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Lie operad

Context

Lie theory

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Background

Smooth structure

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Higher groupoids

Lie theory

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∞-Lie groupoids

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∞-Lie algebroids

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Formal Lie groupoids

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Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

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Higher algebra

Algebraic theories

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Algebras and modules

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Higher algebras

  • symmetric monoidal (∞,1)-category of spectra

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Model category presentations

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Geometry on formal duals of algebras

Theorems

Contents

Idea

Given an ambient additive symmetric monoidal category, the Lie operad is the operad whose algebras over an operad are Lie algebras in that category. It is a quadratic operad? whose Koszul dual is the operad for commutative algebras.

(This Koszul duality is what makes L-∞ algebras be equivalent to (semifree) differential graded coalgebras.)

In the case of the monoidal category of chain complexes, we also say dg-Lie-operad. Its cofibrant resolution in the model structure on operads is the L-infinity operad whose algebras over an operad are L-∞-algebras.

References

Last revised on January 11, 2017 at 14:24:20. See the history of this page for a list of all contributions to it.