Capsules provide a clean algebraic representation of the state of a computation in higher-order functional and imperative languages. They play the same role as closures or heap- or stack-allocated environments but are much simpler. A capsule is essentially a finite coalgebraic representation of a regular closed lambda-coterm. One can give an operational semantics based on capsules for a higher-order programming language with functional and imperative features, including mutable bindings. Lexical scoping is captured purely algebraically without stacks, heaps, or closures. All operations of interest are typable with simple types, yet the language is Turing complete. Recursive functions are represented directly as capsules without the need for unnatural and untypable fixpoint combinators
