Formal ball monads

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The formal ball construction B is a central tool of quasi-metric space theory. We show that it induces monads on certain natural categories of quasi-metric spaces, with 1-Lipschitz maps as morphisms, or with 1-Lipschitz continuous maps as morphisms. Those are left Kock-Zöberlein monads, and that allows us to characterize their algebras exactly. As an application, we study so-called Lipschitz regular spaces, a natural class of spaces that contain all standard algebraic quasi-metric spaces with relatively compact balls, in particular all metric spaces whose closed balls are compact. There are other Lipschitz regular spaces, as we show, and notably all B-algebras. That includes all spaces of formal balls, with their d+-Scott topology. The value of Lipschitz regularity is that, for a Lipschitz regular standard quasi-metric space X,d, the space LX of lower semicontinuous maps from X to R‾+, with the Scott topology, retracts onto each of the spaces Lα(X,d) of α-Lipschitz continuous maps, and that the subspace topology on the latter coincides with the Scott topology.