Simulations between processes can be understood in terms of
coalgebra homomorphisms, with homomorphisms to the final coalgebra
exactly identifying bisimilar processes. The elements of the final
coalgebra are thus natural representatives of bisimilarity classes,
and a denotational semantics of processes can be developed in a
final-coal\-ge\-bra-en\-ri\-ched category where arrows are
processes, canonically represented. In the present paper, we first
describe a general framework for building final-coalgebra-enriched
categories. Every such category is constructed from a multivariant
functor representing a notion of process, much like Moggi's
categories of computations arising from monads as notions of
computation. The ``notion of process'' functors are intended to
capture different flavors of processes as dynamically extended
computations. These functors may involve a computational (co)monad,
os that a process category in many cases contains an associated
computational category as a retract. We further discuss categories
of resumptions and hyperfunctions, which are the main examples of
process categories. Very informally, the resumptions can be
understood as computations extended in time, whereas
hypercomputations are extended in space.