Algebraic & Geometric Topology Volume 20, issue 3 (2020)

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An explicit model for the homotopy theory of finite-type Lie $n$–algebras

Christopher L Rogers

Algebraic & Geometric Topology 20 (2020) 1371–1429

DOI: 10.2140/agt.2020.20.1371

Abstract

Lie $n$ –algebras are the $L_{\infty}$ analogs of chain Lie algebras from rational homotopy theory. Henriques showed that finite-type Lie $n$ –algebras can be integrated to produce certain simplicial Banach manifolds, known as Lie $\infty$ –groups, via a smooth analog of Sullivan’s realization functor. We provide an explicit proof that the category of finite-type Lie $n$ –algebras and (weak) $L_{\infty}$ –morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Roughly speaking, this CFO structure can be thought of as the transfer of the classical projective CFO structure on nonnegatively graded chain complexes via the tangent functor. In particular, the weak equivalences are precisely the $L_{\infty}$ –quasi-isomorphisms. Along the way, we give explicit constructions for pullbacks and factorizations of $L_{\infty}$ –morphisms between finite-type Lie $n$ –algebras. We also analyze Postnikov towers and Maurer–Cartan/deformation functors associated to such Lie $n$ –algebras. The main application of this work is our joint paper with C Zhu (1127–1219), which characterizes the compatibility of Henriques’ integration functor with the homotopy theory of Lie $n$ –algebras and that of Lie $\infty$ –groups.

Keywords

homotopy Lie algebra, Lie $n$–algebra, category of fibrant objects, simplicial manifold

Mathematical Subject Classification 2010

Primary: 17B55, 18G55, 55U35

Secondary: 55P62

References

Publication

Received: 16 September 2018

Revised: 2 June 2019

Accepted: 1 October 2019

Published: 27 May 2020

Authors

Christopher L Rogers
Department of Mathematics and Statistics University of Nevada, Reno Reno, NV United States