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- W2890970281 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. In this paper, 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 non-negatively 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 arXiv:1609.01394 with C. Zhu which characterizes the compatibility of Henriques' integration functor with the homotopy theory of Lie $n$-algebras and that of Lie $infty$-groups." @default.
- W2890970281 created "2018-09-27" @default.
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- W2890970281 date "2020-05-27" @default.
- W2890970281 modified "2023-09-23" @default.
- W2890970281 title "An explicit model for the homotopy theory of finite-type Lie n–algebras" @default.
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- W2890970281 doi "https://doi.org/10.2140/agt.2020.20.1371" @default.
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