This paper presents a functional programming language, based on Moggi’s monadic metalanguage. In the first part of this paper, we show how the language can be regarded as a monad on a category of signatures, and that the resulting category of algebras is equivalent to the category of computationally cartesian closed categories. In the second part, we extend the language to include a nondetermin...

-semilattices with that in the category of modules over a given unital commutative quantale. The resulting structures are called quantale algebroids. We show that their constitute a monadic category and prove a representation theorem for them using the notion of nucleus adjusted for our needs. We also characterize the lattice of nuclei on a free quantale algebroid. At the end of the paper, we p...

This article presents a development of Category Theory in Isabelle. A Category is defined using records and locales in Isabelle/HOL. Functors and Natural Transformations are also defined. The main result that has been formalized is that the Yoneda functor is a full and faithful embedding. We also formalize the completeness of many sorted monadic equational logic. Extensive use is made of the HO...

Given a monad T on Set whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids is defined as the category of monoids in the hom-sets of the Kleisli category of T. The Eilenberg-Moore category of T is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of T moreover forms an order-enriched category,...

Foundations should be designed for the needs of mathematics and not vice versa. We propose a technique for doing this that exploits the correspondence between category theory and logic and is potentially applicable to several mathematical disciplines. This method is applied to devising a new paradigm for general topology, called Abstract Stone Duality. We express the duality between algebra and...

We introduce the notion of a positive face structure. The positive face structures to positive-to-one computads are like simple graphs, c.f. [MZ], to free ω-categories over ω-graphs. In particular, they allow to give an explicit combinatorial description of positive-to-one computads. Using this description we show, among other things, that positive-to-one computads form a presheaf category with...

For a set M of graphs the category CatM of all M-complete categories and all strictly M-continuous functors is known to be monadic over Cat. The question of monadicity of CatM over the category of graphs is known to have an affirmative answer when M specifies either (i) all finite limits, or (ii) all finite products, or (iii) equalizers and terminal objects, or (iv) just terminal objects. We pr...

The concept of iteration theory of Bloom and Ésik summarizes all equational properties that iteration has in usual applications, e.g., in Domain Theory where to every system of recursive equations the least solution is assigned. However, this assignment in Domain Theory is also functorial. Yet, functoriality is not included in the definition of iteration theory. Pitty: functorial iteration theo...

Computational monads offer a powerful way to parameterize functional specifications, but they give rise to exceedingly tedious simplifications to instantiate this “monadic” interpreter. We report on the use of partial evaluation to achieve the following instantiations automatically. • We derive equivalent formulations of the monadic λ-interpreter, based on equivalent specifications of monads fr...

We investigate the extension of Monadic Second Order logic, interpreted over infinite words and trees, with generalized “for almost all” quantifiers interpreted using the notions of Baire category and Lebesgue measure. All three authors were supported by the Polish National Science Centre grant no. 2014-13/B/ST6/03595. The work of the second author has also been supported by the project ANR-16-...