  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.
