Categorical semantics for typeclass hierarchies?

Stream: general

[nrs]

We can use type classes to produce hierarchies of mathematical objects. Ignoring their type-inference and elaboration-related feature, a typeclass considered only type-theoretically seems to be a type constructor for a record type, and instantiation produces said record type. Do hierarchies produced in this way have a mathematical counterpart? i.e. are there categorical semantics for typeclass hierarchies?