Models for Operads

TL;DR

This paper develops a systematic theory of models for differential graded operads modulo weak equivalences, with applications in topology, homological algebra, and string theory, including proving the existence of homotopy structures on physical spaces.

Contribution

It introduces a new framework for understanding dg operads modulo weak equivalences, enabling proofs of homotopy structures in physically relevant spaces.

Findings

Proves the existence of homotopy Lie algebra structures induced by string-field theories.

Provides a systematic method to establish homotopy structures on complex spaces.

Includes new examples illustrating the theory's applications.

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.