Spaces of Operad Structures

TL;DR

This paper develops a derived Morita theory for simplicial multicategories, describing derived mapping spaces via bimodules and establishing the existence of internal Hom-objects in their derived category.

Contribution

It introduces a derived Morita theory for operads and characterizes derived mapping spaces in terms of bimodules, advancing the understanding of their homotopy theory.

Findings

Derived mapping spaces are described via P-Q-bimodules.

The derived category of operads has internal Hom-objects.

Provides a framework for homotopy-theoretic analysis of colored operads.

Abstract

The purpose of this paper is to study the derived category of simplicial multicategories with arbitrary sets of objects (also known as, colored operads in simplicial sets). Our main result is a derived Morita theory for operads-where we describe the derived mapping spaces between two multicategories P and Q in terms of the nerve of a certain category of P-Q-bimodules. As an application, we show that the derived category possesses internal Hom-objects.