Moduli spaces of algebras over non-symmetric operads

TL;DR

This paper investigates the homotopical structure of algebra spaces over non-symmetric operads within symmetric monoidal model categories, extending classical results and constructing geometric moduli stacks in a homotopical algebraic geometry framework.

Contribution

It computes the homotopy fiber of the forgetful functor for operad algebras and constructs moduli stacks of these algebras, including the unital associative case, in a homotopical setting.

Findings

Homotopy fiber of the forgetful functor computed

Moduli stacks of operad algebras constructed

Unital associative algebra stack is a Zariski open substack

Abstract

In this paper we study spaces of algebras over an operad (non-symmetric) in symmetric monoidal model categories. We first compute the homotopy fiber of the forgetful functor sending an algebra to its underlying object, extending a result of Rezk. We then apply this computation to the construction of geometric moduli stacks of algebras over an operad in a homotopical algebraic geometry context in the sense of To\"en and Vezzosi. We show under mild hypotheses that the moduli stack of unital associative algebras is a Zariski open substack of the moduli stack of non-necessarily unital associative algebras. The classical analogue for finite-dimensional vector spaces was noticed by Gabriel.