I will describe an algebraic construction that models the passage from a topological space to its free loop space, without imposing any restrictions on the fundamental group of the underlying space.
The input of the construction is a coalgebra with certain extra structure and the construction to be discussed is a modified version of the coHochschild complex. The construction is invariant with respect to a notion of weak equivalence between coalgebras that is stronger than quasi-isomorphisms. When this construction is applied to the coalgebra of chains, suitably interpreted, of an arbitrary simplicial set X one obtains a chain complex with a rotation operator that is quasi-isomorphic to the chains on the free loop space of the geometric realization of X. This extends classical results regarding models for the free loop space of a simply connected space in terms of Hochschild homology