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

#### Posted by Urs Schreiber

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.

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.

## 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.