Volume 20, issue 3 (2020)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
An explicit model for the homotopy theory of finite-type Lie $n$–algebras

Christopher L Rogers

Algebraic & Geometric Topology 20 (2020) 1371–1429

Lie n–algebras are the L 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 –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–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–quasi-isomorphisms. Along the way, we give explicit constructions for pullbacks and factorizations of L–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 –groups.

homotopy Lie algebra, Lie $n$–algebra, category of fibrant objects, simplicial manifold
Mathematical Subject Classification 2010
Primary: 17B55, 18G55, 55U35
Secondary: 55P62
Received: 16 September 2018
Revised: 2 June 2019
Accepted: 1 October 2019
Published: 27 May 2020
Christopher L Rogers
Department of Mathematics and Statistics
University of Nevada, Reno
Reno, NV
United States