Abstract: Recent progress on the K-theory of "large" categories has raised interest in the algebraic K-theory of sheaves on locally compact Hausdorff spaces, which serves as a central stepping stone to modern approaches to assembly conjectures in K-theory and L-theory. A central ingredient for the computation of the K-theory of these sheaf categories is Verdier duality: The categories of sheaves and cosheaves agree when the target is a stable category. We present a category-theoretic perspective on this computation by analyzing the notion of a continuous algebra of a lax-idempotent monad. As a result we obtain a completely formal generalization of Verdier duality to a larger class of spaces - so-called stably locally compact spaces. We will elaborate on the role of classical Stone duality, as well as sketch a proof of the computation of the algebraic K-theory of a coherent space.