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<p>I added three more references to <a href="https://ncatlab.org/nlab/show/Bredon+cohomology">Bredon cohomology</a>.</p>
<p>two of them, by H. Honkasolo, discuss a sheaf-cohomology version of Bredon cohomology, realized as the cohomology of a topos built from the <a href="https://ncatlab.org/nlab/show/orbit+category">orbit category</a>.</p>
<p>It's too late for me today now to follow this up in detail, but I thought this might be of interest in the light of our discussion at <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=650&page=1">G-equivariant stable homotopy theory</a>.</p>
An advice: It would be good when citing references to really cite them, not only put a blind link to the paper. Especially for offline usage and usage via printouts. I am often doing repairing these blind "citations" and done it this time for the Honkasolo references, but this procedure doubles both the effort, the bandwidth (second person needs also to download the paper to read the reference) and multiplies nlab versions. The link to second Honkasolo reference even frose/killed my browser so I did not succeed to cite it in the first attempt.
okay.
I added the redirect complex cobordism to complex cobordism cohomology theory as one usually just simply says complex cobordism, just like one says complex K-theory for something what pedantically can be called complex K-theory cohomology theory. It seems to me that Honkasolo's references have some overlap with Moerdijk-Svensson approach -- the Grothendieck construction here also plays a role.
I have added an actual Definition-section to Bredon cohomology.
(Of course this overlaps with what ages ago Mike added in the subsection on Bredon cohomology at equivariant cohomology.)
For one I mentioned the chain complex model that Bredon actually wrote down (as opposed to all the high-powered generalizations that now carry his name).
I’m confused by the Definition at Bredon cohomology. What is $\mathbf{H}$ and what is $\mathbf{B}G$? We appear to be defining “$\mathbf{H}_{/\mathbf{B}G}$” to be the $(\infty,1)$-topos of $G$-spaces, but that notation looks like it is a slice category of something, whereas neither of the definitions given ($L_{fpwe}G Top$ and $PSh_\infty(Orb_G)$) are slices.
Yes, that notation is bad, I should change it. How about $\mathbf{H}^{orb_G}$?
That’s better, although it looks like a functor category, and if $\mathbf{H}$ means $\infty Gprd$ then we ought to say contravariant functors, right?
Right, sure. I have changed the notation in the entry 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’.
presumably the ’org’ is ’orb’
Oh, sorry. Fixed now.
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 $Orb_G$.
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 $Orb_G$.
Sort of, but you still have to motivate your particular definition of $G$-CW-complex, don’t you?
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.
Orbifold cohomology is related to Bredon cohomology, see works of Ruan et al.
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)
“locally constant sheaf except for dependence on isotropy groups”
so a constructible sheaf?
Ah, right, thanks. Haven’t thought about this.
Some remarks:
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.(?)
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.
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 ;-)
So here’s the question: what about in the non-global quotient case, as in the second half of my #15 above?
Yeah, that’s what I had sketched out at orbifold cohomology. Polished writeup should be available soon.
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?
No, I didn’t know that there are such lectures on YouTube.
It’s working from his monograph Global homotopy theory. I will add the lectures there as a reference.
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…
I have added full publication data to
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.
added pointer to
added pointer to:
added these references on Kronholm’s freeness theorem:
William C. Kronholm, A Freeness Theorem for $RO(\mathbb{Z}/2)$-graded Cohomology, Topology and its Applications 157 5 (2010) 902-915 [arXiv:0908.3825, doi:10.1016/j.topol.2009.12.006]
Eric Hogle, Clover May, The freeness theorem for equivariant cohomology of $Rep(C_2)$-complexes, Topology and its Applications 285 1 (2020) 107413 [arXiv:2005.07300, doi:10.1016/j.topol.2020.107413]
Daniel Dugger, Christy Hazel, Clover May, Equivariant $\mathbb{Z}/\ell$-modules for the cyclic group $C_2$, Journal of Pure and Applied Algebra 228 3 (2024) 107473 [arXiv:2203.05287, doi:10.1016/j.jpaa.2023.107473]
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