## Algebraic & Geometric Topology

### Moduli spaces of algebras over nonsymmetric operads

Fernando Muro

#### Abstract

In this paper we study spaces of algebras over an operad (nonsymmetric) 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 Toën 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 nonnecessarily unital associative algebras. The classical analogue for finite-dimensional vector spaces was noticed by Gabriel.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 3 (2014), 1489-1539.

Dates
Accepted: 24 October 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715912

Digital Object Identifier
doi:10.2140/agt.2014.14.1489

Mathematical Reviews number (MathSciNet)
MR3190602

Zentralblatt MATH identifier
1305.18036

#### Citation

Muro, Fernando. Moduli spaces of algebras over nonsymmetric operads. Algebr. Geom. Topol. 14 (2014), no. 3, 1489--1539. doi:10.2140/agt.2014.14.1489. https://projecteuclid.org/euclid.agt/1513715912

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