added pointer to:

- Tammo tom Dieck, Section II.9 of:
*Transformation Groups*, de Gruyter 1987 (doi:10.1515/9783110858372)

added pointer to

- Sören Illman,
*Equivariant singular homology and cohomology*, Bull. Amer. Math. Soc. Volume 79, Number 1 (1973), 188-192 (euclid:bams/1183534324)

ah, never mind, I see that it’s Theorem 2.11 in Bredon 67

]]>I have added full publication data to

- John Greenlees, Peter May, pages 9-10 of
*Equivariant stable homotopy theory*, chapter 8, pages 277-323 in: Ioan James (ed.),*Handbook of Algebraic Topology*, North-Holland, Amsterdam, 1995. (doi:10.1016/B978-0-444-81779-2.X5000-7, pdf)

This article stands out in that it admits that Bredon cohomology is equivalently homs into Eilenberg-MacLane $G$-spaces.

Is there any other reference that would expand on this? Greenlees-May just state it in passing.

]]>My interest is more in the/a K-theory version

I should add pointers to traditional orbifold K-theory to the entry *traditional orbifold cohomology – references*.

Adem-Ruan in “Twisted Orbifold K-Theory” arXiv:math/0107168 don’t really define orbifold K-theory. What they really do is (Def. 3.4) say that on global quotient orbifolds of some $X$ by some $G$ it should equal the genuine (Bredon) $G$-equivariant K-theory of $X$, and then they run with that as a definition in that case.

Then there is the constructions by Bunke-Schick and by Freed et al. Should add commented pointers to these…

]]>It’s working from his monograph Global homotopy theory. I will add the lectures there as a reference.

]]>No, I didn’t know that there are such lectures on YouTube.

]]>OK, thanks! My interest is more in the/a K-theory version, so that might be something that we can discuss later/elsewhere.

On a separate note, have you recorded a reference to Schwede’s Global homotopy theory course on YouTube anywhere?

]]>Yeah, that’s what I had sketched out at *orbifold cohomology*. Polished writeup should be available soon.

So here’s the question: what about in the non-global quotient case, as in the second half of my #15 above?

]]>I see. If that subsection on traditional orbifold cohomology does not leave the reader with a deep feeling of dissatisfaction, then I may need to rewrite it ;-)

]]>Nothing in particular. I just happened to see the phrase I quoted in #18 in this thread (I haven’t looked at the other recent, related, threads yet), and wanted to point you to what it seemed to be referring. If you aren’t going down that path then it’s just going to be a side comment in the story you are hoping to tell, that’s all. But thanks for clarifying

the take-away message from traditional orbifold cohomology – references is, in contrast, that none of this business with funny coefficient sheaves on funny quotients is necessary if we’d just work with Bredon cohomology from the get-go. That’s what Honkasalo’s result says.

I had seen the ’funny coefficient sheaves’ in something I was thinking about on and off, in a way that seems rather useful, but maybe I should be thinking about Bredon cohomology (or rather genuinely global equivariant cohomology) instead.

]]>If you want to go down that road now, allow me remark that, to my mind, the take-away message from *traditional orbifold cohomology – references* is, in contrast, that none of this business with funny coefficient sheaves on funny quotients is necessary if we’d just work with Bredon cohomology from the get-go. That’s what Honkasalo’s result says. And if we don’t (mis?)use sheaves on funny auxiliary space to encode genuine equivariance, we still have them available to do their real work on the geometrically meaningful spaces: namely to provide enhancement to differential equivariant cohomology.

Of course I don’t know what piques your interest here. If you are just intrigued that Honkasalo’s and maybe Segal’s and other old results might be prettfied by being recast in the language of constructible sheaves then that might be worthwhile. On the other hand, you went to the thread on Bredon cohomology with this comment, so I am not sure what you are after.(?)

]]>Some remarks:

- Segal has a similar sheaf in his equivariant K-theory paper (prop 5.3), but it’s the sheafification of a presheaf defined using K-theory
- The quotient of a manifold by a proper Lie group action (so with compact stabilisers) is nicely stratified by manifolds, and more generally so is the orbit space of a proper Lie groupoid
- the constructibility will be with respect to this stratification.

Ah, right, thanks. Haven’t thought about this.

]]>“locally constant sheaf except for dependence on isotropy groups”

so a constructible sheaf?

]]>added these pointers to discussion of equivalence of Bredon cohomology of topological G-spaces $X$ to abelian sheaf cohomology of the topological quotient space $X/G$ with coefficients a “locally constant sheaf except for dependence on isotropy groups”:

Hannu Honkasalo,

*Equivariant Alexander-Spanier cohomology*, Mathematica Scandinavia, 63, 179-195, 1988 (doi:10.7146/math.scand.a-12232)(for finite groups)

Hannu Honkasalo,

*Equivariant Alexander-Spanier cohomology for actions of compact Lie groups*, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 (jstor:24492569)(for compact Lie groups)

Hannu Honkasalo,

*Sheaves on fixed point sets and equivariant cohomology*, Math. Scand.**78**(1996), 37–55 (jstor:)(reformulation in topos theory)

Orbifold cohomology is related to Bredon cohomology, see works of Ruan et al.

]]>Is there any sort of generalisation of Bredon cohomology from action groupoids to more general (nice) groupoids, say locally quotient groupoids? That is, does the abstract infinity-topos viewpoint allow something to be constructed?

Carla Farsi told me that she and some others tried to develop this for proper Lie groupoids more or less by hand (or so I gather) without much success.

]]>Sort of, but you still have to motivate your particular definition of $G$-CW-complex, don’t you?

]]>I’ve never really seen a convincing argument for any “the” reason why we see $Orb_G$.

Isn’t “the” reason given by the $G$-Whitehead theorem + Elmendorf’s theorem ?

There is an evident notion of homotopy equivalence between $G$-CW-complexes.

by the G-Whitehead theorem this is equivalent to maps that are weak homotopy equivalences on all $H$-fixed point spaces for all closed subgroups $H$;

by Elmendorf’s theorem this is equivalent to weak equivalences in the functor category over $Orb_G$.

I don’t think that’s “the” reason we see $Orb_G$, because when $V = G Set$ the single object $G/e$ is already dense (since it’s the unique representable when $G Set$ is regarded as a presheaf category. I’ve never really seen a convincing argument for any “the” reason why we see $Orb_G$.

]]>presumably the ’org’ is ’orb’

Oh, sorry. Fixed now.

]]>Is there something from Mike’s comments here worth including? It’s in the context of a discussion of homology in enriched categories, which seems to work generally for semicartesian enrichment:

Note that in the case when $V$ is cartesian monoidal and also extensive with an indecomposable terminal object, we can identify $V$-enriched categories with certain $V$-internal categories. In that case, this construction amounts to taking the internal nerve $Cat(V) \to V^{\Delta^{op}}$ and then applying the Yoneda embedding $V \to Set^{V^{op}}$ levelwise to get a presheaf of simplicial sets on $V$.

In particular, therefore, if $V$ has a small dense subcategory $V_0$, we can use the restricted Yoneda embedding $V \to Set^{V_0^{op}}$ instead without losing information. This is a fancier way of saying why we see nothing new when $V=Set$, because the single object $1$ is dense in $Set$, and its restricted Yoneda embedding is the identity.

Another interesting example of this sort is $V=G Set$ for a group $G$, with $V_0$ the category of orbits $G/H$. Thus, a category enriched over $G$-sets has a homology indexed by orbits, which I think is just the usual equivariant (Bredon) homology of its $G$-equivariant nerve. (Curiously, I don’t think I recall ever seeing the “morphism-graded” version in equivariant homotopy theory. Maybe it’s present implicitly somewhere?)

Is this ’the’ reason why we see $Orb_G$?

By the way, presumably the ’org’ is ’orb’, and there should be consistency as to ’orb’ or ’Orb’.

]]>