We present some tools for providing situations where the generalised Rota formula of Carlier (Algebr Geometr Topol 20:169–213, 2020) applies. As an example this, we compute Möbius function incidence algebra any directed restriction species, free operad, or more generally monad on a finitary polynomial monad.

The Hoare state monad provides a powerful means of structuring the verification of higher-order, stateful programs. This paper defines a new variant of the Hoare state monad, which, rather than being a triple of a pre-condition, a return type, and a post-condition, is a pair of a return type and a predicate transformer. We dub this monad the Dijkstra state monad. Using the Dijkstra state monad,...

Describing systems in terms of choices and their resulting costs rewards offers the promise freeing algorithm designers programmers from specifying how those should be made; implementations, can realized by optimization techniques and, increasingly, machine-learning methods. We study this approach a programming-language perspective. define two small languages that support decision-making abstra...

We continue the study of Capretta’s delay monad as a means of introducing non-termination from iteration into Martin-Löf type theory. In particular, we explain in what sense this monad provides a canonical solution. We discuss a class of monads that we call ω-complete pointed classifying monads. These are monads whose Kleisli category is an ωcomplete pointed restriction category where pure maps...

The center Z(C) of an autonomous category C is monadic over C (if certain coends exist in C). The notion of Hopf monad naturally arises if one tries to reconstruct the structure of Z(C) in terms of its monad Z: we show that Z is a quasitriangular Hopf monad on C and Z(C) is isomorphic to the braided category ZC of Z-modules. More generally, let T be a Hopf monad on an autonomous category C. We ...

Turi and Plotkin's bialgebraic semantics is an abstract approach to specifying the operational of a system, by means distributive law between its syntax (encoded as monad) dynamics (an endofunctor). This setup instrumental in showing that semantic specification (a coalgebra) compositional. In this work, we use derive well-behaved structural string diagrams, graphical increasingly used study int...

Every finitary monad T on the category of sets is described by an algebraic theory whose n-ary operations are the elements of the free algebra Tn generated by n letters. This canonical presentation of the monad (called its Lawvere theory) offers a precious guideline in the search for an intuitive presentation of the monad by generators and relations. Hence, much work has been devoted to extend ...