arXiv subjects

Bryce Clarke

Publications and source records attributed to Bryce Clarke.

11 recordsLinked to original sources

arXiv2025

The free bifibration on a functor

We consider the problem of constructing the free bifibration generated by a functor of categories $p : D \to C$. This problem was previously considered by Lamarche, and is closely related to the problem, considered by Dawson, Par\'e, and Pronk, of ``freely adjoining adjoints'' to a category. We develop a proof-theoretic approach to the problem, beginning with a construction of the free bifibration $\Lambda_p : Bifib(p)\to C$ in which objects of $Bifib(p)$ are formulas of a primitive ``bifibrational logic'', and arrows are derivations in a cut-free sequent calculus modulo a notion of permutation equivalence. We show that instantiating the construction to the identity functor generates a _zigzag double category_ $\mathbb{Z}(C)$, which is also the free double category with companions and conjoints (or fibrant double category) on $C$. The approach adapts smoothly to the more general task of building $(P,N)$-fibrations, where one only asks for pushforwards along arrows in $P$ and pullbacks along arrows in $N$ for some subsets of arrows; this encompasses Kock and Joyal's notion of _ambifibration_ when $(P,N)$ form a factorization system. We establish a series of progressively stronger normal forms, guided by ideas of _focusing_ from proof theory, and obtain a canonicity result under assumption that the base category is factorization preordered relative to $P$ and $N$. This canonicity result allows us to decide the word problem and to enumerate relative homsets without duplicates. Finally, we describe several examples of a combinatorial nature, including a category of plane trees generated as a free bifibration over $\omega$, and a category of increasing forests generated as a free ambifibration over $\Delta$, which contains the lattices of noncrossing partitions as quotients of its fibers by the Beck-Chevalley condition for bicartesian squares.

math.CT↗

arXiv2025

The Grothendieck construction for delta lenses

Delta lenses are functors equipped with a functorial choice of lifts, generalising the notion of split opfibration. In this paper, we introduce a Grothendieck construction (or category of elements) for delta lenses, thus demonstrating a correspondence between delta lenses and certain lax double functors into the double category of sets, functions, and split multivalued functions. We show that the double category of split multivalued functions admits a universal property as a certain kind of limit, and inherits many nice properties from the double category of spans. Applications of this construction to the theory of delta lenses are explored in detail.

math.CT↗

arXiv2024

Lifting twisted coreflections against delta lenses

Delta lenses are functors equipped with a suitable choice of lifts, generalising the notion of split opfibration. In recent work, delta lenses were characterised as the right class of an algebraic weak factorisation system. In this paper, we show that this algebraic weak factorisation system is cofibrantly generated by a small double category, and characterise the left class as split coreflections with a certain property; we call these twisted coreflections. We demonstrate that every twisted coreflection arises as a pushout of an initial functor from a discrete category along a bijective-on-objects functor. Throughout the article, we take advantage of a reformulation of algebraic weak factorisation systems, due to Bourke, based on double-categorical lifting operations.

math.CT↗

arXiv2023

The Algebraic Weak Factorisation System for Delta Lenses

Delta lenses are functors equipped with a suitable choice of lifts, and are used to model bidirectional transformations between systems. In this paper, we construct an algebraic weak factorisation system whose R-algebras are delta lenses. Our approach extends a semi-monad for delta lenses previously introduced by Johnson and Rosebrugh, and generalises to any suitable category equipped with an orthogonal factorisation system and an idempotent comonad. We demonstrate how the framework of an algebraic weak factorisation system provides a natural setting for understanding the lifting operation of a delta lens, and also present an explicit description of the free delta lens on a functor.

math.CT↗

arXiv2022

An introduction to enriched cofunctors

Cofunctors are a kind of map between categories which lift morphisms along an object assignment. In this paper, we introduce cofunctors between categories enriched in a distributive monoidal category. We define a double category of enriched categories, enriched functors, and enriched cofunctors, whose horizontal and vertical 2-categories have 2-cells given by enriched natural transformations between functors and cofunctors, respectively. Enriched lenses are defined as a compatible enriched functor and enriched cofunctor pair; weighted lenses, which were introduced by Perrone, are precisely lenses enriched in weighted sets. Several other examples are also studied in detail.

math.CT↗

arXiv2021

Delta lenses as coalgebras for a comonad

Delta lenses are a kind of morphism between categories which are used to model bidirectional transformations between systems. Classical state-based lenses, also known as very well-behaved lenses, are both algebras for a monad and coalgebras for a comonad. Delta lenses generalise state-based lenses, and while delta lenses have been characterised as certain algebras for a semi-monad, it is natural to ask if they also arise as coalgebras. This short paper establishes that delta lenses are coalgebras for a comonad, through showing that the forgetful functor from the category of delta lenses over a base, to the category of cofunctors over a base, is comonadic. The proof utilises a diagrammatic approach to delta lenses, and clarifies several results in the literature concerning the relationship between delta lenses and cofunctors. Interestingly, while this work does not generalise the corresponding result for state-based lenses, it does provide new avenues for exploring lenses as coalgebras.

math.CT↗

arXiv2021

Limits and Colimits in a Category of Lenses

Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we study the category of small categories and asymmetric delta lenses, and prove that it has several good exactness properties. These properties include the existence of certain limits and colimits, as well as so-called imported limits, such as imported products and imported pullbacks, which have arisen previously in applications. The category is also shown to be extensive, and it has an image factorisation system.

math.CT↗

arXiv2021

A diagrammatic approach to symmetric lenses

Lenses are a mathematical structure for maintaining consistency between a pair of systems. In their ongoing research program, Johnson and Rosebrugh have sought to unify the treatment of symmetric lenses with spans of asymmetric lenses. This paper presents a diagrammatic approach to symmetric lenses between categories, through representing the propagation operations with Mealy morphisms. The central result of this paper is to demonstrate that the bicategory of symmetric lenses is locally adjoint to the bicategory of spans of asymmetric lenses, through constructing an explicit adjoint triple between the hom-categories.

math.CT↗

arXiv2020

Internal lenses as functors and cofunctors

Lenses may be characterised as objects in the category of algebras over a monad, however they are often understood instead as morphisms, which propagate updates between systems. Working internally to a category with pullbacks, we define lenses as simultaneously functors and cofunctors between categories. We show that lenses may be canonically represented as a particular commuting triangle of functors, and unify the classical state-based lenses with both c-lenses and d-lenses in this framework. This new treatment of lenses leads to considerable simplifications that are important in applications, including a clear interpretation of lens composition.

math.CT↗

arXiv2020

Internal split opfibrations and cofunctors

Split opfibrations are functors equipped with a suitable choice of opcartesian lifts. The purpose of this paper is to characterise internal split opfibrations through separating the structure of a suitable choice of lifts from the property of these lifts being opcartesian. The underlying structure of an internal split opfibration is captured by an internal functor equipped with an internal cofunctor, while the property may be expressed as a pullback condition, akin to the simple condition on an internal functor to be an internal discrete opfibration. Furthermore, this approach provides two additional characterisations of internal split opfibrations, via the d\'ecalage construction and strict factorisation systems. For small categories, this theory clarifies several aspects of delta lenses which arise in computer science.

math.CT↗

arXiv2020

Profunctor Optics, a Categorical Update

Optics are bidirectional data accessors that capture data transformation patterns such as accessing subfields or iterating over containers. Profunctor optics are a particular choice of representation supporting modularity, meaning that we can construct accessors for complex structures by combining simpler ones. Profunctor optics have previously been studied only in an unenriched and non-mixed setting, in which both directions of access are modelled in the same category. However, functional programming languages are arguably better described by enriched categories; and we have found that some structures in the literature are actually mixed optics, with access directions modelled in different categories. Our work generalizes a classic result by Pastro and Street on Tambara theory and uses it to describe mixed V-enriched profunctor optics and to endow them with V-category structure. We provide some original families of optics and derivations, including an elementary one for traversals. Finally, we discuss a Haskell implementation.

cs.PL↗