Moduli spaces of algebras over nonsymmetric operads

Muro, Fernando¹

¹Universidad de Sevilla, Facultad de Matemáticas, Departamento de Álgebra, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain

Algebraic and Geometric Topology, Tome 14 (2014) no. 3, pp. 1489-1539

Résumé

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.

Détail

DOI : 10.2140/agt.2014.14.1489

Classification : 18D50, 14K10, 55U35
Keywords: operad, algebra, associative algebra, unital algebra, model category, mapping space, moduli stack

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