  Let kQ be the path algebra of a quiver Q with its standard grading. We show
that the category of graded kQ-modules modulo those that are the sum of their
finite dimensional submodules, QGr(kQ), is equivalent to several other
categories: the graded modules over a suitable Leavitt path algebra, the
modules over a certain direct limit of finite dimensional multi-matrix
algebras, QGr(kQ') where Q' is the quiver whose incidence matrix is the n^{th}
power of that for Q, and others. A relation with a suitable Cuntz-Krieger
algebra is established. All short exact sequences in the full subcategory of
finitely presented objects in QGr(kQ), split so that subcategory can be given
the structure of a triangulated category with suspension functor the Serre
degree twist (-1); it is shown that this triangulated category is equivalent to
the "singularity category" for the radical square zero algebra kQ/kQ_{\ge 2}.
