Models for Operads

Abstract

We study properties of differential graded (dg) operads modulo weak equivalences, that is, modulo the relation given by the existence of a chain of dg operad maps inducing a homology isomorphism. This approach, naturally arising in string theory, leads us to consider various versions of models. Besides of some applications in topology and homological algebra we show that our theory enables one to prove the existence of homotopy structures on physically relevant spaces. For example, we prove that a closed string-field theory induces a homotopy Lie algebra structure on the space of relative states, which is one of main results of T. Kimura, A. Voronov and J. Stasheff (see [16]). Our theory gives a systematic way to prove statements of this type. The paper is a corrected version of a preprint which began to circulate in March 1994, with some new examples added.