  Operator systems connect operator algebra, free semialgebraic geometry and
quantum information theory. In this work we generalize operator systems and
many of their theorems. While positive semidefinite matrices form the
underlying structure of operator systems, our work shows that these can be
promoted to far more general structures. For instance, we prove a general
extension theorem which unifies the well-known homomorphism theorem, Riesz'
extension theorem, Farkas' lemma and Arveson's extension theorem. On the other
hand, the same theorem gives rise to new vector-valued extension theorems, even
for invariant maps, when applied to other underlying structures. We also prove
generalized versions of the Choi-Kraus representation, Choi-Effros theorem,
duality of operator systems, factorizations of completely positive maps, and
more, leading to new results even for operator systems themselves. In addition,
our proofs are shorter and simpler, revealing the interplay between cones and
tensor products, captured elegantly in terms of star autonomous categories.
This perspective gives rise to new connections between group representations,
mapping cones and topological quantum field theory, as they correspond to
different instances of our framework and are thus siblings of operator systems.
