Added reference to final published version of (Waldorf 2013)

]]>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)

]]>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!!

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

]]>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 $DG_{\mathbb{Z}}Sp(k)$ is described, which is Quillen equivalent to $dgcu(k)$, and whise fibrant objects are precisely $L_\infty$-algebras over $k$. Moreover a morphism of $L_\infty$-algebras is a fibration in $DG_{\mathbb{Z}}Sp(k)$ 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 $DG_{\mathbb{Z}}Sp(k)$ 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 $L_\infty$-algebra morphism $\mathfrak{so}(n)\to \mathbf{b}^2\mathfrak{u}(1)$ factors as

$\mathfrak{so}(n)\to cone(\mathbf{b}\mathfrak{u}(1)\to \mathfrak{string}(n))\to\mathbf{b}^2\mathfrak{u}(1)$where the first morphism is a weak equivalence and the second one is a fibration.

]]>Hi Urs,

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

$\array{ \mathbf{b} \mathfrak{u}(1)&\to& 0 \\ \downarrow&& \downarrow \\ 0 &\to& \mathbf{b}^2 \mathfrak{u}(1) } \,$is a homotopy pullback in $\mathbf{dgla}$. 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

$\array{ \mathbf{b}\mathfrak{u}(1)&\to&\mathfrak{string}&\to& 0 \\ \downarrow&&\downarrow&& \downarrow \\ 0&\to&\mathfrak{so}(n) &\stackrel{\mu}{\rightarrow}& \mathbf{b}^2 \mathfrak{u}(1) } \,$tells us that also

$\array{ \mathbf{b}\mathfrak{u}(1)&\to&\mathfrak{string} \\ \downarrow&&\downarrow \\ 0&\to&\mathfrak{so}(n) } \,$is a pullback diagram. In particular this gives a canonical morphism from the homotopy pushout of the diagram

$\array{ \mathbf{b}\mathfrak{u}(1)&\to&\mathfrak{string} \\ \downarrow&& \\ 0&& } \,$to $\mathfrak{so}(n)$. This homotopy pushout should be nothing but $cone(\mathbf{b}\mathfrak{u}(1)\to\mathfrak{string})$.

Have to run, now. More on this later.

]]>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.

]]>Hi Urs,

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

$\array{ && 0 \\ && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\rightarrow}& \mathbf{b}^2 \mathfrak{u}(1) } \,.$and replaces $\mu$ 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 $\mathbf{dgcu}$ model structure)

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

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

$\array{ && 0 \\ && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\rightarrow}& \mathbf{b}^2 \mathfrak{u}(1) } \,.$By this we mean that we are considering the following diagram in $\mathbf{dgcu}$:

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

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

$\array{ && CE\mathbf{eb}\mathfrak{u}(1) \\ && \uparrow \\ CE\mathfrak{so}(n) &\stackrel{\mu}{\leftarrow}& CE\mathbf{b}^2 \mathfrak{u}(1) } \,,$and this pushout is $CE(\mathfrak{string})$.

]]>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.

]]>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

$\exp((b \mathbb{R} \to \mathfrak{g}_\mu)) \stackrel{\simeq}{\to} \exp(\mathfrak{g}) \,.$We have shown that then

$\exp(b \mathbb{R} \to \mathfrak{g}_\mu) \stackrel{\mu}{\to} \exp(b^2 \mathbb{R})$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 $b \mathbb{R} \to \mathfrak{g}_\mu$ to 0 and what remains are the generators of $\mathfrak{g}_\mu$.

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

$\array{ CE(\mathfrak{string}_\mu(n)) &\leftarrow& * \\ \uparrow && \uparrow \\ CE(\mathfrak{so}(n)) &\stackrel{\mu}{\leftarrow}& CE(\mathbf{b}^2 \mathfrak{u}(1)) } \,.$as at string Lie 2-algebra. Is it then true that the diagram of simplicial presheaves over CartSp

$\array{ exp(\mathfrak{string}_\mu(n)) &\to& * \\ \downarrow && \downarrow \\ exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& exp(\mathbf{b}^2 \mathfrak{u}(1)) } \,$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 ${}_conn$ refinement), by identifying $exp(\mathfrak{string}_\mu(n))$ with an explicit model for the homotopy fiber of $exp(\mathfrak{so}(n)) \stackrel{\mu}{\to} exp(\mathbf{b}^2 \mathfrak{u}(1))$, am I right?

]]>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 ]]>

Even if there is a very general construction available, still I think we should be careful: it seems to me that apart from $\mathfrak{strings}$ there are a lot of $L_\infty$ 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 $inn(\mathfrak{g})$, which is the cone over the identity of $\mathfrak{g}$.

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 $\mthbb{R}$ 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.

]]>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.

]]>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 $L_\infty$-algebras over $\mathbb{R}$. This seems to be too a strong restriction to me.

]]>but it contrasts

Ah, I should change that.

]]>model structure on dg-algebras

on commutative dg-algebras ?

]]>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 $(\infty,1)$-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”.

]]>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 $L_\infty$-algebras into simplicial presheaves over the site $CartSp_{synthdiff}$ 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 $L_\infty$-algebra literally as a first order infinitesimal $\infty$-groupoid. And the $\exp(..)$-construction then turns it into an “$\infty$-order” formal $\infty$-groupoid.

]]>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, $\mathbf{B}String$ and $\mathfrak{string}$ 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 $\mathbf{B}Strting_{conn}$, so a possibility is to define it as the homotopy fiber of $\mathbf{B}Spin \to \mathbf{B}^3U(1)$; on the other hand, on the exp side we can do two things, either consider $exp(\mathfrak{string})_{conn}$, or consider the homotopy fiber of $exp(\mathfrak{so})_{conn}\to \exp(b^2\mathfrak{u}_1)$. the fact that these two are equivalent is a proposition. this brings with itself a canonical morphism $\exp(\mathfrak{string})_{conn}\to \mathbf{B}String_{conn}$ and taking 3-coskeleton one gets an equivalence (another poposition). So one can look at $cosk_3\exp(\mathfrak{string})_{conn}$ (which is entirely defined in terms of the Lie 2-algebra $\mathfrak{string}$ as a Lie algebra integration realization of $\mathbf{B}String_{conn}$ (which is defined as a homotopy fiber).

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

do you mean by $exp: L_\infty$-$algebras\to simplicial presheaves$?, i.e., defining $\mathfrak{g}\to \mathfrak{h}$ to be a fibration/cofibration/weak equivalence iff $exp(\mathfrak{g})\to exp(\mathfrak{h})$ 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 $L_\infty$-algebras.

on the other hand we have an embedding $CE: L_\infty$-$algebras^{op}\to dg$-$algebras$, so I would find it more reasonable to induce on $L_\infty$-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

$\array{ \mathfrak{string}(n) &\to& * \\ \downarrow && \downarrow \\ \mathfrak{so}(n) &\stackrel{\mu}{\to}& \mathbf{b}^2 \mathfrak{u}(1) } \,.$is a homotopy pullback: what one is saying is that

$\array{ CE(\mathfrak{string}(n)) &\leftarrow& * \\ \uparrow && \uparrow \\ CE(\mathfrak{so}(n)) &\stackrel{\mu}{\rightarrow}& CE(\mathbf{b}^2 \mathfrak{u}(1)) } \,.$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 $\mathfrak{string}$, and I find a “definition/proposition” sequence like this:

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

*Proposition. The diagram*

*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 $exp(\mathfrak{string})$ and $\mathbf{B}String$ (defined as the homotopy fiber of $\mathbf{B}Spin\to \mathbf{B}^3U(1))$ can be expressed as

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

and similarly for the $\mathbf{B}G_{conn}$ 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 $exp(\mathfrak{g})$’s as an homotopy pullback. Rather one would prove that something naturally defined by the exp construction is indeed a pullback.