Moduli spaces of algebras over nonsymmetric operads

Muro, Fernando

doi:10.2140/agt.2014.14.1489

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Moduli spaces of algebras over nonsymmetric operads

Fernando Muro

Algebraic & Geometric Topology 14 (2014) 1489–1539

DOI: 10.2140/agt.2014.14.1489

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.

Keywords

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

Mathematical Subject Classification 2010

Primary: 18D50, 14K10

Secondary: 55U35

References

Publication

Received: 21 August 2013

Accepted: 24 October 2013

Published: 7 April 2014

Authors

Fernando Muro
Universidad de Sevilla Facultad de Matemáticas Departamento de Álgebra Avda. Reina Mercedes s/n 41012 Sevilla Spain
http://personal.us.es/fmuro

Volume 14, issue 3 (2014)