  In this paper we study the homogenization of a class of energies concentrated
on lines. In dimension $2$ (i.e., in codimension $1$) the problem reduces to
the homogenization of partition energies studied by \cite{AB}. There, the key
tool is the representation of partitions in terms of $BV$ functions with values
in a discrete set. In our general case the key ingredient is the representation
of closed loops with discrete multiplicity either as divergence-free
matrix-valued measures supported on curves or with $1$-currents with
multiplicity in a lattice. In the $3$ dimensional case the main motivation for
the analysis of this class of energies is the study of line defects in
crystals, the so called dislocations.
