### Roytenberg on Weak Lie 2-Algebras

#### Posted by John Baez

It’s been circulating informally since October, but now it’s available on the arXiv — a proposal for the definition of a *fully general* categorified Lie algebra!

- Dmitry Roytenberg, On weak Lie 2-algebras.

**Abstract:** A Lie 2-algebra is a linear category equipped with a functorial bilinear operation satisfying skew-symmetry and Jacobi identity up to natural transformations which themselves obey coherence laws of their own. Functors and natural transformations between Lie 2-algebras can also be defined, yielding a 2-category. Passing to the normalized chain complex gives an equivalence of 2-categories between Lie 2-algebras and 2-term “homotopy everything” Lie algebras; for strictly skew-symmetric Lie 2-algebras, these reduce to $L_\infty$-algebras, by a result of Baez and Crans. Lie 2-algebras appear naturally as infinitesimal symmetries of solutions of the Maurer–Cartan equation in some differential graded Lie algebras and $L_\infty$-algebras. In particular, (quasi-) Poisson manifolds, (quasi-) Lie bialgebroids and Courant algebroids provide large classes of examples.

When categorifying an algebraic notion, we have various choices, since we can “weaken” any equational law, replacing it by an isomorphism, or demand that it still hold “strictly” as an equation. For Lie algebras the two main equational laws are the antisymmetry of the bracket:

$[x,y] = -[y,x]$

and the Jacobi identity

$[x,[y,z]] = [[x,y],z] + [y, [x,z]]$

When Alissa Crans and I categorified the concept of Lie algebra in HDA6, we weakened the Jacobi identity, replacing it by an isomorphism we called the **Jacobiator** — but we kept the antisymmetry holding strictly. We called the resulting gadgets ‘semistrict Lie 2-algebras’. We hoped that these might be ‘sufficiently general’, i.e. that any fully weak Lie 2-algebra, when these were finally defined, would turn out to be equivalent to a semistrict one.

This appears not to be true! In this paper, Dmitry Roytenberg weakens not only the Jacobi identity but also the antisymmetry, replacing the equation

$[x,y] = -[y,x]$

by an isomorphism he calls the **alternator**. He classifies the resulting ‘weak Lie 2-algebras’, and it seems not all of them are equivalent to our old ‘semistrict’ ones… although they can all be *made* semistrict by a certain potentially destructive process.

There’s a lot more to say… but for now, if you’re a fan of higher gauge theory, 2-groups, and the like, I urge you to read the paper yourself!

I’m very happy this paper is out, since my grad student Alex Hoffnung and Chris Rogers are already running into weak Lie 2-algebras that aren’t semistrict in their work on categorified classical mechanics.

## Re: Roytenberg on Weak Lie 2-Algebras

In case anyone is interested, I had discussed aspects of this in Categorified Clifford Algebra and weak Lie $n$-Algebras.

Lie $\infty$-algebras with strict skew symmetry and weak Jacobi identity are the same as graded commutative codifferential coalgebras.

In the above entry I suggested that the right generalization to fully weak Lie $\infty$-algebras should correspond to allowing

arbitrarycodifferential coalgebras, not necessarily graded commutative.I still don’t have much more (but a little bit more) than the gut feeling “hey, that

mustbe right” to support this, but my recent conversation with Todd Trimble about the general nice properties of the category of codifferential coalgebras (which need not be not restricted to commutative coalgebras) just strenghtened my belief.I am willing to take high bets that this suggestion is right. Anyone?