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If generalized nonabelian cohomology, from the nPOV, means a hom-space in some $(\infty,1)$-topos, then it can equivalently be characterized as global sections of an object in some $(\infty,1)$-topos, since for any $X,A$ in an $(\infty,1)$-topos $E$ we have $Hom_E(X,A) = Hom_{E/X}(1,X^*A) = \Gamma_{E/X}(X^*A)$. Recall that traditional “abelian” sheaf cohomology $H^n(X;A)$ is the case when $A$ is an $n$-fold delooping of a discrete abelian group object, and when $A$ is locally constant (whatever that means) it reduces to “cohomology with local coefficients” and further to the most traditional algebraic-topology sort of cohomology when $A$ is constant. Generalizing in a different direction, if $A$ is constant, not on a discrete abelian group, but on a spectrum, then we obtain classical “generalized cohomology”, and if we further generalize to spectrum objects in $E$ then we have “generalized sheaf cohomology” with coefficients in a sheaf of spectra. Note that $\Gamma$ preserves abelian group objects and spectrum objects, so that with these definitions, abelian cohomology theories always produce abelian objects.
In another thread I asked a question about homology from the nPOV (and David C kindly supplied some links to past discussions). A couple of answers were given, but I just thought of a slightly different way of stating it, which I like. Suppose our $(\infty,1)$-topos is locally ∞-connected, so that in addition to a right adjoint $\Gamma$, the constant stack functor $\Delta$ has a left adjoint $\Pi$. Now $\Pi$ won’t preserve abelian and spectrum objects, but by general “adjoint lifting theorems” I would nevertheless expect to be able to construct from it a functor $\Pi_{spectra}: Spectra(E) \to Spectra$ which is left adjoint to $\Delta_{spectra}: Spectra \to Spectra(E)$. It seems to me that it would make sense to regard $\Pi_{spectra}$ as a notion of “sheaf homology” with coefficients in a sheaf of spectra (perhaps a constant one).
It could be that this is way off-base, but I’m getting my intuition from the May-Sigurdsson theory of “parametrized” spectra, which should morally (I believe) be identifiable with “locally constant” sheaves of spectra over nice spaces. In their context, the pullback functor $r^* : Spectra \to Spectra/X$ always has both a left adjoint $r_!$ and a right adjoint $r_*$, and the left adjoint is homology while the right adjoint is cohomology. In particular, $r_! r^* M$, for a spectrum $M$, can be identified with the generalized homology theory $H_*(X;M)$.
Thoughts? Is this obviously true? Obviously false?
@Mike: I think that requiring that we be in an $(\infty,1)$ topos is overly strict. For instance, in any pointed model category (hell, pointed homotopical category), we can get generalized homology and cohomology groups by looking at homotopy groups of the DK-localization.
Good idea. I had missed that adjunction in May-Sigurdsson. (I had only looked at small subsections of it.)
On a $\infty$-connected $\infty$-topos $\Pi$ is homotopy . Homology (at least the ordinary non-generalized notion) should be abelianized homotopy (pass from the singular simplicial complex to the free abelian group of singular chains that it generates).
So we just transplant the homotopy/nonabelian-cohomology adjunction of an $\infty$-connected $\infty$-topos
$Sh(C, \infty Grpd) \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd$to the homology/abelian-cohomology adjunction
$Sh(C, Sp) \stackrel{\overset{\Pi_{Ab}}{\to}}{\stackrel{\overset{LConst_{Ab}}{\leftarrow}}{\underset{\Gamma_{Ab}}{\to}}} Sp$on a locally presentable stable $\infty$-category, where we use the stable Giraud theorem to identifiy every locally presentable stable $(\infty,1)$-category with one of spectrum-valued sheaves.
These will be presented by a stable model category and also Harry’s comment is taken care of. Looks good. Let’s try to make precise that May-Sigurdsson’s adjunction is secretly $\Pi_{Ab} \dashv LConst_{Ab} \dashv \Gamma_{ab})$.
Another viewpoint of a homological bent, is to say cohomology is the derived form of Hom so homology should be a derived tensor!
Another viewpoint of a homological bent, is to say cohomology is the derived form of Hom so homology should be a derived tensor!
That’s what i said in the earlier discussion that Mike pointed to. But as Mike points out now, the above way of looking at it (if it is indeed correct) might have its virtues.
This reminds me of my old plan to consider $\Pi$ on $\infty$-operad valued sheaves
$Sh(C, \infty Op) \stackrel{\overset{\Pi_{Op}}{\to}}{\stackrel{\overset{LConst_{Op}}{\leftarrow}}{\underset{\Gamma_{Op}}{\to}}} \infty Op$and check if indeed for $X$ a topological space $\Pi_{Op}(X)$ is something like $Bord_{(\infty,\infty)}(X)^\otimes$ as in (infinity,n)-category of cobordisms or maybe possibly something like $Fact_X$ as in factorization algebra.
I agree, but perhaps what we need also is some notion of what relationship a homology should have to the other invariants and my own feeling is that we do not yet have a unobstructed view of the question. Ronnie and I sometimes mused about needing a whole range of intermediate theories between homotopy and homology not just homology = abelianisation of homotopy. Then there is the question of what is duality in this setting? That relates to the blockbundle geometric approach that I mentioned earlier, so the homology reflects the geometry more.
The idea of using cobordisms in someway is again attractive.
One thought that I forgot to mention is that non-abelian cohomology should perhaps have a non-abelian homology which was related to a non-abelian tensor???
Well, there is a whole range of abelianness – instead of spectra (= $E_\infty$-objects, if connective) we could have a version of $\Pi$ acting on $E_n$-objects.
The idea I sketched above suggests that the duality between homology and cohomology is not tensor versus hom (or, at least, it need not be), but rather mapping-in versus mapping-out, so that “non-abelian homology” is really just homotopy.
Hi. This is funny and pleasant (and showing how nLab is beginning to create a common point of view among its memers, maybe one day we should post an arXiv preprint signed as nLab as author): I was discussing Chern-Simons and cobordism with Urs on the other thread; there at a point I said that Thom spectrum should have played a role, and Urs addressed me to this thread here. Since my browser had firmly decided not to load this page then, I wrote Urs an e-mail with what I had vaguely in mind.. and now I see it was already here, and better developed than in my email, which was the following:
Hi Urs,
I’ve a few problems with nforum today (it is horribly slow, at least to me), so I’ll use e-mail. If later nforum will be working for me again I’ll copy this there.
The first remark is that the Thom spectrum is a topological $E_\infty$ ring spectrum, and this is what makes it meaningful to consder higher symmetric monoidal functors out of it: these will be nothing but $E_\infty$-morphisms.
But this is only a partial vision: it tells us that if $E$ is a topological $E_\infty$ ring spectrum to any element $e$ of $E$ is associated an object $X_e$ and given a pair of elements $(e,f)$ we have a morphism $X_e\otimes X_f \to X_{e f}$, but says nothing about “functoriality in $e$” (obviously, since $E$ is trivial from a categorical point of view). But then let us add morphisms, i.e. let us go from $E$ to $\Pi_1(E)$ or even better to $\Pi(E)$. Then one would hope that $\Pi(E)$ is a sort of $E_\infty$ object in oo-categories, and be interested in $E_\infty$ representations of this.
Best,
d.
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