$L_{\infty }$-Morphisms

Abstract

In Chap. 10 we introduced the décalage isomorphisms, which can be interpreted as a canonical equivalence between the categories of $L_{\infty }$ and $L_{\infty }[1]$-algebras, both endowed with linear morphisms. Here we want to enrich these categories by enhancing their sets of morphisms; more precisely, for every pair (V, q), (W, r) of $L_{\infty }[1]$-algebras we introduce the notion of $L_{\infty }$-morphism $f:(V,q)\rightsquigarrow (W,r)$ together with the composition rule of two of them. The $L_{\infty }$-morphisms between $L_{\infty }$-algebras are then defined by imposing that the décalage isomorphisms give again an equivalence of categories. The key point in the above construction is the interpretation of every $L_{\infty }[1]$-algebra as formal pointed DG-manifolds equipped with a fixed cogenerator onto its tangential complex.