What do homotopy algebras form?

In paper [4], we constructed a symmetric monoidal category SLie∞MC whose objects are shifted (and filtered) L∞-algebras. Here, we fix a cooperad C and show that algebras over the operad Cobar(C) naturally form a category enriched over SLie∞MC. Following [4], we “integrate” this SLie∞MC-enriched cate...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2015-04, Vol.274, p.562-605
Main Authors: Dolgushev, Vasily A., Hoffnung, Alexander E., Rogers, Christopher L.
Format: Article
Language:English
Subjects:
Higher categories
Homotopy algebras
Operads
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Summary:In paper [4], we constructed a symmetric monoidal category SLie∞MC whose objects are shifted (and filtered) L∞-algebras. Here, we fix a cooperad C and show that algebras over the operad Cobar(C) naturally form a category enriched over SLie∞MC. Following [4], we “integrate” this SLie∞MC-enriched category to a simplicial category HoAlgCΔ whose mapping spaces are Kan complexes. The simplicial category HoAlgCΔ gives us a particularly nice model of an (∞,1)-category of Cobar(C)-algebras. We show that the homotopy category of HoAlgCΔ is the localization of the category of Cobar(C)-algebras and ∞-morphisms with respect to ∞-quasi-isomorphisms. Finally, we show that the Homotopy Transfer Theorem is a simple consequence of the Goldman–Millson theorem.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2015.01.014