  The oriented skein category $OS(z,t)$ is a ribbon category which underpins
the definition of the HOMFLY-PT invariant of an oriented link, in the same way
that the Temperley-Lieb category underpins the Jones polynomial. In this
article, we develop its representation theory using a highest weight theory
approach. This allows us to determine the Grothendieck ring of its additive
Karoubi envelope for all possible choices of parameters, including the (already
well-known) semisimple case, and all non-semisimple situations. Then we
construct a graded lift of $OS(z,t)$ by realizing it as a 2-representation of a
Kac-Moody 2-category. We also discuss the degenerate analog of $OS(z,t)$, which
is the oriented Brauer category $OB(\delta)$.
