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We are finalizing an article:
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Abstract. We extend the Chern character on K-theory, in its generalization to the Chern-Dold character on generalized cohomology theories, further to twisted non-abelian cohomology theories, where its target is a non-abelian de Rham cohomology of twisted L-∞ algebra valued differential forms. The construction amounts to leveraging the fundamental theorem of dg-algebraic rational homotopy theory, which we review in streamlined form, to a twisted non-abelian generalization of the de Rham theorem. We show that this non-abelian character reproduces the Chern-Weil homomorphism on non-abelian cohomology in degree 1, represented by principal bundles; and thus generalizes it to higher non-abelian cohomology, represented by higher bundles/higher gerbes. As a fundamental example we discuss the character map on twistorial Cohomotopy theory over 8-manifolds, which is a twisted non-abelian enhancement of the Chern-Dold character on topological modular forms (tmf) in degree 4. This turns out to exhibit a list of subtle topological relations that in high energy physics are thought to govern the charge quantization of fluxes in M-theory.
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Comments are welcome. Please grab the latest version of the file from behind the above link.
I look forward to having a proper read, but a couple of typos for now:
where is simply; Su[llivan
Thanks! Fixed now.
Can we think of some of this in terms of what happens to the differential cohomology diagram when a non-stable type is chosen?
Typos:
Remark 2.27, you have
see [?, §4.4])
On p.16, $t_{\mathbb{H}}$ becomes $t_{\mathbb{C}}$ (twice), and there’s an extra ’s’ in the middle term of the factorization.
Elsewhere
correspondonds, equivatiant, terminoloy, fnite, equialent
Thanks! Fixed now.
I don’t know what the hexagon means in the non-abelian case. For one, it’s not exact anymore, so that it doesn’t even characterize the middle object in general.
Sure. I guess I’m just looking for an abstract general take on what you’re doing.
So that character map is the combination of (a) a general abstract construction (rationalization) with (b) passage to minimal models. The second step is all about maximally compressing the data to something tractable.
Its motivation is in it being the link between abstract homotopy theory and differential form data seen in practice.
We have been alluding to non-abelian characters all along in saying what it means to charge-quantize, say, 4-forms in J-twisted Cohomotopy theory. This here now to expand on what this means in detail and to put it into context.
Thanks! From the abstract point of view is there a reason why rationalization, the “coarsest of the localization approximations”, should connect to differential geometric data?
Another typo:
characer
It’s de Rham’s theorem which says that over smooth manifolds, rational cohomogy, rationalized over the reals, equals the cohomology of smooth differential forms.
The generalization of de Rham’s theorem to twisted and non-abelian cohomology is the bulk of the work in the article. It’s curious that the non-abelian de Rham theorem ends up being, up to some mild reconceptualization, the fundamental theorem of dg-algebraic rational homotopy theory, enhanced by the observation in section 9 of Griffith-Morgan, that when rationalizing over the real numbers, Sullivan’s de Rham complex of piecewise polynomial forms is equivalent to piecewise smooth differential forms.
For abelian cohomology this is no news: the construction of generalized differential cohomology by Hopkins-Singer is by pullback of curvature differential form data along the rationalization map, coupled via a de Rham-theorem like identification – Dold-Buchstaber’s equivalence really (though they attribute it just by the word “recall”).
The generalization of this to non-abelian differential cohomology is section 4.3 of our article. We did this already in 2015 when constructing differential Cohomotopy, but the new article means to lay out the construction principle more comprehensively.
OK, so I guess I’m then wondering how abstract general things are. If twisted cohomology is the intrinsic cohomology of $\infty$-toposes, $\mathbf{H}^I$, the use of rationalization there suggests that rationalization in a general $\infty$-topos should be interesting.
Naturally enough we have a page on this – rational homotopy theory in an (infinity,1)-topos – but unfortunately the content was removed 10 years ago. It points us instead to the brief section function algebras on infinity-stacks which is in terms of the Lawvere theory of $\mathbb{Q}$-algebras.
That got me wondering about whether the Morava K-theory story generalizes to general $\infty$-toposes. Perhaps a path to that is via the prime spectrum of a symmetric monoidal stable (∞,1)-category.
Or is the situation such that since, as per (29), you can just work in a slice, then for that first map in (220) one doesn’t need to think about rationalization in more general settings?
Many aspects could be explored here. Just to note that specifically for Hopkins-Singer-style differential nonabelian cohomology (section 4.3, twisted or not), which is one way of enhancing a given bare cohomology theory, the rationalization always happens in the plain base infinity-topos, embedded under Disc into the cohesive infinity topos. It is only the differential form datum being pulled back, and the homotopy in that homotopy pullback which picks up the geometric dara.
We have expanded a fair bit:
Now the full concept of twisted differential non-abelian cohomology is introduced. As examples, the Cheeger-Simons homomorphism is realized as a secondary non-abelian cohomology operation, the twisted differential Chern character is realized as a twisted secondary cohomology operation, and differential twistorial Cohomotopy is made explicit.
Also the title got expanded, accordingly:
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The commutative square after the text “sits in a cohomology operation diagram (3) of this form” in the last section before the appendix has some lonely $\frac{1}{4}$s with no $p_1$s.
Thanks! Fixed now.
Third sentence has that disallowed (in British English anyway) “X allows to Y” construction.
that allow to approximate generalized cohomology in consecutive stages
’Allow’ needs an object, so perhaps
that allow generalized cohomology to be approximated in consecutive stages; or, in order to approximate generalized cohomology in consecutive stages
Later there’s the word ’to’ to be deleted in
finds to its
In the top left term of the right hand square in (3), the writing in blue seems to be just a copy of the left hand entry, e.g., lacks ’twisted’ and has the same def number.
Hoŕava-Witten
should be
Hořava–Witten
The sentence beginning
It is the crucial appearance of quadratic functions…
doesn’t have a main verb.
is an iterated loop spaces; deloopig
In
$A \simeq B^3 A$
the second $A$ should be $G$.
less deloopings
’fewer deloopings’ is better.
this space carries is itself
With
Mac Lane/MacLane
twice you have the space, 15 times you don’t.
one can from
Thanks! All fixed now.
(That sentence, seemingly without verb, continued after the displayed maths. I have rearranged to make it clearer, hopefully.)
It’s still not right
It is the crucial appearance of quadratic functions…in the Cohomotopical character map (4) which are brought about by the non-abelian nature of (twisted) Cohomotopy theory
The subject is ’it’ or ’the crucial appearance’. Where is the singular verb ?
A few more:
In (25) you have a comma before the final coefficient term $A_2$.
Example 2.2, presumably after “their rationalization” the last term should be $L_{\mathbb{R}} A$.
Cohomtopy
In (246), second coefficient should be $A_2$. (Also, the pointer in the following line to (24) is wrong.)
Oh, I see: should have been “is brought about” (namely: the appearance of quadratic functions in the character), not “are brought about”.
Have fixed it (hopefully!) Also the other items. Thanks again.
Pointer to (24) was intentional, but in lack of an anlogous equation of structured cohomology operations. Have added that now (29).
OK, but the ’It is…’ still sounds odd.
Note that the crucial appearance…
would sound more natural.
On a more interesting note, presumably there should be some ways for pure mathematicians to put the framework of this article to use.
Thanks for your patience.
I have now reworked the paragraph in question, and then expanded the p. 5 that it sits on to conclude the introduction with an actual punchline and some outlook.
one may regard the non-abelian Boardman homomorphism (7) as a non-abelian but K-theory valued character
So if, as at higher chromatic Chern character,
higher chromatic Chern characters that map a generalized cohomology theory of some chromatic level to another of some other level.
One could have non-abelian higher chromatic Chern characters, presumably, from non-abelian cohomology to K-theory, etc.
Typos
quantizazing
Then
deloopig
is still there.
All constructions on non-abelian have…
@ David,
true, it makes sense to think of the Boardman homomorphisms to K-theory, to elliptic cohomologies etc. as chromatic characters of sorts. Though I don’t know that the chromatic community would speak this way?
@ Chris,
sure, thanks. I had looked at you article when it came out, but should remind myself.
Chris,
is there anything to be said in comparison to the category of fibrant objects that underlies Pridham’s model structure on $L_\infty$-algebras? I once made a note on the latter in Prop. 3.25 here (though now I see that I left some gap in the justification there, need to check).
Impressive to see Buchstaber of [BU70] still going strong 50 years later
Chris, thanks for the pointers. That’s useful, I will have another look. (If maybe not right now.)
Anything to be said about the relation of homotopy categories between the opposite of the Bousfield-Gugenheim model structure on dgc-algebras and that of finite type connective L-infinity algebras?
David, thanks for highlighting that recent preprint by Buchstaber. That’s good to know.
Hi Urs,
regarding comment #30 and the relationship to the opposite of the Bousfield-Gugenheim model structure on cdga’s, I can’t point you to a concise optimal statement at the moment, although it shouldn’t be too hard to figure one out. On the other hand, I can point to two related facts, which maybe you already know, but:
1) the class of weq between finite-type Lie n-algebras (connective L-infty algebras) is a proper subclass of those L-infty morphisms which induce quasi-isomorphisms between their corresponding CE coalgebras. (I work out the “classic” example involving a map out of the 2 dim solvable (non-nilpotent) Lie algebra in Remark 3.7 (4) in my paper.)
2) The model structure given by Hess and Shipley on bounded below conilpotent dg cocommutative coalgebras (in which the weq are quasi-isomorphisms, and cofibrations are degreewise monomorphisms – so roughly the “opposite” of the Bousfield-Gugenheim model structure) is the left Bousfield localization of the Hinich model structure on the same category of coalgebras. This is proved in Prop. 2.1.2 of the aforementioned paper by Bruno.
Addendum: incidentally, the homotopy category of “my CFO” structure on Lie_n-Alg is very easy to describe. Every acyclic fibration admits a right inverse, so for any finite-type Lie n-algebras $L$ and $L'$, we have
$hom_{Ho(Lie nalg)}(L,L') \cong \hom_{Lie nalg}(L,L')/ \simeq$where $\simeq$ is right-homotopy equivalence with respect to any path object. (In other words, every object in this CFO is also “cofibrant”.)
Yeah, we have run into this question a few times, but never sorted it out. I was wondering if you meanwhile knew.
For instance when we (you, Domenico and me) proved that Sullivan cell attachment in the BG model structure remains, dually, a homotopy fiber construction inside the model structure on all $L_\infty$-algebras (I mean our Theorem B.0.8 here). This would imply that the BG homotopy theory of nilpotent $L_\infty$-algebras is fully faithful in the Hinich/Valette model structure as soon as it is known that derived hom spaces out of nilpotent $L_\infty$-algebras into abelian $L_\infty$-algebras coincide on both sides. Which seems rather plausible, but maybe there is a catch.
In any case, it should be fairly immediate to see with a good model for a simplicial framing on the Hinich/Valette structure. Which is probably known? I see that Valette’s note has comments on cylinder objects in his note which go in this direction. But I haven’t dug into it yet.
Hi Urs,
since your last comment mentioned simplicial framing, and then the nilpotent finite-type connective case, I can add the following observations:
1) I know how to enrich both my CFO and the Hinich/Vallette CFO over Kan complexes (which gives the correct homotopy category, of course), and in such a way that Quillen’s SM7 holds. (In both CFOs, take cofibration to mean an L-\infty morphism whose linear term i.e. tangent map is a monomorphism between the underlying cochain complexes.)
To do this, I first enrich over filtered L-infty algebras (as described here https://arxiv.org/abs/1406.1751 ), then use the fact that filtered L-infty algebras also admit a CFO structure. I verify SM7 in filtered L-infty algebras, and then use the fact that the simplicial Maurer-Cartan functor from this CFO to KanCmplx is exact.
I prove all these facts about the CFO structure on filtered L-infty algebras and the exactness of the simplicial Maurer-Cartan functor in a recent preprint here: https://arxiv.org/abs/2008.01706 .
I haven’t bothered yet to type up the proof that axiom SM7 holds. Also I think that it is unlikely that axiom SM0 holds.
2) For the nilpotent finite-type case, I have a sketch of a proof that implies the statement you mentioned relating the BG homotopy theory to the Hinch/Vallette one. Here’s a brief summary:
The filtered L-infty model I use for mapping spaces in part (1) is the same one considered by Alexander Berglund in
https://arxiv.org/abs/1110.6145v2
to study rational models of mapping spaces. In particular, for mapping spaces between finite type degree-wise nilpotent Lie n-algebras, he provides a homotopy equivalence between the simplicial Maurer-Cartan set of the L-infty mapping space (or a smaller homotopy equivalent version of it) and the simplicial mapping space between their Sullivan realizations. (Theorem 6.3 in Alexander’s paper.) The latter simplicial mapping space is homotopy equivalent to the mapping space between the corresponding CE algebras in the BG simplicial model structure (via Brown and Szczarba’s theorem.)
Anyway, I have no time at the moment to flesh this out into a public document. (If I knew it would be useful to somebody, maybe I could write it up over the next couple months or so.)
Berglund’s paper might be worth referencing in your draft for background material on nilpotent L-infty models vs. Sullivan models. Some of what he works out is quite explicit, and overlaps with your introductory/setup material (the statements after Example 3.29, I think.). See his Thm 2.3, for example.
I won’t say anymore here unless prompted; I didn’t mean to hijack your thread and steer it into L-infty algebra technicalities.
Hi Chris,
just to clarify that I am asking on this point to satisfy your original request:
If we knew the statement, we could add to our note a remark like this:
Notice that the Bousfield-Gugenheim homotopy theory of nilpotent $L_\infty$-algebras used here is homotopy fully faithfully embedded into the Hinich/Valette homotopy theory of all $L_\infty$-algebras, for convenient presentations of the latter see also Rogers…
By the way, I think I see that every quasi-isomorphism between CE-algebras of nilpotent $L_\infty$-algebras (i.e.: between Sullivan algebras) is dually a quasi-iso on unary chain complexes:
By Prop. 7.11 in Bousfield-Gugenheim, any such CE-algebra is the tensor product of a minimal one with the contractible CE-algebra of an $L_\infty$-algebra whose unary chain complex is contractible.
Thus, composing the quasi-iso between the CE-algebras with the injection and projection of the minimal tensor factors (which themselves are quasi-isos both on CE-algebras as well as on unary chain complexes), we get a quasi-iso between minimal algebras. By BG Prop. 7.8 this is actually an isomorphism, hence in particular is a quasi-isomorphism on unary chain complexes. But then, by 2-out-of-3, and by that Prop. 7.11, the same holds for the original quasi-iso.
What I don’t see yet is the converse implication. But that you took as obvious or established?
If we knew the statement, we could add to our note a remark like this:
I’m sorry. I misunderstood what precisely you were asking about.
Yes, for finite-type nilpotent Lie n-algebras, an $L_\infty$ morphism $f=(f_1,f_2,\ldots)$ induces a quasi-isomorphism between CE algebras $\Leftrightarrow$ $f$ is a weak equivalence (i.e., $f_1$ is a quasi-isomorphism of complexes).
Your proof of the $\Rightarrow$ direction via minimal models looks right to me, and is basically the one given for arbitrary Sullivan algebras in Felix, Halperin, and Thomas’ first book (See Prop 14.13.) Note that they also give the $\Leftarrow$ direction there as well.
What I don’t see yet is the converse implication. But that you took as obvious or established?
Yes, the $\Leftarrow$ direction is one of those “well known…” sort of things, which holds for arbitrary finite-type Lie $n$-algebras, no nilpotence assumption needed. Here is a proof (I have no reference):
Let $f=f_1,f_2,\ldots \colon (L,\ell) \to (L',\ell^\prime)$ be a morphism between finite type Lie $n$-algebras such that $f_1 \colon (L,\ell_1) \to (L',\ell'_1)$ is a quasi-isomorphism. Let $F \colon (S(V'),\delta') \to (S(V),\delta)$ be the induced map on CE algebras, with $V=L^\vee[1]$, and $V'=L^{' \vee}[1]$, concentrated in positive degrees. The CE algebra $(S(V'),\delta')$ admits a decreasing filtration of sub-cochain complexes by “tensor length”: $\mathcal{F}_{n} S(V') = \bigoplus_{n \leq k}^\infty S^k(V')$. This filtration is clearly bounded on the left by $S(V')$, and also locally bounded on the right since all generators are concentrated in positive degrees. Of course, $(S(V), \delta)$ also admits such a filtration, and it is not difficult to see that the morphism $F$ is filtration preserving.
So in order to show $F$ is a quasi-isomorphism, a standard spectral sequence argument implies that it is sufficient to show that the induced map $Gr(F)$ on the associated graded complexes is a quasi-isomorphism. It’s fairly easy to see that $(Gr(S(V)), \delta_{Gr})$ as a cochain complex is isomorphic to $(S(V),\delta^1)$, where $\delta^1$ is just the differential induced by the dual of the “unary differential” $\ell_1$ on $L$. Similarly, the morphism $Gr(F) \colon (S(V'),\delta^{\prime 1}) \to (S(V),\delta^1)$ is just the unique algebra map $F^1$ induced by the dual of the linear map $f_1$.
Since we are working over a field, the Kunneth theorem implies that $H^{\ast}(T(V),\delta^1) \cong T(H^\ast(V,\delta^{1}_1))$, where $\delta^1_1$ is the suspended dual of the differential $\ell_1$. Since we are in characteristic zero, cohomology commutes with invariants, so we further have $H^{\ast}(S(V),\delta^1_1) \cong S(H^\ast(V,\delta^{1}_1))$. Since $f_1$ is a quasi-iso, its suspended dual $F^1_1$ is a quasi-iso. So, by the above (natural) identifications in cohomology, we conclude $Gr(F)=F^1$ is a quasi-isomorphism. $\square$
just to clarify that I am asking on this point to satisfy your original request:
I appreciate that, but no pressure to cite that work if it isn’t relevant to your discussion. (My original message wasn’t a veiled “cite my work!” request. It was “maybe this is helpful for what you are working on” observation.)
Ah, thanks, I had missed that Prop. 14.13 in FHT.
Okay, there is now a Remark 3.46 on this point, in the pdf linked to here.
Following a detailed referee report (thanks to the anonymous referee!, that was useful) we have expanded out details on a number of points in the article. Version 2 is now available as pdf here.
The only request not yet reacted to is to expand the Introduction. Happy to do so, but first some vacation next week.
In 3.51(i), you are asking that to drop nilpotency, one needs to include an action of the fundamental group. But surely even in the nilpotent case, there is a (no trivial, in general) action, since in 3.50(ii) the higher homotopy groups are already modules over $\pi_1$.
Remark 3.73, “solid ring” — not to be confused with solid abelian group or solid modules (!)
Remark 4.39 has some indexing subscripts off by 1: there are two lots of duplicated -1.
On p. 2, it wasn’t obvious to me at first which was
The equivalence on the right of (1)
Also 2 lines above that, what counts as “the first map in (1)” isn’t so obvious either.
One instance of “cocyle” and one of “cocyles”.
Thanks, fixed now (here).
The “equivalence on the right” in (1), namely Dold’s equivalence, is now on the top (since the diagram had been reworked). The rationalization map is that on the left (hence the first in order of composition, but I see how that wasn’t made clear).
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