  We provide an effect system CatEff based on a category-graded extension of
algebraic theories that correspond to category-graded monads. CatEff has
category-graded operations and handlers. Effects in CatEff are graded by
morphisms of the grading category. Grading morphisms represent fine structures
of effects such as dependencies or sorts of states. Handlers in CatEff are
regarded as an implementation of category-graded effects. We define the notion
of category-graded algebraic theory to give semantics of CatEff and prove
soundness and adequacy. We also give an example using category-graded effects
to express protocols for sending receiving typed data.
