Algebraic & Geometric Topology

Moduli spaces of algebras over nonsymmetric operads

Fernando Muro

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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

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

Received: 21 August 2013
Accepted: 24 October 2013
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D50: Operads [See also 55P48] 14K10: Algebraic moduli, classification [See also 11G15]
Secondary: 55U35: Abstract and axiomatic homotopy theory

operad algebra associative algebra unital algebra model category mapping space moduli stack


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.

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