Semi-infinite highest weight categories
We develop axiomatics of highest weight categories and quasi-hereditary algebras in order to incorporate two semi-infinite situations which are in Ringel duality with each other; the underlying posets are either upper finite or lower finite. We also consider various more general sorts of signed highest weight categories and stratified categories. In the upper finite cases, we give an alternative characterization of these categories in terms of quasi-hereditary and stratified algebras equipped with idempotent-adapted cellular bases. Finally, we explain a general construction which produces many explicit examples of such algebras starting from naturally-occurring locally unital algebras which admit Cartan or triangular decompositions.