Graduated categories were introduced in [1] in order to prove that finitary endofunctors are right adjoints iff they preserve countable limits. Recently, these categories turned out to be useful in categorical colagebra: every finitary endofunctor preserving finite intersections has a terminal colagebra obtained in countably many steps [2].

[1] J. Adamek and L. Sousa, A finitary adjoint functor theorem, TAC 41 (2024), 1919-1936.

[2] J. Adamek, S. Milius and L. Moss, Terminal coalgebras in countably many steps, submitted.