Related papers: Calculating monad transformers with category theor…
Via the adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Span}(\mathcal V) \to \mathcal V \text{-} \mathsf{Mat} $ and a cartesian monad $ T $ on an extensive category $ \mathcal V $ with finite limits, we construct…
We introduce a generalization of monads, called relative monads, allowing for underlying functors between different categories. Examples include finite-dimensional vector spaces, untyped and typed lambda-calculus syntax and indexed…
We propose an intersection type system for an imperative lambda-calculus based on a state monad and equipped with algebraic operations to read and write to the store. The system is derived by solving a suitable domain equation in the…
We investigate the phenomenon that "every monad is a linear state monad". We do this by studying a fully-complete state-passing translation from an impure call-by-value language to a new linear type theory: the enriched call-by-value…
This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition…
We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means…
This is the second paper in a series that aims to provide mathematical descriptions of objects and constructions related to the first few steps of the semantical theory of dependent type systems. We construct for any pair $(R,LM)$, where…
The principle behind algebraic language theory for various kinds of structures, such as words or trees, is to use a compositional function from the structures into a finite set. To talk about compositionality, one needs some way of…
A transduction provides us with a way of using the monadic second-order language of a structure to make statements about a derived structure. Any transduction induces a relation on the set of these structures. This article presents a…
This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…
Monads are a popular tool for the working functional programmer to structure effectful computations. This paper presents polymonads, a generalization of monads. Polymonads give the familiar monadic bind the more general type forall a,b. L a…
Inspired by the classical theory of modules over a monoid, we give a first account of the natural notion of module over a monad. The associated notion of morphism of left modules ("Linear" natural transformations) captures an important…
Understanding how torsion theories are described and constructed is crucial to the study of torsion theory. Mutations of torsion theories have been studied as a method of constructing another torsion theory from a given one. We have already…
We present the formalization of a theory of syntax with bindings that has been developed and refined over the last decade to support several large formalization efforts. Terms are defined for an arbitrary number of constructors of varying…
Monadic programming presents a significant challenge for many programmers. In light of category theory, we offer a new perspective on the use of monads in functional programming. This perspective is clarified through numerous examples coded…
The basic concepts in category theory are representables, adjoints, limits, and monads. In this talk, we define the notion of a Kan extension and show that this notion encompasses these concepts.
Monads in category theory are algebraic structures that can be used to model computational effects in programming languages. We show how the notion of "centre", and more generally "centrality", i.e. the property for an effect to commute…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…
We give an introduction to constructive category theory by answering two guiding computational questions. The first question is: how do we compute the set of all natural transformations between two finitely presented functors like…