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<p>I added three more references to <a href="http://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="http://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.
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