then the global orbit category is just the $(2,1)$-category of homotopy 1-types with finite $\pi_1$. If we did this for “2-equivariance” as in “2-equivariant elliptic cohomology” then we’d be looking at 2-groupoids with finite homotopy groups. Generally then we could consider the $\infty$-category of homotopy types with finite homotopy groups as a site for “global $\infty$-equivariance”.

This reminds us of the (opposite) of finite homotopy types, which is the site for the classifying topos for an object. As we are discussing elsewhere, this is the origin of Goodwillie calculus.

Now, of course, despite the similarity in name, finite homotopy types are different, in fact pretty much complementary to, homotopy types with finite homotopy groups. So maybe we’d want to combine them, somehow, to merge Goodwillie theory with global equivariant homotopy theory. And maybe the $\infty$-category which suitably subsumes both finite homotopy types and homotopy types with finite homotopy groups is the site for a good classifying $\infty$-topos, i.e. maybe that $\infty$-topos has a good internal characterization.

Just speculating.

]]>Well anyway, it would interest me to see how renderings in HoTT allow expression in something closer to natural language, as Mike does in Univalence for inverse EI diagrams, example 7.5:

from a propositions-as-types point of view, we might say that A consists of a type with a G-action together with, for each fixed point of this action, a type of “special reasons” why that point should be considered fixed (which might be empty). That is, in passing from (the naive homotopy theory of) G-spaces to $O^{op}G$ -diagrams, we make “being a fixed point” from a property into data.

Is ’naive’ being used in the sense as opposed to ’genuine’ here? So then, especially with a $G$ with many subgroups, the expression of the genuine would be rather complicated. Is there a reason why we need to range over all subgroups?

]]>re #16: sorry, I didn’t put that well after all.

I had been prompted by the paragraphs in Goodwillie 03, p.5, 6 (of 67) but maybe some care is due here. In any case, I have rephrased a bit more, just to bring out the “G-Whitehead theorem” better, which says that those fixed-point wise weak homotopy equivalences are, on G-CW-complexes, the evident G-equivariant homotopy equivalences.

]]>Maybe you’re warning against this here

For G a discrete group (geometrically discrete) the homotopy theory of G-spaces which enters Elmendorf’s theorem is different (finer) than the standard homotopy theory of G-∞-actions, which is presented by the Borel model structure

But then what allows the expression of the finer aspects? Something geometric? Cohesive, at least?

]]>That’s the clearest description I’ve seen of the naive/genuine distinction. Is the difference expressible in HoTT in terms of working in the context $\mathbf{B} G$, dependent sum/product for co(invariants), etc.? Perhaps the genuine version is more simply expressed?

]]>In the Idea-section both at *equivariant homotopy theory* and at *topological G-spaces* I have added a paragraph that makes more explicit where the fixed-point wise homotopy equivalences come from

]]>The canonical homomorphisms of topological $G$-spaces are $G$-equivariant continuous functions, and the canonical choice of homotopies between these are $G$-equivariant continuous homotopies (for trivial $G$-action on the interval). A $G$-equivariant version of the Whitehead theorem says that on G-CW complexes these $G$-equivariant homotopy equivalences are equivalently those maps that induce weak homotopy equivalences on all fixed point spaces for all subgroups of $G$ (compact subgroups, if $G$ is allowed to be a Lie group). By Elmendorf’s theorem, this, in turn, is equivalent to the (∞,1)-presheaves over the orbit category of $G$. See below at

In topological spaces – Homotopy theory.

Yeah, I am wrong, it’s not. Sorry for the distraction.

]]>Can you explain why $Glo_{\mathbb{B}G}$ is an inverse EI-$\infty$-category?

]]>Right it isn’t, but I don’t see what you mean to imply by saying so.

What Charles Rezk discusses in 5.2, 5.3 is an $\infty$-topos that he writes $Top_{Glo}/\mathbb{B}G \simeq PSh_\infty(Glo_{\mathbb{B}G})$. What I said in #8 is that $Glo_{\mathbb{B}G}$ is an EI-$\infty$-category. And I was guessing that that finiteness condition holds here, too. If so, then by Mike’s preprint $Top_{Glo}/\mathbb{B}G$ would be a model of HoTT with univalent strict universes. Unless I am missing something.

Now it is true that on top of that Charles has the nice statement that $Top_{Glo}/\mathbb{B}G$ is cohesive over $G Top \simeq PSh_{\infty}(\mathcal{O}_G)$. The latter is *also* presheaves over an EI-$\infty$-category, that’s the example that Mike makes explicit in his preprint.

Put together, these two statements would seem to imply that this gives a model for cohesive HoTT with univalent strict universes.

]]>I just meant, from my cursory glance, it didn’t seem to me that Mike’s paper was motivated by anything to do with cohesion.

]]>Here I am not sure what your “but” refers to.

Regarding cohesion, Mike has a note in preparation with a comprehensive account of cohesive HoTT. Combing that with what seems (unless I am mixed up) a consequence of his present preprint should say that Charles’ sliced version of the equivariant cohesion is a model for cohesive HoTT with strict univalent universes.

]]>