arXiv subjects

Robin Cockett

Publications and source records attributed to Robin Cockett.

At least 19 recordsLinked to original sources

arXiv2026

Unitary, Inner product, and Dagger categories

This article provides an alternate characterization of dagger categories, which are central to the study of categorical quantum mechanics, in terms of inner product categories. An inner product category is an "achiral involutive" category with an inner product combinator. Inner product categories are, in turn, precisely the same as unitary categories, which are a weaker form of dagger categories. In unitary categories, there is an isomorphism between an object and its dagger, instead of the identity function as in the case of dagger categories. Every unitary category is equipped with a global inner product structure, which allows one to strictify the involutive structure on the unitary category to obtain a dagger category, making unitary categories 2-categorically equivalent to dagger categories. By regarding the inner product as an abstract metric on an (achiral) involutive category, one can define metric-preserving maps (isometries) in inner product categories, and also develop other notions of special maps -- unitary, Hermitian, positive, and normal maps -- in this setting.

math.CT↗

arXiv2026

Generalized Inverses of Quantum Channels: a categorical perspective

A quantum channel is defined as being completely positive (CP) and trace preserving (TP). While not every quantum channel is invertible or reversible, every quantum channel admits various kinds of generalized inverses such as the Moore-Penrose inverse and the Drazin inverse. A generalized inverse of a quantum channel may not itself be a quantum channel: it often fails to be CP. However, generalized inverses still play an important role in quantum error mitigation. Here, because it is often desirable for the generalized inverse of a quantum channel to be at least TP, the Drazin inverse, which is TP, is favoured over the Moore-Penrose inverse, which is not in general TP. In this paper, we take a categorical perspective on generalized inverses of quantum channels. This allows us to give a simple proof of the fact that the Drazin inverse of a quantum channel is always TP. It also allows us to show that for unital quantum channels, the Drazin inverse is also unital. We then generalize this result to dagger Drazin inverses, which allows us to show that for unital quantum channels, the Moore-Penrose inverse is both TP and unital as well. This opens the door to new applications of both the Drazin inverse and Moore-Penrose inverse in quantum information theory and, in particular, in quantum error mitigation.

quant-ph↗

arXiv2025

Lie groups in tangent join restriction categories

Principal bundles have at least three different definitions, depending on the category of geometric objects studied. In Differential Geometry, they are defined as locally trivial projection map of smooth manifolds with an atlas whose transition maps are given by group multiplication. In Topology they are $G$-equivariantly trivial $G$-spaces. In Algebraic Geometry, they are \'Etale locally isotrivial geometric quotients of $G$-varieties. The goal of this work is to have a categorical notion that recovers all of them. While they are different structures, they are all locally isomorphic to the Cartesian product of a base space with a group. There are a variety of other results on group objects and their tangent bundle. In particular we show that the tangent bundle is the product of the tangent space and the group object and that the tangent space has an external Lie algebra structure, generalizing the correspondence between Lie groups and Lie algebras. In order to give a purely categorical definition of a principal bundle, we formulate this notion in the language of join restriction categories. Restriction categories were developed by Cockett and Lack to generalize partial maps (maps defined only on a subset of the domain) and have since then found applications in mathematics and computer science. Join restriction categories, as described by Guo are restriction categories where local restrictions can be joined to obtain a global map. Together with a manifold construction due to Grandis, that allows us to glue together objects, we can describe principal bundles entirely in the language of join-restriction categories.

math.CT↗

arXiv2025

Itegories

An itegory is a restriction category with a Kleene wand. Cockett, D{\'\i}az-Bo{\"\i}ls, Gallagher, and Hrube{\v{s}} briefly introduced Kleene wands to capture iteration in restriction categories arising from complexity theory. The purpose of this paper is to develop in more detail the theory of Kleene wands and itegories. A Kleene wand is a binary operator which takes in two disjoint partial maps, one of type ${X \to X}$ and the other of type ${X \to A}$, and produces a partial map $X \to A$. This latter map is interpreted as iterating the endomorphism until it lands in the domain of definition of the second map. In a setting with infinite disjoint joins, there is always a canonical Kleene wand realizing this intuition. The standard categorical interpretation of iteration is via trace operators on coproducts. For extensive restriction categories, we explain in detail how having a Kleene wand is equivalent to this standard interpretation of iteration. This suggests that Kleene wands can be used to replace parametrized iteration and traces in restriction categories which lack coproducts. Further evidence of this is exhibited by providing a matrix construction which embeds an itegory into a traced extensive restriction category. We also consider Kleene wands in classical restriction categories and show how, in this case, a Kleene wand is completely determined by its endomorphism component.

math.CT↗

arXiv2025

Actegories, Copowers, and Higher-Order Message Passing Semantics

In this paper we prove that giving a right actegory with hom-objects is equivalent to giving a right-enriched category with copowers. While this result is known in the closed symmetric setting, our contribution extends the equivalence to non-closed and non-symmetric monoidal bases. This generalization is motivated by the semantics of higher-order message passing in the Categorical Message Passing Language (CaMPL), a concurrent language whose semantics is given by a linear actegory. A desirable feature for this language is the support of higher-order processes: processes that are passed as first class citizens between processes. While this ability is already present in any closed linear type systems -- such as CaMPL's -- to support arbitrary recursive process definitions requires the ability to reuse passed processes. Concurrent resources in CaMPL, however, cannot be duplicated, thus, passing processes as linear closures does not provide the required flexibility. This means processes must be passed as sequential data and the concurrent side must be enriched in the sequential side, motivating the technical result of this paper.

cs.PL↗

arXiv2025

Dagger-Drazin Inverses

Drazin inverses are a special kind of generalized inverses that can be defined for endomorphisms in any category. A natural question to ask is whether one can somehow extend the notion of Drazin inverse to arbitrary maps - not simply endomorphisms. It turns out that this is possible and, indeed, natural to do so for dagger categories. This paper, thus, introduces the notion of a dagger-Drazin inverse, which is a new kind of generalized inverse appropriate for arbitrary maps in a dagger category. This inverse is closely related to the Drazin inverse, for having dagger-Drazin inverses is equivalent to asking that positive maps have Drazin inverses. Moreover, dagger-Drazin inverses are also closely related to Moore-Penrose inverses as we observe that a map has a Moore-Penrose inverse if and only if it is a Drazin inverse. Furthermore, we explain how Drazin inverses of opposing pairs correspond precisely to dagger-Drazin inverses in cofree dagger categories. We also give examples of dagger-Drazin inverses for matrices over (involutive) fields, bounded linear operators, and partial injections.

math.CT↗

arXiv2024

Drazin Inverses in Categories

Drazin inverses are a fundamental algebraic structure which have been extensively deployed in semigroup theory, ring theory, and matrix theory. Drazin inverses can also be defined for endomorphisms in any category. However, beyond a paper by Puystjens and Robinson from 1987, there has been almost no further development of Drazin inverses in category theory. Here we provide a survey of the theory of Drazin inverses from a categorical perspective. We introduce Drazin categories, in which every endomorphism has a Drazin inverse, and provide various examples including the category of matrices over a field, the category of finite length modules over a ring, and finite set enriched categories. We also introduce the notion of expressive rank and prove that a category with expressive rank is Drazin. Moreover, we not only study Drazin inverses in mere categories, but also in additive categories and dagger categories. In an arbitrary category, we show how a Drazin inverse corresponds to an isomorphism in the idempotent splitting, as well as explain how Drazin inverses relate to Leinster's notion of eventual image duality. In additive categories, we consider core-nilpotent decompositions, image-kernel decompositions, and Fitting decompositions. We also develop the notion of Drazin inverses for pairs of opposing maps, generalizing the usual notion of Drazin inverse for endomorphisms. As an application of this new kind of Drazin inverse, for dagger categories, we provide a novel characterization of the Moore-Penrose inverse in terms of being a Drazin inverse of the pair of a map and its adjoint.

math.CT↗

arXiv2023

Moore-Penrose Dagger Categories

The notion of a Moore-Penrose inverse (M-P inverse) was introduced by Moore in 1920 and rediscovered by Penrose in 1955. The M-P inverse of a complex matrix is a special type of inverse which is unique, always exists, and can be computed using singular value decomposition. In a series of papers in the 1980s, Puystjens and Robinson studied M-P inverses more abstractly in the context of dagger categories. Despite the fact that dagger categories are now a fundamental notion in categorical quantum mechanics, the notion of a M-P inverse has not (to our knowledge) been revisited since their work. One purpose of this paper is, thus, to renew the study of M-P inverses in dagger categories. Here we introduce the notion of a Moore-Penrose dagger category and provide many examples including complex matrices, finite Hilbert spaces, dagger groupoids, and inverse categories. We also introduce generalized versions of singular value decomposition, compact singular value decomposition, and polar decomposition for maps in a dagger category, and show how, having such a decomposition is equivalent to having M-P inverses. This allows us to provide precise characterizations of which maps have M-P inverses in a dagger idempotent complete category, a dagger kernel category with dagger biproducts (and negatives), and a dagger category with unique square roots.

cs.LO↗

arXiv2023

Classical Distributive Restriction Categories

In the category of sets and partial functions, $\mathsf{PAR}$, while the disjoint union $\sqcup$ is the usual categorical coproduct, the Cartesian product $\times$ becomes a restriction categorical analogue of the categorical product: a restriction product. Nevertheless, $\mathsf{PAR}$ does have a usual categorical product as well in the form $A \& B := A \sqcup B \sqcup (A \times B)$. Surprisingly, asking that a distributive restriction category (a restriction category with restriction products $\times$ and coproducts $\oplus$) has $A \& B$ a categorical product is enough to imply that the category is a classical restriction category. This is a restriction category which has joins and relative complements and, thus, supports classical Boolean reasoning. The first and main observation of the paper is that a distributive restriction category is classical if and only if $A \& B := A \oplus B \oplus (A \times B)$ is a categorical product in which case we call $\&$ the ''classical'' product. In fact, a distributive restriction category has a categorical product if and only if it is a classified restriction category. This is in the sense that every map $A \to B$ factors uniquely through a total map $A \to B \oplus \mathsf{1}$, where $\mathsf{1}$ is the restriction terminal object. This implies the second significant observation of the paper, namely, that a distributive restriction category has a classical product if and only if it is the Kleisli category of the exception monad $\_ \oplus \mathsf{1}$ for an ordinary distributive category. Thus having a classical product has a significant structural effect on a distributive restriction category. In particular, the classical product not only provides an alternative axiomatization for being classical but also for being the Kleisli category of the exception monad on an ordinary distributive category.

math.CT↗

arXiv2023

Normalizing Resistor Networks

Star to mesh transformations are well-known in electrical engineering, and are reminiscent of local complementation for graph states in qudit stabilizer quantum mechanics. This paper describes a rewriting system for resistor circuits over any positive division rig using general star to mesh transformations. We show how these transformations can be organized into a confluent and terminating rewriting system on the category of resistor circuits. Furthermore, based on the recently established connections between quantum and electrical circuits, this paper pushes forward the quest for approachable normal forms for stabilizer quantum circuits.

math.CT↗

arXiv2022

Extending Resource Monotones using Kan Extensions

In this paper we generalize the framework proposed by Gour and Tomamichel regarding extensions of monotones for resource theories. A monotone for a resource theory assigns a real number to each resource in the theory signifying the utility or the value of the resource. Gour and Tomamichel studied the problem of extending monotones using set-theoretical framework when a resource theory embeds fully and faithfully into the larger theory. One can generalize the problem of computing monotone extensions to scenarios when there exists a functorial transformation of one resource theory to another instead of just a full and faithful inclusion. In this article, we show that (point-wise) Kan extensions provide a precise categorical framework to describe and compute such extensions of monotones. To set up monotone extensions using Kan extensions, we introduce partitioned categories (pCat)as a framework for resource theories and pCat functors to formalize relationship between resource theories. We describe monotones as pCat functors into the preorder of non-negative real numbers, and describe extending monotones along any pCat functor using Kan extensions. We show how our framework works by applying it to extend entanglement monotones for bipartite pure states to bipartite mixed states, to extend classical divergences to the quantum setting, and to extend a non-uniformity monotone from classical probabilistic theory to quantum theory.

quant-ph↗

arXiv2022

Categories of Kirchhoff relations

It is known that the category of affine Lagrangian relations, AffLagRel_F, over a field, F, of integers modulo a prime p (with p > 2) is isomorphic to the category of stabilizer quantum circuits for p-dits. Furthermore, it is known that electrical circuits (generalized for the field F) occur as a natural subcategory of AffLagRel_F. The purpose of this paper is to provide a characterization of the relations in this subcategory -- and in important subcategories thereof -- in terms of parity-check and generator matrices as used in error detection. In particular, we introduce the subcategory consisting of Kirchhoff relations to be (affinely) those Lagrangian relations that conserve total momentum or equivalently satisfy Kirchhoff's current law. Maps in this subcategory can be generated by electrical components (generalized for the field F): namely resistors, current dividers, and current and voltage sources. The "source" electrical components deliver the affine nature of the maps while current dividers add an interesting quasi-stochastic aspect. We characterize these Kirchhoff relations in terms of parity-check matrices and in addition, characterizes two important subcategories: the deterministic Kirchhoff relations and the lossless relations. The category of deterministic Kirchhoff relations as electrical circuits are generated by resistors circuits. Lossless relations, which are deterministic Kirchhoff, provide exactly the basic hyper-categorical structure of these settings.

quant-ph↗

arXiv2021

Exponential Modalities and Complementarity (extended abstract)

The exponential modalities of linear logic have been used by various authors to model infinite-dimensional quantum systems. This paper explains how these modalities can also give rise to the complementarity principle of quantum mechanics. The paper uses a formulation of quantum systems based on dagger-linear logic, whose categorical semantics lies in mixed unitary categories, and a formulation of measurement therein. The main result exhibits a complementary system as the result of measurements on free exponential modalities. Recalling that, in linear logic, exponential modalities have two distinct but dual components, ! and ?, this shows how these components under measurement become "compacted" into the usual notion of complementary Frobenius algebras from categorical quantum mechanics.

math.CT↗

arXiv2020

Linearizing Combinators

In 2017, Bauer, Johnson, Osborne, Riehl, and Tebbe (BJORT) showed that the Abelian functor calculus provides an example of a Cartesian differential category. The definition of a Cartesian differential category is based on a differential combinator which directly formalizes the total derivative from multivariable calculus. However, in the aforementioned work the authors used techniques from Goodwillie's functor calculus to establish a linearization process from which they then derived a differential combinator. This raised the question of what the precise relationship between linearization and having a differential combinator might be. In this paper, we introduce the notion of a linearizing combinator which abstracts linearization in the Abelian functor calculus. We then use it to provide an alternative axiomatization of a Cartesian differential category. Every Cartesian differential category comes equipped with a canonical linearizing combinator obtained by differentiation at zero. Conversely, a differential combinator can be constructed \`a la BJORT when one has a system of partial linearizing combinators in each context. Thus, while linearizing combinators do provide an alternative axiomatization of Cartesian differential categories, an explicit notion of partial linearization is required. This is in contrast to the situation for differential combinators where partial differentiation is automatic in the presence of total differentiation. The ability to form a system of partial linearizing combinators from a total linearizing combinator, while not being possible in general, is possible when the setting is Cartesian closed.

math.CT↗

arXiv2020

Latent Fibrations: Fibrations for Categories of Partial Maps

Latent fibrations are an adaptation, appropriate for categories of partial maps (as presented by restriction categories), of the usual notion of fibration. The paper initiates the development of the basic theory of latent fibrations and explores some key examples. Latent fibrations cover a wide variety of examples, some of which are partial versions of standard fibrations, and some of which are particular to partial map categories (particularly those that arise in computational settings). Latent fibrations with various special properties are identified: hyperconnected latent fibrations, in particular, are shown to support the construction of a fibrational dual; this is important to reverse differential programming and, more generally, in the theory of lenses.

math.CT↗

arXiv2020

Generalising the \'etale groupoid--complete pseudogroup correspondence

We prove a generalisation of the correspondence, due to Resende and Lawson--Lenz, between \'etale groupoids---which are topological groupoids whose source map is a local homeomorphisms---and complete pseudogroups---which are inverse monoids equipped with a particularly nice representation on a topological space. Our generalisation improves on the existing functorial correspondence in four ways. Firstly, we enlarge the classes of maps appearing to each side. Secondly, we generalise on one side from inverse monoids to inverse categories, and on the other side, from \'etale groupoids to what we call partite \'etale groupoids. Thirdly, we generalise from \'etale groupoids to source-\'etale categories, and on the other side, from inverse monoids to restriction monoids. Fourthly, and most far-reachingly, we generalise from topological \'etale groupoids to \'etale groupoids internal to any join restriction category C with local glueings; and on the other side, from complete pseudogroups to ``complete C-pseudogroups'', i.e., inverse monoids with a nice representation on an object of C. Taken together, our results yield an equivalence, for a join restriction category C with local glueings, between join restriction categories with a well-behaved functor to C, and partite source-\'etale internal categories in C. In fact, we obtain this by cutting down a larger adjunction between arbitrary restriction categories over C, and partite internal categories in C. Beyond proving this main result, numerous applications are given, which reconstruct and extend existing correspondences in the literature, and provide general formulations of completion processes.

math.CT↗

arXiv2019

Reverse derivative categories

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

cs.LO↗

arXiv2019

Tangent Categories from the Coalgebras of Differential Categories

Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science. This is an extended version of a conference paper for CSL2020.

math.CT↗