Moduli spaces of algebras over non-symmetric operads

Abstract

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

38 pages. Many improvements and generalizations with respect to the previous version

References in corpus (3)

• Homotopy theory of non-symmetric operads
• Homotopy units in A-infinity algebras
• Homotopy theory of non-symmetric operads II: change of base category and left properness

Cited by in corpus (6)

• Homotopy theory for algebras over polynomial monads
• Admissibility and rectification of colored symmetric operads
• Homotopy units in A-infinity algebras
• Factorization homology and calculus à la Kontsevich Soibelman
• Enhanced $A$-infinity obstruction theory
• Operads, modules and topological field theories