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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2010

    am starting differential string structure, but not much there yet

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2010

    I have expanded differential string structure. There is now a detailed derivation of the cocycles, culminating in the formula for the Green-Schwarz mechanism. More later.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 4th 2011
    • (edited Feb 4th 2011)

    Months ago I had written above

    More later.

    I am now getting around to this. I have expanded, polished and reworked differential string structure.

    Now that the series of definitions and propositions at smooth infinity-groupoid is typed up, I could formalize the discussion more properly.

    There is now the previous homotopy fiber definition with full precise qualification and then a systematic analysis of the computation of the homotopy fiber via construction of a fibration presentation of the differential characteristic class and a brief analysis of its ordinary fibers.

    (at varous points I currently say: see Chern-Weil theory in Smooth∞Grpd for more details. This is currently a bit of a lie, as I still need to polsih that latter entry. The details are really at smooth infinity-groupoid for the time being).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 5th 2011

    have expanded the section on th GS-mechanism at differential string structure.

    Also have fixed some signs and prefactors and also an annoying mixup of the symbols cc versus hh in the computation of the presentation of differential string structures by Lie integrated simplicial presheaves.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2011
    • (edited Apr 23rd 2011)

    Added to differential string structure statement and proof of the characterization of twisted differential string structures by globally defined 3-forms in the case that the underlying class of the twist is trivial: In the section Properties – general

  1. There is something I’m missing at a very basic level, here. Let us look at the untwisted case for clarity. Then we should have a natural notion of String diff(X)String_{diff}(X) as the 2-groupoid of principal String-bundles with String-connection on XX. And so the fact that we have a homotopy pullback diagram

    String diff(X) * H conn(X,BSpin) H conn(X,B 3U(1)) \array{String_{diff}(X)&\to&*\\ \downarrow&&\downarrow\\ \mathbf{H}_{conn}(X,\mathbf{B}Spin)&\to& \mathbf{H}_{conn}(X,\mathbf{B}^3U(1)) }

    should then be a theorem rather than a definition. In other words I’m wondering whether these two operations one can do starting with the morphism 12p 1:BSpinB 3U(1)\frac{1}{2}\mathbf{p}_1: \mathbf{B}Spin\to \mathbf{B}^3U(1) do commute: i) take the differential refinement; ii) take the homotopy fiber.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2011

    Hi Domenico,

    good to see you back!

    String bundles with connection is a priori a different notion than differential string structures, and I don’t think that they should be or would be equivalent.

    But one should state a natural comparison morphism between them. This I need to think about.

    • CommentRowNumber8.
    • CommentAuthordomenico_fiorenza
    • CommentTimeApr 30th 2011
    • (edited Apr 30th 2011)

    Hi Urs,

    sorry to have been mostly off forum lately; it has been a really exhausting period for me, bu now it’s over, and I’m back :)

    One on the main difficulties I have in clearly understanding the point above is that I have not a clear picture of what a string connection should be by itself. For instance, in Konrad Waldorf’s arXiv:0906.0117, string bundles with string connections over XX are defined as the homotopy fiber of H conn(X,BSpin)H conn(X,B 3U(1))\mathbf{H}_{conn}(X,\mathbf{B}Spin) \to \mathbf{H}_{conn}(X,\mathbf{B}^3U(1)), by using the fact that one has a notion of connections on Spin-bundles and on 2-gerbes, and the fact that the string group is the homotopy fiber of BSpinB 3U(1)\mathbf{B}Spin \to \mathbf{B}^3U(1). Tis, hoever is not (in my opinion) a good definition: it is not the specialization to String of a general notion of principal GG-bundle with connection, where GG is a smooth nn-group, but rather an ad hoc definition.

    Concerning the general notion, I guess it should be naively formulated in terms of the paths nn-groupoids of XX, and then formalized in terms of differential forms: naively, we should have kk-dimensional parallel transport for any 0kn0\leq k\leq n, and the glueing rules for local connections should precisely say that the parlallel transport along a kk-simplex which is contained in an intersection of trivializing open charts for the GG-bundle does not depend on the local chart chosen.

    One should then check that this give the usual notion of conenctions for GG a 1-group (this is well known), and for G=B nU(1)G=\mathbf{B}^n U(1), i.e., for the (higher) gerbes case. And then, finally, one should check the compatibility of this notion with homotopy pullbacks, thus relatinh H conn(X,BString)\mathbf{H}_{conn}(X,\mathbf{B}String) to the homotopy fiber of H conn(X,BSpin)H conn(X,B 3U(1))\mathbf{H}_{conn}(X,\mathbf{B}Spin) \to \mathbf{H}_{conn}(X,\mathbf{B}^3U(1)).

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2011

    Yes, that’s what I tried to say: our definition of \infty-connection specialized to string-connections (expressed in terms of parallel transport or not) is not the same as the definition of differential string structure.

    The coefficient of \infty-connections provides universally a lift of the unrefined \infty-Chern-Weil homomorphism (with values in de Rham cocycles) to the refined \infty-Chern-Weil homomorphism (with values in differential cocycles). Whereas the definition of (twisted) string structure finds the locus of spin-connections where the refined Chern-Weil homomorphism of a certain class has a certain value.

    So its different, but of course related. There should be a comparison morphism, and one should work out what it does.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2011
    • (edited May 1st 2011)

    So here is a way to go about building the comparison homomorphism:

    from the discussion at differential string structure (following SaScSt) we may compute the homotopy fibers that give (twisted) string structures by considering \infty-connections with values in the Lie 3-algebra

    (b𝔤 μ), (b \mathbb{R} \to \mathfrak{g}_{\mu}) \,,

    the mapping cone of the extension

    b𝔤 μ𝔤, b \mathbb{R} \to \mathfrak{g}_\mu \to \mathfrak{g} \,,

    where 𝔤=𝔰𝔬(n)\mathfrak{g} = \mathfrak{so}(n) and 𝔤 μ\mathfrak{g}_\mu is the string Lie 2-algebra.

    As shown there, this Lie 3-algebra is weakly equivalent to just 𝔤\mathfrak{g}: it is a suitable resolution of that on which the homotopy fiber becomes an ordinary fiber. The shifted piece bb \mathbb{R} \to \cdots is the piece that picks up the failure of the lift from 𝔤\mathfrak{g} to b𝔤 μb \mathbb{R} \to \mathfrak{g}_\mu to factor through a genuine lift throuh 𝔤 μ\mathfrak{g}_\mu. So among all (b𝔤 μ)(b \mathbb{R} \to \mathfrak{g}_\mu)-3-connections, the string connections are precisely those whose components in that shifted piece vanish.

    This should be the genuinely untwisted string structures, i.e. those for which not only the 4-class of the Chern-Simons 2-gerbe vanishes, but also its connection 3-form (that globally defined 3-form) . I’ll try to write out the formal statement tomorrow. It’s too late tonight.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2011
    • (edited May 1st 2011)

    Okay, I have added to differential string structure statement and proof that

    String diff,tw=0(X)String2Bund (X). String_{diff, tw = 0}(X) \simeq String 2Bund_\nabla(X) \,.

    In the new section Relation to string 2-connections.

    Thanks for pushing me, Domenico. That statement had been the whole motivation in SaScSt to consider the mapping cone Lie 3-algebra b𝔤 μb \mathbb{R} \to \mathfrak{g}_\mu in the first place. But with the long cohesive-topos-excursion in between, I have apparently almost forgotten some of the original motivations.

    But have a careful look at the proof. It is supposed to be an immediate simple consequence of the fibration presentation of the homotopy fibers that is established in the section further above, which we did discuss in detail a while ago. But nevertheless check if you agree with my argument.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 1st 2011

    For better exposition, I have now also discussed more explicitly how the long fiber sequence of L L_\infty-algebras extending the string extension to the right controls the entire situation : here

  2. Hi Urs,

    in a couple of days I’ll go and check the details. In the meanwhile I’ve been thinking to this, which is fairly obvious but I’m not sure we have already stressed it somewhere. Let GG be a Lie group, and PXP\to X a princiapl GG bundle. What is a GG-connection on this GG-bundle? A way of recovering the classical definition is the following: consider the oo-sheaves BG\mathbf{B}G and BG conn\mathbf{B}G_{conn} of principal GG-bundles and of principal GG-bundles with connections, respectively. There is an obvious “forget the connection” morphism BG connBG\mathbf{B}G_{conn}\to \mathbf{B}G. Then, the principal bundle PP is a morphism P:*H(X,BG)P:*\to \mathbf{H}(X,\mathbf{B}G), and the groupoid of GG-connections over PP can naturally be defined as the homotopy fiber of H(X,BG conn)H(X,BG)\mathbf{H}(X,\mathbf{B}G_{conn})\to \mathbf{H}(X,\mathbf{B}G). Once a defining cocycle for PP is chosen, from this one-line abstract definition one can spell out the local charts description of a connection on PP and gauge transformations between connections on PP.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 3rd 2011

    from this one-line abstract definition one can spell out the local charts description

    True. On the other hand one could argue that the real work with making the definition is in defining BG conn\mathbf{B}G_{conn} itself, in the first place.

  3. I’m now going into the details of the proof of

    String diff,tw=0(X)String2Bund (X). String_{diff, tw = 0}(X) \simeq String 2Bund_\nabla(X) \,.

    there’s still something in the general picture I’m missing: we start with the pullback diagram

    𝔰𝔱𝔯𝔦𝔫𝔤(n) * 𝔰𝔬(n) μ b 2𝔲(1). \array{ \mathfrak{string}(n) &\to& * \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& \mathbf{b}^2 \mathfrak{u}(1) } \,.

    as a definition of 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string}. then we have a diagram

    exp(𝔰𝔱𝔯𝔦𝔫𝔤(n)) * exp(𝔰𝔬(n)) μ exp(b 2𝔲(1)). \array{ exp(\mathfrak{string}(n)) &\to& * \\ \downarrow && \downarrow \\ exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& exp(\mathbf{b}^2 \mathfrak{u}(1)) } \,.

    and saying that this too is a pullback diagram is now a proposition, right? similarly for the next step:

    exp(𝔰𝔱𝔯𝔦𝔫𝔤(n)) conn * exp(𝔰𝔬(n)) conn μ exp(b 2𝔲(1)) conn. \array{ exp(\mathfrak{string}(n))_{conn} &\to& * \\ \downarrow && \downarrow \\ exp(\mathfrak{so}(n))_{conn} &\stackrel{\mu}{\to}& exp(\mathbf{b}^2 \mathfrak{u}(1))_{conn} } \,.

    so my question is: is this fact a general fact, i.e., expexp and exp connexp_{conn} preserve homotopy pullbacks, or there is some specific miracle happening here? (I would be for the first hypothesis). then there is the coskeletization issue, and also there I expect by a general argument that taking 3-coskeleta we should still have a fibration sequence.

    If this is correct, this should give the equivalence String diff,tw=0(X)String2Bund (X) String_{diff, tw = 0}(X) \simeq String 2Bund_\nabla(X) on general grounds.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2011

    Hi Domenico,

    yes, that’s exactly right. I think we talked about this by email once: it looks like it ought to be true that exp(...)exp(...) and similar is something like a right Quillen functor for a suitable model structure on L L_\infty-algebras. All the detailed computations at differetial string structure just confirm what one would expect on these grounds generally.

    But I haven’t yet tried to prove it in generality. I guess I should. But it looks tedious.

    Notice that in the purely algebraic case, exp(...)exp(...) is known to be as nice as you would hope, in fact its just the enriched hom on dg-algebras, in that case, as described here model structure on dg-algebra – derived hom-functor.

    But notice also that this uses the standard model structure on dg-algebras, and it looks like this is not the right one to use when we think of these dg-algebras as Chevalley-Eilenberg algebras. (For instance an ordinary Lie algebra 𝔤\mathfrak{g} is not in general fibrant in that case, but we do seem to want exp(𝔤)exp(\mathfrak{g}) to be homotopy-sensible ). It rather seems that the right model structure to use is that induced from that on all simplicial presheaves after embedding L L_\infty-algebras into these. This is also the model structure in which the square of L L_\infty-algebras that you display above is actually a homotopy pullback diagram.

  4. It rather seems that the right model structure to use is that induced from that on all simplicial presheaves after embedding L L_\infty-algebras into these

    do you mean by exp:L exp: L_\infty-algebrassimplicialpresheavesalgebras\to simplicial presheaves?, i.e., defining 𝔤𝔥\mathfrak{g}\to \mathfrak{h} to be a fibration/cofibration/weak equivalence iff exp(𝔤)exp(𝔥)exp(\mathfrak{g})\to exp(\mathfrak{h}) is? that can be fine, but it is a site-dependent notion: presheaves over which site? Cartesian spaces is the site we have in mind for what we’re after here, but it seems to me that restricting to this would be too restrictive if we are after a general noion of model structure on L L_\infty-algebras.

    on the other hand we have an embedding CE:L CE: L_\infty-algebras opdgalgebras^{op}\to dg-algebrasalgebras, so I would find it more reasonable to induce on L L_\infty-algebras the opposite model structure from the standard one on dg-algebras. note that, according to the discussion at string lie 2-algebra, this is precisely how one says that

    𝔰𝔱𝔯𝔦𝔫𝔤(n) * 𝔰𝔬(n) μ b 2𝔲(1). \array{ \mathfrak{string}(n) &\to& * \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& \mathbf{b}^2 \mathfrak{u}(1) } \,.

    is a homotopy pullback: what one is saying is that

    CE(𝔰𝔱𝔯𝔦𝔫𝔤(n)) * CE(𝔰𝔬(n)) μ CE(b 2𝔲(1)). \array{ CE(\mathfrak{string}(n)) &\leftarrow& * \\ \uparrow && \uparrow \\ CE(\mathfrak{so}(n)) &\stackrel{\mu}{\rightarrow}& CE(\mathbf{b}^2 \mathfrak{u}(1)) } \,.

    is a homotopy pushout.

    then comes the exp thing. here we could have a fantastic dream-theorem saying that exp preserves homotopy pullbacks, but as far as we know this could (should?) be false in such a generality. but still this is true for 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string}, and I find a “definition/proposition” sequence like this:

    Definition. The L L_\infty-algebra 𝔰𝔱𝔯𝔦𝔫𝔤 μ\mathfrak{string}_\mu is the homotopy pullback

    𝔰𝔱𝔯𝔦𝔫𝔤 μ(n) * 𝔰𝔬(n) μ b 2𝔲(1). \array{ \mathfrak{string}_\mu(n) &\to& * \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& \mathbf{b}^2 \mathfrak{u}(1) } \,.

    Proposition. The diagram

    exp(𝔰𝔱𝔯𝔦𝔫𝔤 μ(n)) * exp(𝔰𝔬(n)) μ exp(b 2𝔲(1)). \array{ exp(\mathfrak{string}_\mu(n)) &\to& * \\ \downarrow && \downarrow \\ exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& exp(\mathbf{b}^2 \mathfrak{u}(1)) } \,.

    is a homotopy pullback of simplicial presheaves on CartSp in the local projective model structure.

    to be a neat way of presenting this issue. Next, the relation between exp(𝔰𝔱𝔯𝔦𝔫𝔤)exp(\mathfrak{string}) and BString\mathbf{B}String (defined as the homotopy fiber of BSpinB 3U(1))\mathbf{B}Spin\to \mathbf{B}^3U(1)) can be expressed as

    Proposition. The natural morphism exp(𝔰𝔱𝔯𝔦𝔫𝔤)BStringexp(\mathfrak{string})\to \mathbf{B}String induced by the universal property of the pullback factors as exp(𝔰𝔱𝔯𝔦𝔫𝔤)cosk 3(exp(𝔰𝔱𝔯𝔦𝔫𝔤))BStringexp(\mathfrak{string})\to cosk_3(exp(\mathfrak{string}))\stackrel{\sim}{\to}\mathbf{B}String.

    and similarly for the BG conn\mathbf{B}G_{conn} version.

    Telling the story this way, one would have homotopy pullbacks taken as definitions only on the Lie group side, whereas on the Lie algebra side one would only use the exp construction. That is one would never define anything the exp(𝔤)exp(\mathfrak{g})’s as an homotopy pullback. Rather one would prove that something naturally defined by the exp construction is indeed a pullback.

    • CommentRowNumber18.
    • CommentAuthorjim_stasheff
    • CommentTimeMay 4th 2011
    I'm against turning theorems into definitions.
  5. Hi Jim,

    that’s exactly what I’m after: a way of telling the whole differential string structures story having a minimal set of definitions and a few good theorems. namely, BString\mathbf{B}String and 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string} would be defined as pullbacks, and then they would be related by the exp construction. on the connection side the situation is a bit more ambiguous: on the one hand we have no a priori notion of BStrting conn\mathbf{B}Strting_{conn}, so a possibility is to define it as the homotopy fiber of BSpinB 3U(1)\mathbf{B}Spin \to \mathbf{B}^3U(1); on the other hand, on the exp side we can do two things, either consider exp(𝔰𝔱𝔯𝔦𝔫𝔤) connexp(\mathfrak{string})_{conn}, or consider the homotopy fiber of exp(𝔰𝔬) connexp(b 2𝔲 1)exp(\mathfrak{so})_{conn}\to \exp(b^2\mathfrak{u}_1). the fact that these two are equivalent is a proposition. this brings with itself a canonical morphism exp(𝔰𝔱𝔯𝔦𝔫𝔤) connBString conn\exp(\mathfrak{string})_{conn}\to \mathbf{B}String_{conn} and taking 3-coskeleton one gets an equivalence (another poposition). So one can look at cosk 3exp(𝔰𝔱𝔯𝔦𝔫𝔤) conncosk_3\exp(\mathfrak{string})_{conn} (which is entirely defined in terms of the Lie 2-algebra 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string} as a Lie algebra integration realization of BString conn\mathbf{B}String_{conn} (which is defined as a homotopy fiber).

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2011

    Hi Domenico,

    right, I noticed I should have been more explict after I had posted my message, but then I had to run and do something else:

    I mean the mebedding of L L_\infty-algebras into simplicial presheaves over the site CartSp synthdiffCartSp_{synthdiff} of infinitesimally extended Cartesian spaces as those presheaves that have infinitesimal hom-spaces. This is described in detail by now at infinity-Lie algebroid.

    So in this perspective we regard an L L_\infty-algebra literally as a first order infinitesimal \infty-groupoid. And the exp(..)\exp(..)-construction then turns it into an “\infty-order” formal \infty-groupoid.

  6. Hi Urs,

    I can agree with this, but it contrasts with what is currently said at string Lie 2-algebra, where one takes “by definition the (,1)(\infty,1)-category of \infty-Lie algebroids to be that presented by the opposite (after passing to Chevalley-Eilenberg algebras) of the model structure on dg-algebras”.

    • CommentRowNumber22.
    • CommentAuthorzskoda
    • CommentTimeMay 4th 2011

    model structure on dg-algebras

    on commutative dg-algebras ?

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2011

    but it contrasts

    Ah, I should change that.

  7. Hi Urs,

    I think you should wait before changing that: the opposite commutative dg-algebras description is fairly universal, whereas the infinitesimal Cartesian spaces description seems to be quite bound to th realm of L L_\infty-algebras over \mathbb{R}. This seems to be too a strong restriction to me.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMay 4th 2011

    Hi Domenico,

    that is not really a restriction, but just a reflection of my current preferences. The construction at infinity-Lie algebroid is much more general than I state there.

    So in any case, I should eventually add some discussion somewhere.

  8. Even if there is a very general construction available, still I think we should be careful: it seems to me that apart from 𝔰𝔱𝔯𝔦𝔫𝔤𝔰\mathfrak{strings} there are a lot of L L_\infty algebras out there that we are defining as homotopy limits using the naive (i.e., the opposite commutative dg-algebra) model structure. The first example coming to my mind is inn(𝔤)inn(\mathfrak{g}), which is the cone over the identity of 𝔤\mathfrak{g}.

    Anyway, the main difference between the two approaches is that the opposite commutative dg-algebra approach is site independent, whereas the presheaf approach is site dependent. The situation seems to me to be similar to what happens with classical Lie algebras. There one has an abstract (and simple) bilinear algebra definition, and thyen one can prove that in finite dimension over mthbbR\mthbb{R} this is the same thing as an infinitesimal Lie group. Or, one can directly focus on \mathbb{R} and define a Lie algebra as an infinitesimal Lie group, but this will not suggest a definition of Lie algebras over an arbitrary field (at least, not a simple one) until one does not work out the real case to the poit one recovers the bilinear algebra definition.

    • CommentRowNumber27.
    • CommentAuthorjim_stasheff
    • CommentTimeMay 5th 2011
    Hi Domenico,

    The situation seems to me to be similar to what happens with classical Lie algebras. There one has an abstract (and simple) bilinear algebra definition, and thyen one can prove that in finite dimension over
    mthbbR this is the same thing as an infinitesimal Lie group.

    YES! a good model to follow
    • CommentRowNumber28.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 6th 2011
    • (edited May 16th 2011)

    Urs, I’m still confused: assume we define 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string} as the Lie 2-algebra given by the homotopy pushout

    CE(𝔰𝔱𝔯𝔦𝔫𝔤 μ(n)) * CE(𝔰𝔬(n)) μ CE(b 2𝔲(1)). \array{ CE(\mathfrak{string}_\mu(n)) &\leftarrow& * \\ \uparrow && \uparrow \\ CE(\mathfrak{so}(n)) &\stackrel{\mu}{\leftarrow}& CE(\mathbf{b}^2 \mathfrak{u}(1)) } \,.

    as at string Lie 2-algebra. Is it then true that the diagram of simplicial presheaves over CartSp

    exp(𝔰𝔱𝔯𝔦𝔫𝔤 μ(n)) * exp(𝔰𝔬(n)) μ exp(b 2𝔲(1)) \array{ exp(\mathfrak{string}_\mu(n)) &\to& * \\ \downarrow && \downarrow \\ exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& exp(\mathbf{b}^2 \mathfrak{u}(1)) } \,

    ia a homotopy pullback (in the local projective model structure)? if this is true, where is it proven? it seems to me that Proposition 10 at differential string structure should actually be a proof of this fact (together with the conn{}_conn refinement), by identifying exp(𝔰𝔱𝔯𝔦𝔫𝔤 μ(n))exp(\mathfrak{string}_\mu(n)) with an explicit model for the homotopy fiber of exp(𝔰𝔬(n))μexp(b 2𝔲(1))exp(\mathfrak{so}(n)) \stackrel{\mu}{\to} exp(\mathbf{b}^2 \mathfrak{u}(1)), am I right?

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeMay 6th 2011
    • (edited May 6th 2011)

    Right, maybe I should include that version of the statement explicitly.

    So the homotopy pullback is modeled by an ordinary pullback after passing to the resolution

    exp((b𝔤 μ))exp(𝔤). \exp((b \mathbb{R} \to \mathfrak{g}_\mu)) \stackrel{\simeq}{\to} \exp(\mathfrak{g}) \,.

    We have shown that then

    exp(b𝔤 μ)μexp(b 2) \exp(b \mathbb{R} \to \mathfrak{g}_\mu) \stackrel{\mu}{\to} \exp(b^2 \mathbb{R})

    is a fibration, and therefore the homotopy fiber in question is the ordinary fiber of this morphism over 0. But over 0 this just sets the extra generators in b𝔤 μb \mathbb{R} \to \mathfrak{g}_\mu to 0 and what remains are the generators of 𝔤 μ\mathfrak{g}_\mu.

    • CommentRowNumber30.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 16th 2011
    • (edited May 16th 2011)

    Hi Urs,

    I’m now suspecting that the relevant model category structure here is the one on unital commutative dg-coalgebras described by Vladimir Hinich in Theorem 3.1 in DG coalgebras as formal stacks.

    • CommentRowNumber31.
    • CommentAuthordomenico_fiorenza
    • CommentTimeMay 16th 2011
    • (edited May 18th 2011)

    Things could possibly go as follows: let kk be a characteristic zero field, and dgla\mathbf{dgla} and dgcu\mathbf{dgcu} the categories of differential graded Lie algebras and unital cocommutative coalgebras over kk, respectively. Then there is a well known cofree coalgebra functor 𝒞:dgladgcu\mathcal{C}:\mathbf{dgla}\to \mathbf{dgcu} and model category structures on dgla\mathbf{dgla} and dgcu\mathbf{dgcu} making 𝒞\mathcal{C} a Quillen equivalence (see Hinich, DG coalgebras as formal stacks). Moreover, the model category structure on dgla\mathbf{dgla} is extremely natural: fibrations are surjective morphisms and weak equivalences are quasi-isomorphisms. The model category strcuture on dgcu\mathbf{dgcu} in more subtle, but Hinich proves (Lemma 5.2.3 in loc. cit.) that 𝒞\mathcal{C} preserves fibrations and acyclic fibrations. Now consider the following homotopy pullback diagram of L L_\infty-algebras:

    0 𝔰𝔬(n) μ b 2𝔲(1). \array{ && 0 \\ && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\rightarrow}& \mathbf{b}^2 \mathfrak{u}(1) } \,.

    By this we mean that we are considering the following diagram in dgcu\mathbf{dgcu}:

    k 𝒞𝔰𝔬(n) μ 𝒞b 2𝔲(1). \array{ && k \\ && \downarrow \\ \mathcal{C}\mathfrak{so}(n) &\stackrel{\mu}{\rightarrow}& \mathcal{C}\mathbf{b}^2 \mathfrak{u}(1) } \,.

    Here the point is that despite all the objects in the first diagram are dglas, the morphism μ\mu is not a dgla morphism, but a L L_\infty-algebra morphism, i.e., precisely a morphism 𝒞𝔰𝔬(n)𝒞b 2𝔲(1)\mathcal{C}\mathfrak{so}(n)\to \mathcal{C}\mathbf{b}^2 \mathfrak{u}(1) in in dgcu\mathbf{dgcu}. Since all objects in dgla\mathbf{dgla} are fibrant, and 𝒞\mathcal{C} preserves fibrations (Hinich’s lemma), all the dgcu’s in the above diagram are fibrant and so, to compute this homotopy pullback we just have to replace k𝒞b 2𝔲(1)k\to \mathcal{C}\mathbf{b}^2\mathfrak{u}(1) by a fibration and then compute an ordinary pullback. This is easily done: the surjective morphism of complexes eb𝔲(1):=b 2(𝔲(1)id𝔲(1)[1])b 2𝔲(1)\mathbf{eb}\mathfrak{u}(1):=\mathbf{b}^2(\mathfrak{u}(1)\stackrel{id}{\to}\mathfrak{u}(1)[-1])\to \mathbf{b}^2\mathfrak{u}(1) is a fibration of (abelian) dglas, and is a fibrant replacement of 0b 2𝔲(1)0\to \mathbf{b}^2\mathfrak{u}(1). Let us now apply the functor 𝒞\mathcal{C} to this morphism: by Hinich’s lemma we get a fibration. We have this arrived to the diagram of dgcu\mathbf{dgcu}s

    𝒞eb𝔲(1) 𝒞𝔰𝔬(n) μ 𝒞b 2𝔲(1). \array{ && \mathcal{C}\mathbf{eb}\mathfrak{u}(1) \\ && \downarrow \\ \mathcal{C}\mathfrak{so}(n) &\stackrel{\mu}{\rightarrow}& \mathcal{C}\mathbf{b}^2 \mathfrak{u}(1) } \,.

    where all dgcu’s in the diagram are fibrant, and the right most arrow is a fibration, so we can compute its homotopy pullback simply by taking the ordinary pullback. Since all dgcu’s in this diagram are finite dimensional in each degree, we can compute this pullback by dualizing the diagram and taking the pushout. Thereore we are reduced to comput the ordinary pushout

    CEeb𝔲(1) CE𝔰𝔬(n) μ CEb 2𝔲(1), \array{ && CE\mathbf{eb}\mathfrak{u}(1) \\ && \uparrow \\ CE\mathfrak{so}(n) &\stackrel{\mu}{\leftarrow}& CE\mathbf{b}^2 \mathfrak{u}(1) } \,,

    and this pushout is CE(𝔰𝔱𝔯𝔦𝔫𝔤)CE(\mathfrak{string}).

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2011
    • (edited May 16th 2011)

    Thanks, Domenico, that looks good. I have to think then about whether the composite functor dgcudgAlg k opsAlg op[CartSp synthdiff op,sSet] proj\mathbf{dgcu} \to dgAlg_k^{op} \to sAlg^{op} \to [CartSp_{synthdiff}^{op}, sSet]_{proj} is right Quillen, where the second functor in the composite is monoidal DK functor in one direction. I’ll get back to you…

  9. Hi Urs,

    I cleaned a bit the post above. Now I’m working along the same lines on the other possible description of 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string} namely the one that starting with the diagram

    0 𝔰𝔬(n) μ b 2𝔲(1). \array{ && 0 \\ && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\rightarrow}& \mathbf{b}^2 \mathfrak{u}(1) } \,.

    and replaces μ\mu with a fibration (you have already written how this should work at differential string structure, I’m only trying to see how we can write it using the dgcu\mathbf{dgcu} model structure)

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2011
    • (edited May 18th 2011)

    Thanks, Domenico, that’s really nice.

    I’ll try to get back to you on this as soon as possible, but:

    I will be absorbed today with preparing a lecture, then tomorrow with teaching and having a guest, on Friday with running our QVEST seminar. And then I need to start reading a master thesis for which I am second reader. But THEN I get back to you on this. :-) Sorry.

  10. Hi Urs,

    absolutely nothing to be sorry! In the meanwhile I’ll go on writing here. A first very simple remark is that

    b𝔲(1) 0 0 b 2𝔲(1) \array{ \mathbf{b} \mathfrak{u}(1)&\to& 0 \\ \downarrow&& \downarrow \\ 0 &\to& \mathbf{b}^2 \mathfrak{u}(1) } \,

    is a homotopy pullback in dgla\mathbf{dgla}. this is easily proven along the lines in the above post. Then the general argument for fibration sequences applies, i.e., the 2-out-of-3 rule applied to the diagram

    b𝔲(1) 𝔰𝔱𝔯𝔦𝔫𝔤 0 0 𝔰𝔬(n) μ b 2𝔲(1) \array{ \mathbf{b}\mathfrak{u}(1)&\to&\mathfrak{string}&\to& 0 \\ \downarrow&&\downarrow&& \downarrow \\ 0&\to&\mathfrak{so}(n) &\stackrel{\mu}{\rightarrow}& \mathbf{b}^2 \mathfrak{u}(1) } \,

    tells us that also

    b𝔲(1) 𝔰𝔱𝔯𝔦𝔫𝔤 0 𝔰𝔬(n) \array{ \mathbf{b}\mathfrak{u}(1)&\to&\mathfrak{string} \\ \downarrow&&\downarrow \\ 0&\to&\mathfrak{so}(n) } \,

    is a pullback diagram. In particular this gives a canonical morphism from the homotopy pushout of the diagram

    b𝔲(1) 𝔰𝔱𝔯𝔦𝔫𝔤 0 \array{ \mathbf{b}\mathfrak{u}(1)&\to&\mathfrak{string} \\ \downarrow&& \\ 0&& } \,

    to 𝔰𝔬(n)\mathfrak{so}(n). This homotopy pushout should be nothing but cone(b𝔲(1)𝔰𝔱𝔯𝔦𝔫𝔤)cone(\mathbf{b}\mathfrak{u}(1)\to\mathfrak{string}).

    Have to run, now. More on this later.

  11. Hi Urs,

    for everyone’s convenience let me recall here what I’ve been writing you via email during the last couple of days. Discussing this topic with Jonathan Pridham I was addressed to his Unifying derived deformation theories. there, a model category DG Sp(k)DG_{\mathbb{Z}}Sp(k) is described, which is Quillen equivalent to dgcu(k)dgcu(k), and whise fibrant objects are precisely L L_\infty-algebras over kk. Moreover a morphism of L L_\infty-algebras is a fibration in DG Sp(k)DG_{\mathbb{Z}}Sp(k) if its linear part is a fibration (i.e., a surjective map) of complexes and is a weak equivalence if its linear part is a weak equivalence (i.e., a quasi-isomorphism) of complexes. Finally, the model structure on DG Sp(k)DG_{\mathbb{Z}}Sp(k) is right proper, so homotopy pullbacks can be computed by replacing only one of the morphisms with a fibration.

    with this in mind it is immediate to see that the L L_\infty-algebra morphism 𝔰𝔬(n)b 2𝔲(1)\mathfrak{so}(n)\to \mathbf{b}^2\mathfrak{u}(1) factors as

    𝔰𝔬(n)cone(b𝔲(1)𝔰𝔱𝔯𝔦𝔫𝔤(n))b 2𝔲(1) \mathfrak{so}(n)\to cone(\mathbf{b}\mathfrak{u}(1)\to \mathfrak{string}(n))\to\mathbf{b}^2\mathfrak{u}(1)

    where the first morphism is a weak equivalence and the second one is a fibration.

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2011

    Great, thanks. I am sitting in a conference right now. Will get back to you later.

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeMay 20th 2011
    • (edited May 20th 2011)

    That’s good. I need to look into this in detail. Since I feel time pressured, maybe you can help me a little with extracting the important pieces of the definitions and Quillen equivalences to an nLab entry. I am still bound in talks, but I have started an nLab page

    model structure for L-infinity algebras

    This is just a template for the moment. But I have pasted Jonathan’s proposition into there already.

    Also notice that model structure on dg-coalgebras and model structure on dg-Lie algebras already exist.

    Thanks for looking into all this!!

    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeAug 5th 2011

    Of course we are being asked to polish our article. So I have created now a personal nLab page where I keep the latest version:

    Twisted Differential String and Fivebrane Structures (schreiber)

    • CommentRowNumber40.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 24th 2019

    Added reference to final published version of (Waldorf 2013)

    diff, v54, current

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