Higher Morphisms of Lie n-Algebras and L-infinity Algebras

February 15, 2007

Higher Morphisms of Lie n-Algebras and L-infinity Algebras

Posted by Urs Schreiber

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A Lie algebra is a vector space $V$ with a skew linear product map $V\otimes V \to V$ satisfying the Jacobi identity.

A Lie 2-algebra is a 2-vector space $V$ with a skew linear product functor $V \otimes V \to V$ having a Jacobi isomorphism satisfying a coherence condition.

For the case that the 2-vector spaces in question are Baez-Crans 2-vector spaces, i.e. $\mathrm{Disc}(k)$ -module categories, the (“semistrict”) Lie 2-algebras obtained this way are equivalent to 2-term $L_\infty$ -algebras.

All that is the content of

J. Baez, A. Crans, HDA VI: Lie 2-Algebra

$L_\infty$ -algebras have been extensively studied, long before John and Alissa dreamed up Lie 2-algebras, by people like Jim Stasheff, Tom Lada, Martin Markl and many others. I once collected some literature that I found helpful here.

So what’s the point of Lie $n$ -algebras, then? What’s the point of reformulating $n$ -term $L_\infty$ -algebras as linear categories with a skew-linear product functor on them satisfying a Jacobi identity up to higher coherent isomorphism?

One point is that, while equivalent to $n$ -term $L_\infty$ -algebras, the Lie $n$ -algebras are conceptually more transparent. The equivalence tells us that and how $L_\infty$ -algebras are indeed a categorified notion of Lie algebra.

That’s more than just a philosophical point. In particular, since Lie $n$ -algebras are a kind of monoidal $n$ -categories, there is a canonical way to see what the right notion of morphisms between them are. And, crucially, what the right higher morphisms are.

When John and Alissa wrote their paper, they could not locate in the literature a definition of 2-morphism of $L_\infty$ -algebras #. And it is not really easy to just guess the right definition.

But what is rather straightforward, (though getting increasingly tedious as $n$ increases), is to work out the right notion of 1- and 2-morphisms of Lie $2$ -algebras. These come from functors and natural transformations of the underlying categories, respecting the available structure in a suitable way. It’s clear what these should look like. (def. 37, p. 36 in the above paper.)

The issue of higher morphisms is of course most crucial for understanding the structure of our categorified Lie algebras, since they determine the notion of equivalence between these.

So : what is known about higher morphisms and about the notion of equivalence for $L_\infty$ -algebras?

Behind the scenes there is a long dicussion going on about this. Here I collect some remarks and some literature.

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As John explains here, the issue is in principle completely solved in the old text

J. M. Boardman, R. M. Vogt
Homotopy invariant algebraic structures
Lecture Notes in Mathematics 347, Springer, 1973

Though it seems to be less clear what exactly that implies in practice.

I am very grateful to Danny Stevenson for pointing out the following two references to me:

Marco Grandis
On the homotopy structure of strongly homotopy associative algebras
Journal of Pure and Applied Algebra, Volume 134, Number 1, 5

Martin Markl
Homotopy Diagrams of Algebras
math.AT/0103052

Markl here defines a homotopy between strongly homotopy homomorphisms of $L_\infty$ -algebras (page 17 and also Proposition 29 and 30) and proves that this gives rise to an equivalence relation. In particular he shows that this gives rise to the same notion of homotopy as that considered by Grandis in the former paper.

If one is just interested in equivalence of $L_\infty$ -algebras, and not in how this equivalence is realized in terms of morphisms going back and forth, there is supposed to be a shortcut:

a 1-morphism of $L_\infty$ -algebras is regarded as an equivalence, if, when regarded just as a chain map of the underlying cochain complex, it is a quasi-isomorphism of chain complexes, i.e. if it induces an isomorphism on cohomology.

This can be found on p. 13 of the seminal

Maxim Kontsevich
Deformation quantization of Poisson manifolds, I
q-alg/9709040

As a theorem following this definition, Kontsevich says that for any such quasi-isomorphism of $L_\infty$ -algebras one can find an $L_\infty$ -morphism going the other way, which is the inverse morphism on the level of the underlying cohomology of cochain complexes.

One is hence naturally lead to wonder whether these two morphisms, $L \stackrel{f}{\to} L' \stackrel{g}{\to} L$ say, that go back and forth and that are inverses on the level of cohomology, do extend to a true equivalence at the level of $L_\infty$ morphisms.

I was asked to forward the following reply to this question by Johannes Hübschmann to the $n$ -Café:

I doubt that, the query

$f\circ g$ and $g\circ f$ are homotopic to the identities of $L'$ and $L$ respectively through $L_\infty$ morphisms”

has an affirmative answer. For the special case of a contraction, the basic difficulty is explained on p. 855, just after (2.5), of

J. Huebschmann and J. Stasheff:
Formal solution of the master equation via HPT and deformation theory,
Forum mathematicum 14 (2002), 847–868,
math.AG/9906036.

For the (co)free case (tensor coalgebra case), one can always arrive at a tensor coalgebra contraction in the sense that the homotopy is compatible with the coalgebra structure. This can be achieved for example by means of the methods developed in

J. Huebschmann and T. Kadeishvili:
Small models for chain algebras
Math. Z. 207(1991), 245–280.

In the situation discussed below (symmetric coalgebra), one can get a filtered contraction of coaugmented differential graded coalgebras by applying symmetrization to the corresponding contraction involving the tensor coalgebras. Even though this tensor coalgebra contraction will be one of coalgebras, that is, the homotopy is compatible with the coalgebra structure, symmetrization spoils this property, and the homotopy coming into play in the symmetrized contraction is no longer compatible with the coalgebra structures.

The notion of homotopy for cocommutative coalgebras developed in Schlessinger-Stasheff is more subtle.

Posted at February 15, 2007 3:06 PM UTC

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Re: Higher Morphisms of Lie n-Algebras and L-infinity Algebras

The work of Boardman and Vogt mentioned above has found a recent reformulation and generalization in the papers of Katherine Hess, Jonathan Scott, Paul-Eugene Parent and several other collaborators (Tonks, Baues.) Basically they point out that certain bimodules over operads can in some sense universally characterize the morphisms between algebras of those operads. Module here is in the traditional sense of an object acted upon by a monoid, here another symmetric sequence which is acted upon by the operad. In Boardman and Vogt the module of two-colored trees is acted upon by the operad of trees, by the usual grafting.

The “certain bimodules” are those equipped with a diagonal, which allows composition to be well defined. These are called co-rings of the operad.
The exposition in Jonathan’s talk at the Alpine Operads Workshop (organized by the three authors) is very understandable: Co-rings over operads.

Here is the paper: Co-rings over operads characterize morphisms. Here is Katherine’s talk at Category Theory 06 in
Nova Scotia, at White Point.

Posted by: Stefan on February 15, 2007 10:55 PM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras and L-infinity Algebras

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I at least suffered from language problems. Since Lie 2-algebras are $L_\infty$ -algebras with only $V_0$ and $V_1$ non-zero (I was slow to pick up on the crucial indexing), then the standard definition of $L_\infty$ -morphism from $V$ to $W$ truncates: the maps $f_i: V^{\otimes i}\to W$ are of degree i-1 and hence must be 0 in the Lie 2-algebra case for $i \geq 3$ . Now how should I say that in 2-categorical language?

With regard to Johannes’ message, it’s perhaps worth remarking that the associative or $A_\infty$ version is much better behaved and ancient.

jim

Posted by: jim stasheff on February 16, 2007 2:19 PM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras and L-infinity Algebras

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Sorry I didn’t answer this better sooner. I still don’t speak category fluently. I’m beginning to realize (correct me if I’m wrong) that an $n$ -morphism for $L_\infty$ -algebras is just the $n$ th stage of an $L_\infty$ morphism, which is a coherent sequence of maps $f_i: V^{\otimes n} \to W$ of degree n-1 (assuming that $d=m_1$ is of degree -1. In particular, $f_1$ need not respect the brackets but does so up to homotopy. So $f_n$ is an n-morphism?

If so, then the existing literature together with your assumption that $V$ and $W$ have only the 0 and 1 terms non-zero, gives what you worked out on your own - how did you do that? categorically?

My problem with language is that we speak of a morphism of $L_\infty$ -algebras, meaning the full sequence. We also talk about $L_n$ -algebras if the k-fold brackets exist only for $k\leq n$ and $L_n$ -morphisms.

Simllarly, watch out for $\infty$ -categorify as I can think of at least 3 meanings:

create the (strict) category of $- - -$ algebras up to strong (all higher) homotopies

create one of Fukaya’s infinity categories in a particular context

create an $\infinity$ -category in a particular context

Posted by: jim stasheff on February 17, 2007 6:56 PM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras and L-infinity Algebras

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Jim Stasheff wrote:

I’m beginning to realize (correct me if I’m wrong) that an $n$ -morphism for $L_\infty$ -algebras is just the $n$ -th stage of an $L_\infty$ morphism, which is a coherent sequence of maps…

No! An $L_\infty$ -morphism is just a 1-morphism between $L_\infty$ algebras. A 2-morphism is the suitable sort of homotopy between $L_\infty$ -morphisms, and so on.

You’re more familiar with this stuff in the context of plain old chain complexes:

A 1-morphism between chain complexes is a chain map.
A 2-morphism is a chain homotopy between chain maps.
A 3-morphism is a chain homotopy between chain homotopies between chain maps…

and so on.

When our chain complex is equipped with the structure of an $L_\infty$ -algebra, all these maps and higher homotopies need to be equipped with a lot of extra structure of their own to become $n$ -morphisms.

In the case of 1-morphisms, this extra structure is precisely that of an $L_\infty$ -morphism!

So: an $L_\infty$ -morphism is a 1-morphism between $L_\infty$ -algebras. What Urs wants is a concrete description of the $n$ -morphisms for $n > 1$ . It’s all implicit in Boardman and Vogt, but he wants something explicit.

Alissa Crans and I worked it out for $n = 2$ — but only in the special case of Lie 2-algebras, which are severely truncated $L_\infty$ -algebras.

(By the way, you need to turn on TeX before using TeX in your comments, if you want it to work. Do this by choosing a ‘text filter’ like ‘itex to MathML with parbreaks’. I fixed this for your above post, but fixing it is sort of complicated: I had to delete your post and post it again.)

Posted by: John Baez on February 17, 2007 7:08 PM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras and L-infinity Algebras

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Apologies for being slow 1) to catch on to the language and B) to use the right filter.

Correspondence with Urs has raised the following issue in re: even 2-morphisms aka homtopies

If $A\to B$ is a 1-morphism of dg algebras (graded commutative if you prefer), how would you like the homotopy to respect the algebra structure:

as a derivation?

as an algebra map $A \to B[[t,dt]]$ ?

other?

jim

Posted by: jim stasheff on February 20, 2007 2:41 AM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras and L-infinity Algebras

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Correspondence with Urs has raised the following issue in re: even 2-morphisms aka homtopies

If $A \to B$ is a 1-morphism of dg algebras (graded commutative if you prefer), how would you like the homotopy to respect the algebra structure

I’ll provide some background to this question:

As I recalled in the entry above, by realizing 2-term $L_\infty$ -algebras as categories internal to vector spaces and equipped with an antisymmetric monoidal functor having a coherent Jacobi isomorphism, one naturally finds a notion of 1- and 2-morphism between them: 1-morphisms are suitable monoidal functors between these categories and 2-morphisms are suitable natural transformations between those.

Alissa and John worked out what this notion of 1- and 2-morphism means in terms of $L_\infty$ -data.

For 1-morphisms it reproduces the ordinary notion of morphism of $L_\infty$ -algebras.

For 2-morphisms it produces a formula that has apparently not appeared explicitly (?) elsewhere in the literature.

Now, one can simply go ahead and translate that formula into one of the languages used to deal with $L_\infty$ -algebras – like that of quasi-free differential graded algebras (“qFDAs”) or, dually, of codifferential coalgebras – and see what it means there.

When I do so, I find (in the qFDA language) the situation that is described on pages 5 and 6 of my FDA laboratory:

the 2-morphism given by Alissa and John is something like a linearized chain homotopy between FDA morphism chain maps.

In math.DG/0406445 such morphisms are addressed $\Phi$ -Leibnitz operators (and this “linearized” version is also what I am using in the last section of my thesis).

Posted by: urs on February 20, 2007 8:24 AM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras and L-infinity Algebras

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One is hence naturally lead to wonder whether these two morphisms, $L \stackrel{f}{\to} L' \stackrel{g}{\to} L$ say, that go back and forth and that are inverses on the level of cohomology, do extend to a true equivalence at the level of $L_\infty$ morphisms.

As I wrote, Hübschmann answers this to the negative. But if this is so, then my question is:

how do we know, then, that $L_\infty$ -morphisms that are isomorphisms on cohomology do supply the right notion of equivalence of $L_\infty$ -algebras?

Posted by: urs on February 18, 2007 1:19 PM | Permalink | Reply to this

Re: Higher Morphisms of Lie n-Algebras and L-infinity Algebras

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I think this problem can be treated very categorically as follows. First consider the bicategory of simplicial distributors (= bimodules = profunctors between simplicial categories with values in simplicial sets). Then we should “derived” it as follows. Instead of simplicial transformation we cosider homotopy coherent transformations between simplicial bimodules. Instead of usual composition of bimodules we consider its derived composition. This has been done in my papers “Categorical strong shape theory”. This bicategory is biclosed i.e. has left and right internal homs. Every simplicial functor $F: A\rightarrow B$ induces a distributor as for usual bicategory of distributors. So, for a simplicial functor $F$ between two simplicial categories one can consider its left and right extension along itself (i.e. $Lan(F,F)$ and $Ran(F,F)$ ). These two distributors are both monads in this bicategory and one can consider the Kleisli categories of these monads.

If $F$ satisfies an additional condition which I call formality (coformality) and which I will not state here (any full functor is formal and coformal, for example) , then these two monads can be characterised axiomatically quite nicely. In the case of right extension, for examle, this is a unique simplicial distributor $T$ equipped with a map from a hom distributor of $B$ to $T$ with the properties that for any object $X$ of $A$ there is a weak equivalence $B(Y,F(X)) \rightarrow T(Y,F(X))$ and $T$ is continuous with respect to $T.$ There is a dual characterisation for left extensions. I call the Kleisli categories of these monads strong shape and coshape categories of $F.$

Now what it has to do with $L_{\infty}$ morphisms? First, the category of chain complexes can be enrched in simplicial sets. The operad of $L_{\infty}$ -algebras (or any other operad with some cofibrancy conditions (again I will not state them here) in chain complexes or topological spaces or simplicial sets) induces a simplicial monad and we also have a simplicial structure on its category of algebras. So it makes sense to consider a full subcategory of free algebra in its category of algebra. The coshape category of this iclusion is equivalent to the category of algebras localised with respect to the morphisms which are quasiisomorphisms as morphism of chain complexes. Since the monad of left extension is given by a simplicial distributor its $0$ -simplices should be treated as strong homotopy morphisms of $L_{\infty}$ -algebras, $1$ -simplices as their homotopies etc.. And we have its composition ( $Lan(F,F)$ is an $A_{\infty}$ -monad, indeed) and a morphism (strong or strict) between two algebras is a quasiisomorphism if and only if it has a strong invers and two homotopies between identities etc. etc..

Since the comonad is defined up to strong homotopy equivalence between simplicial distributors it admits many different explicit forms. One can use bar construction to describe it or corings in bimodules of operads or Boardman-Vogt resolutions etc.etc. and thus obtain different explicit descriptions of what a strong morphism and a homotopy is. It is not technically simple, so I will not describe it here.

By the way, the idea to use corings of bimodules between operads for description of weak functors between weak $\omega$ -categories is in my paper “Monoidal globular categories …” and Jim Dolan has done interesting calculation with this idea, which were presented in Minesota in 2004.

Posted by: Michael on February 19, 2007 1:08 AM | Permalink | Reply to this

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