Variations on Algebra: monadicity and generalisations of equational theories

Dedicated to Rod Burstal

Axiomatics for Data Refinement in Call by Value Programming Languages

AbstractWe give a systematic category theoretic axiomatics for modelling data refinement in call by value programming languages. Our leading examples of call by value languages are extensions of the computational λ-calculus, such as FPC and languages for modelling nondeterminism, and extensions of the first order fragment of the computational λ-calculus, such as a CPS language. We give a category theoretic account of the basic setting, then show how to model contexts, then arbitrary type and term constructors, then signatures, and finally data refinement. This extends and clarifies Kinoshita and Power's work on lax logical relations for call by value languages

Enhanced 2-categories and limits for lax morphisms

We study limits in 2-categories whose objects are categories with extra structure and whose morphisms are functors preserving the structure only up to a coherent comparison map, which may or may not be required to be invertible. This is done using the framework of 2-monads. In order to characterize the limits which exist in this context, we need to consider also the functors which do strictly preserve the extra structure. We show how such a 2-category of weak morphisms which is "enhanced", by specifying which of these weak morphisms are actually strict, can be thought of as category enriched over a particular base cartesian closed category F. We give a complete characterization, in terms of F-enriched category theory, of the limits which exist in such 2-categories of categories with extra structure.Comment: 77 pages; v2 minor changes only, to appear in Advance

What is the universal property of the 2-category of monads?

For a 2-category $\mathcal{K}$, we consider Street's 2-category Mnd($\mathcal{K}$) of monads in $\mathcal{K}$, along with Lack and Street's 2-category EM($\mathcal{K}$) and the identity-on-objects-and-1-cells 2-functor Mnd($\mathcal{K}$) $\to$ EM($\mathcal{K}$) between them. We show that this 2-functor can be obtained as a ``free completion'' of the 2-functor $1\colon \mathcal{K} \to \mathcal{K}$. We do this by regarding 2-functors which act as the identity on both objects and 1-cells as categories enriched a cartesian closed category $\mathbf{BO}$ whose objects are identity-on-objects functors. We also develop some of the theory of $\mathbf{BO}$-enriched categories

Promonads and String Diagrams for Effectful Categories

Premonoidal and Freyd categories are both generalized by non-cartesian Freyd categories: effectful categories. We construct string diagrams for effectful categories in terms of the string diagrams for a monoidal category with a freely added object. We show that effectful categories are pseudomonoids in a monoidal bicategory of promonads with a suitable tensor product.Comment: In Proceedings ACT 2022, arXiv:2307.1551

Enriched categories as a free cocompletion

This paper has two objectives. The first is to develop the theory of bicategories enriched in a monoidal bicategory -- categorifying the classical theory of categories enriched in a monoidal category -- up to a description of the free cocompletion of an enriched bicategory under a class of weighted bicolimits. The second objective is to describe a universal property of the process assigning to a monoidal category V the equipment of V-enriched categories, functors, transformations, and modules; we do so by considering, more generally, the assignation sending an equipment C to the equipment of C-enriched categories, functors, transformations, and modules, and exhibiting this as the free cocompletion of a certain kind of enriched bicategory under a certain class of weighted bicolimits.Comment: 80 pages; final journal versio

Strong pseudomonads and premonoidal bicategories

Strong monads and premonoidal categories play a central role in clarifying the denotational semantics of effectful programming languages. Unfortunately, this theory excludes many modern semantic models in which the associativity and unit laws only hold up to coherent isomorphism: for instance, because composition is defined using a universal property. This paper remedies the situation. We define premonoidal bicategories and a notion of strength for pseudomonads, and show that the Kleisli bicategory of a strong pseudomonad is premonoidal. As often in 2-dimensional category theory, the main difficulty is to find the correct coherence axioms on 2-cells. We therefore justify our definitions with numerous examples and by proving a correspondence theorem between actions and strengths, generalizing a well-known category-theoretic result.Comment: Comments and feedback welcome

Birkhoff's variety theorem for relative algebraic theories

An algebraic theory, sometimes called an equational theory, is a theory defined by finitary operations and equations, such as the theories of groups and of rings. It is well known that algebraic theories are equivalent to finitary monads on $\mathbf{Set}$. In this paper, we generalize this phenomenon to locally finitely presentable categories using partial Horn logic. For each locally finitely presentable category $\mathscr{A}$, we define an "algebraic concept" relative to $\mathscr{A}$, which will be called an $\mathscr{A}$-relative algebraic theory, and show that $\mathscr{A}$-relative algebraic theories are equivalent to finitary monads on $\mathscr{A}$. Finally, we generalize Birkhoff's variety theorem for classical algebraic theories to our relative algebraic theories.Comment: 34 page