In this paper, we develop and study the following perspective – just as higher-order functions give exponentials, higher-order continuations give coexponentials. From this, we design a language that combines exponentials and coexponentials, producing a duality of abstraction.

We formalise this language by giving an extension of a call-by-value simply-typed lambda-calculus with covalues, coabstraction and coapplication. We develop the semantics of this language using the axiomatic structure of continuations, which we use to produce an equational theory, that gives a complete axiomatisation of control effects. We give a computational interpretation to this language using speculative execution and backtracking, and use this to derive the classical control operators and computational interpretation of classical logic, and encode common patterns of control flow using continuations. By dualising functional completeness, we further develop duals of first-order arrow languages using coexponentials, Finally, we discuss the implementation of this duality as control operators in programming, and develop some applications.