What do homotopy algebras form?

In paper arXiv:1406.1744, we constructed a symmetric monoidal category \(LIE^{MC}\) whose objects are shifted (and filtered) L-infinity algebras. Here, we fix a cooperad \(C\) and show that algebras over the operad \(Cobar(C)\) naturally form a category enriched over \(LIE^{MC}\). Following arXiv:14...

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Bibliographic Details
Published in:arXiv.org 2015-03
Main Authors: Dolgushev, Vasily A, Hoffnung, Alexander E, Rogers, Christopher L
Format: Article
Language:English
Subjects:
Algebra
Infinity
Isomorphism
Mapping
Theorems
Online Access:Get full text
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Summary:In paper arXiv:1406.1744, we constructed a symmetric monoidal category \(LIE^{MC}\) whose objects are shifted (and filtered) L-infinity algebras. Here, we fix a cooperad \(C\) and show that algebras over the operad \(Cobar(C)\) naturally form a category enriched over \(LIE^{MC}\). Following arXiv:1406.1744, we "integrate" this \(LIE^{MC}\)-enriched category to a simplicial category \(HoAlg^{\Delta}_C\) whose mapping spaces are Kan complexes. The simplicial category \(HoAlg^{\Delta}_C\) gives us a particularly nice model of an \((\infty,1)\)-category of \(Cobar(C)\)-algebras. We show that the homotopy category of \(HoAlg^{\Delta}_C\) is the localization of the category of \(Cobar(C)\)-algebras and infinity morphisms with respect to infinity quasi-isomorphisms. Finally, we show that the Homotopy Transfer Theorem is a simple consequence of the Goldman-Millson theorem.
ISSN:2331-8422