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Moduli spaces of algebras over nonsymmetric operads

Fernando Muro

Algebraic & Geometric Topology 14 (2014) 1489–1539

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.

operad, algebra, associative algebra, unital algebra, model category, mapping space, moduli stack
Mathematical Subject Classification 2010
Primary: 18D50, 14K10
Secondary: 55U35
Received: 21 August 2013
Accepted: 24 October 2013
Published: 7 April 2014
Fernando Muro
Universidad de Sevilla
Facultad de Matemáticas
Departamento de Álgebra
Avda. Reina Mercedes s/n
41012 Sevilla