Codensity Lifting of Monads and its Dual

We introduce a method to lift monads on the base category of a fibration to its total category. This method, which we call codensity lifting, is applicable to various fibrations which were not supported by its precursor, categorical TT-lifting. After introducing the codensity lifting, we illustrate some examples of codensity liftings of monads along the fibrations from the category of preorders, topological spaces and extended pseudometric spaces to the category of sets, and also the fibration from the category of binary relations between measurable spaces. We also introduce the dual method called density lifting of comonads. We next study the liftings of algebraic operations to the codensity liftings of monads. We also give a characterisation of the class of liftings of monads along posetal fibrations with fibred small meets as a limit of a certain large diagram.Comment: Extended version of the paper presented at CALCO 2015, accepted for publication in LMC

Codensity Liftings of Monads

We introduce a method to lift monads on the base category of a fibration to its total category using codensity monads. This method, called codensity lifting, is applicable to various fibrations which were not supported by the categorical >>-lifting. After introducing the codensity lifting, we illustrate some examples of codensity liftings of monads along the fibrations from the category of preorders, topological spaces and extended psuedometric spaces to the category of sets, and also the fibration from the category of binary relations between measurable spaces. We next study the liftings of algebraic operations to the codensity-lifted monads. We also give a characterisation of the class of liftings (along posetal fibrations with fibred small limits) as a limit of a certain large diagram

Categories for Me, and You?

A non-self-contained gathering of notes on category theory, including the definition of locally cartesian closed category, of the cartesian structure in slice categories, or of the “pseudo-cartesian structure” on Eilenberg–Moore categories. References and proofs are provided, sometimes, to my knowledge, for the first time

Fully abstract models for effectful λ-calculi via category-theoretic logical relations

We present a construction which, under suitable assumptions, takes a model of Moggi’s computational λ-calculus with sum types, effect operations and primitives, and yields a model that is adequate and fully abstract. The construction, which uses the theory of fibrations, categorical glueing, ⊤⊤-lifting, and ⊤⊤-closure, takes inspiration from O’Hearn & Riecke’s fully abstract model for PCF. Our construction can be applied in the category of sets and functions, as well as the category of diffeological spaces and smooth maps and the category of quasi-Borel spaces, which have been studied as semantics for differentiable and probabilistic programming

Dijkstra monads for all

This paper proposes a general semantic framework for verifying programs with arbitrary monadic side-effects using Dijkstra monads, which we define as monad-like structures indexed by a specification monad. We prove that any monad morphism between a computational monad and a specification monad gives rise to a Dijkstra monad, which provides great flexibility for obtaining Dijkstra monads tailored to the verification task at hand. We moreover show that a large variety of specification monads can be obtained by applying monad transformers to various base specification monads, including predicate transformers and Hoare-style pre- and postconditions. For defining correct monad transformers, we propose a language inspired by Moggi's monadic metalanguage that is parameterized by a dependent type theory. We also develop a notion of algebraic operations for Dijkstra monads, and start to investigate two ways of also accommodating effect handlers. We implement our framework in both Coq and F*, and illustrate that it supports a wide variety of verification styles for effects such as exceptions, nondeterminism, state, input-output, and general recursion

Automatic differentiation via effects and handlers

Machine learning, artificial intelligence, scientific modelling, information analysis, and other data heavy fields have driven the demand for tools that enable derivative based optimization. Automatic differentiation is a family of algorithms used to calculate the derivatives of programs with only a constant factor slowdown. There are many implementation strategies, some built into a language and some outside of it, and there are many different members of the family. The utility of automatic differentiation makes it worthwhile to implement it in as many languages as possible. Effects and handlers are a powerful control flow construct in programming languages based upon delimited continuations. They are a structured method of including side effects into programs, and have found many uses including nondeterminism, state management, and concurrency. Effects and handlers excel in facilitating non-local control flow and also provide methods of abstracting and composing effects. Mainstream programming languages are increasingly incorporating effects and handlers, notably OCaml and WebAssembly. We show that effects and handlers are well-suited for implementing automatic differentiation algorithms while maintaining the desirable asymptotic efficiency. In particular, effects and handlers allow for succinctness in the presence of complex control flow. On a practical level, we implement eight automatic differentiation algorithms in four languages with effects and handlers. The implementations range from standard AD algorithms such as forward mode and continuation-based reverse mode, to more advanced modes such as checkpointed reverse mode. We benchmark the standard modes to empirically show that we can reach the correct asymptotic complexity. Furthermore, we build up a mathematical framework in which to prove correctness of selected standard modes. To do so, we extend the set-theoretic denotational semantics of a simple effect and handler language to a category-theoretic semantics. We then describe how to perform a generalized proof by logical relations in this setting, and identify sufficient conditions for our proof method to apply. Equipped with our conditions, we show that diffeological spaces (a generalization of Euclidean spaces) admit proof by logical relations. Ultimately, this enables us to prove our implementations of forward mode and continuation reverse mode correct