The String Coffee Table

Given a simplicial set $X_{•}$, we can consider the horn ${\Lambda }_k^i$ of the $k$-simplex $\Delta$. If the maps ${\xi }_k^i:X_k\to \mathrm{Hom}({\Lambda }_k^i,X_k)$ are surjective and for $k>n$ bijective, then $X_k$ is the nerve of an $n$-groupoid. We are far more interested in Lie $n$-groupoids and so need $X_{•}$ 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 $L_{\mathrm{\infty }}$ algebras have been discussed before on this blog (see here for links to various papers), but I’ll give a lightning definition:

An $L_{\mathrm{\infty }}$ algebra is a $Z$-graded vector space $g^{•}=\{\dots ,V^{-2},V^{-1},V^0,V^1,V^2,\dots \}$ 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 $k>2$ 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 $L_{\mathrm{\infty }}$ algebra concentratd in degrees $-n,\dots ,0$.

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 $g^{•}$ 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 $>$:(. 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