On an Enhancement of the Category of Shifted L<inf>∞</inf>-Algebras

Abstract

© 2016, Springer Science+Business Media Dordrecht. We construct a symmetric monoidal category SLie∞MC whose objects are shifted L∞-algebras equipped with a complete descending filtration. Morphisms of this category are “enhanced” infinity morphisms between shifted L∞-algebras. We prove that any category enriched over SLie∞MC can be integrated to a simplicial category whose mapping spaces are Kan complexes. The advantage gained by using enhanced morphisms is that we can see much more of the simplicial world from the L∞-algebra point of view. We use this construction in a subsequent paper (Dolgushev et al. Adv. Math. 274, 562–605, 2015) to produce a simplicial model of a (∞,1)-category whose objects are homotopy algebras of a fixed type.