This paper considers some extensions of the notion filter to quantale-valued context, including saturated prefilter, ?-filter and bounded prefilter. The question is whether these constructions give rise monads on category sets. It shown that answer depends structure quantale. Specifically, if quantale unit interval equipped with a continuous t-norm, then only implication operator corresponding ...

We study the construction of preorders on Set-monads by the semantic ⊤⊤-lifting. We show the universal property of this construction, and characterise the class of preorders on a monad as a limit of a Card-chain. We apply these theoretical results to identifying preorders on some concrete monads, including the powerset monad, maybe monad, and their composite monad. We also relate the constructi...

Higher-order functions that are polymorphic in a monad make highly flexible modular components. Unfortunately, the combination of an unknown function parameter and a polymorphic monad are detrimental to reasoning. This paper shows how to eliminate both the function parameter and the polymorphism. The resulting characterization is amenable to reasoning. The approach is based on a judicious combi...

We extend our correspondence between evaluators and abstract machines from the pure setting of the λ-calculus to the impure setting of the computational λ-calculus. We show how to derive new abstract machines from monadic evaluators for the computational λ-calculus. Starting from (1) a generic evaluator parameterized by a monad and (2) a monad specifying a computational effect, we inline the co...

The paper proposes the notions of topological platform and quantalic topological theory for the presentation and investigation of categories of interest beyond the realm of algebra. These notions are nevertheless grounded in algebra, through the notions of monad and distributive law. The paper shows how they entail previously proposed concepts with similar goals.

I exhibit a pair of non-symmetric operads that, while not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if ‘algebraic theory’ is understood as meaning ‘monad’ then operads cannot be regarded as algebraic theories of a special kind.

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures. It turns out to sit in between known monads: on the one hand the distribution and ultrafilter monad, and on the other hand the continuation monad. This expectation monad is used in two probabilistic analogues of fundamental results of Manes and Gelfand for the ultrafilter monad: alge...

Using the structure of a KZ-monad we create a general categorical workspace in which diagrams can be formally constructed. In particular this abstract framework of category theory is shown to provide a precise semantics for constructing the speciications of complex systems from their component parts.

Davydov-Yetter cohomology classifies infinitesimal deformations of tensor categories and functors. Our first result is that for finite equivalent to the a comonad arising from central Hopf monad. This has several applications: First, we obtain short conceptual proof Ocneanu rigidity. Second, it allows use standard methods theory compute family non-semisimple finite-dimensional algebras generali...