An algebraic model for the free loop space

Abstract

We describe an algebraic chain level construction that models the passage from an arbitrary topological space to its free loop space. The input of the construction is a categorical coalgebra, i.e., a curved coalgebra satisfying certain properties, and the output is a chain complex. The construction is a modified version of the co-Hochschild complex of a differential graded (dg) coalgebra. When applied to the chains on an arbitrary simplicial set X X , appropriately interpreted, it yields a chain complex that is naturally quasi-isomorphic to the singular chains on the free loop space of the geometric realization of X X . We relate this construction to a twisted tensor product model for the free loop space constructed using the adjoint action of a dg Hopf algebra model for the based loop space.