The String Coffee Table

Given a simplicial set $X_{\bullet}$, we can consider the horn $\Lambda^i_k$ of the $k$-simplex $\Delta$. If the maps $\xi^i_k:X_k \to \mathrm{Hom}(\Lambda^i_k,X_k)$ are surjective and for $k\gt n$ bijective, then $X_k$ is the nerve of an $n$-groupoid. We are far more interested in Lie $n$-groupoids and so need $X_{\bullet}$ to be a simplicial manifold and the various $\xi$ maps to be surjective submersions. This agrees with the usual defintion of Lie groupoid when $n=1$ and we call this beast a Li $n$-group if $X_0=*$ (i.e. has space of objects a point).

I believe $L_{\infty}$ algebras have been discussed before on this blog (see here for links to various papers), but I’ll give a lightning definition:

An $L_{\infty}$ algebra is a $\mathbf{Z}$-graded vector space $g^{\bullet}=\{\ldots, V^{-2}, V^{-1}, V^0, V^1, V^2, \ldots\}$ with $k$-brackets (functions on $k$-fold tensor product) satisfying a generalised Jacobi property, which encodes the Leibniz rule for $k=2$, makes the 1-bracket a differential, among others. If the $k$-brackets are all zero for $k \gt 2$ then we are only talking about a dg-Lie algebra.

We then found out that if the graded vector space is concentrated in degrees -1 and 0, i.e. $$0 \to V^{-1} \to V^0 \to 0,$$ then $V^0$ is a Lie algebra. More generally a Lie n-algebra is an $L_{\infty}$ algebra concentratd in degrees $-n, \ldots, 0$.

He then went to define differential forms on an $n$-simplex such that the complex of such things is a simplicial dg-algebra. After some nested defintions, we got to the Maurer-Cartan simplicial set of $g^{\bullet}$ which is the same as the nerve of a dg-Lie algebra.

The next talk is starting. More…

D

PS - Fri 1.42PM local - Many apologies for not getting this one up, or even finishing it off - I just couldn’t get the XHTML to work -actually it was a LaTeX problem $\gt$:(. I’ll just give some references. Sorry Urs - this was one you would have liked.

Hinich, V and Schechtman, V, Homotopy Lie algebras

Hinich, V, DG coalgebras as formal stacks

There was also reference to the work of J. Duskin and T. Beke, but a search just now on MathSciNet didn’t turn up anything I could conceptually link to the talk.

Must dash

David