  We introduce a matricial analogue of an Archimedean order unit space, which
we call a $k$-AOU space. We develop the category of $k$-AOU spaces and
$k$-positive maps and exhibit functors from this category to the category of
operator systems and completely positive maps. We also demonstrate the
existence of injective envelopes and C*-envelopes in the category of $k$-AOU
spaces. Finally, we show that finite-dimensional quantum correlations can be
characterized in terms of states on finite-dimensional $k$-AOU spaces. Combined
with previous work, this yields a reformulation of Tsirelson's conjecture in
terms of operator systems and $k$-AOU spaces.
