Monads have become a fundamental tool for structuring denotational semantics and programs by abstracting a wide variety of computational features such as side-effects, input/output, exceptions, continuations and non-determinism. In this setting, the notion of a monad is equipped with operations that allow programmers to manipulate these computational effects. For example, a monad for side-effec...

In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove tha...

The contravariant powerset, and its generalisations Σ to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) = φ ∧ F (>). Conversely, when the adjunction Σ(−) a Σ(−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) = (∃x.φ(x...

In this paper, we introduce and investigate monadic NM-algebras: a variety of NM-algebras equipped with universal quantifiers. Also, we obtain some conditions under which monadic NM-algebras become monadic Boolean algebras. Besides, we show that the variety of monadic NM-algebras faithfully the axioms on quantifiers in monadic predicate NM logic. Furthermore, we discuss relations between monadi...