  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 stratified
categories. In the upper finite cases, we give an alternative characterization
of these categories in terms of based quasi-hereditary algebras and based
stratified algebras, which are certain locally unital algebras possessing
triangular bases.

Representation Theory