Higher categorified algebras versus bounded homotopy algebras

On the infinity category of homotopy Leibniz algebras

We discuss various concepts of $\infty$-homotopies, as well as the relations between them (focussing on the Leibniz type). In particular $\infty$-$n$-homotopies appear as the $n$-simplices of the nerve of a complete Lie ${\infty}$-algebra. In the nilpotent case, this nerve is known to be a Kan complex \cite{Get09}. We argue that there is a quasi-category of $\infty$-algebras and show that for truncated $\infty$-algebras, i.e. categorified algebras, this $\infty$-categorical structure projects to a strict 2-categorical one. The paper contains a shortcut to $(\infty,1)$-categories, as well as a …

Tree- versus graph-level quasilocal Poincare duality on S^1

Among its many corollaries, Poincare duality implies that the de Rham cohomology of a compact oriented manifold is a shifted commutative Frobenius algebra --- a commutative Frobenius algebra in which the comultiplication has cohomological degree equal to the dimension of the manifold. We study the question of whether this structure lifts to a "homotopy" shifted commutative Frobenius algebra structure at the cochain level. To make this question nontrivial, we impose a mild locality-type condition that we call "quasilocality": strict locality at the cochain level is unreasonable, but it is reaso…

从环路群到2-群

From Loop Groups to 2-Groups

本文构造了与李2-代数$\mathfrak{g}_k$等价的无限维李2-群，其对象为基于路径，自同构为环路群的Kac-Moody中心扩张，并由此得到$\mathrm{String}(n)$群。