An algebraic model for the free loop space

TL;DR

This paper introduces an algebraic chain complex model for free loop spaces derived from categorical coalgebras, providing a new algebraic perspective that aligns with topological constructions.

Contribution

It presents a novel algebraic chain-level construction for free loop spaces using curved coalgebras, extending the coHochschild complex framework.

Findings

The chain complex is quasi-isomorphic to the singular chains on the free loop space.

The construction relates to a twisted tensor product model involving dg Hopf algebras.

Provides an algebraic model applicable to chains on simplicial sets.

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 coHochschild complex of a differential graded (dg) coalgebra. When applied to the chains on an arbitrary simplicial set $X$, 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 $X$. 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.