## 2020

### The nucleus: Mining concepts from adjunctions

Categories are many things to many people. We like to think of category theory as the study of certain types of grammar: those that generate the languages of mathematical constructions. This view seems to explain why categories persist in many parts of mathematics despite the ongoing objections that they do not help solve hard problems. Grammars do not help us tell better stories, but they help us tell stories better, and understand them better. Moreover, a language has grammar whether the speakers are aware of it or not.

A grammatical schema that generates many mathematical constructions is that of an *adjunction*. Lawvere noticed that all standard logical operations, quantifiers, and even the set-theoretic comprehension, can be viewed as adjunctions. Lambek observed that natural language grammars also decompose into certain adjunctions, spawning an entire branch of linguistics. Adjunctions are everywhere; all of the category theory can be reconstructed in terms of adjunctions. If category theory is a grammar of mathematical constructions, then adjunctions are the one rule to rule them all.

In Lambek’s grammars, adjunctions can be construed as pairs of brackets, enclosing words into phrases, phrases into sentences, perhaps sentences into narratives. In mathematics, adjunctions bracket structural correspondences across categories. Parsing the syntax of an adjunction uncovers the semantical links between remote objects. A bracket enclosure can be loose or tight; implicit or explicit; the concepts in it can be mixed or separated. Many systems of equations may contain the same information, but the tight system is what we call the solution. The nucleus is the tight form of an adjunction.

Any adjunction induces a monad and a comonad, which respectively display its algebraic and coalgebraic (analytic, state-dependent) aspects. The nucleus construction induces a monad over adjunctions themselves (or equivalently over the induced monads, or comonads), which distills both aspects. It is tight because it is idempotent. The two sides of a nuclear adjunction completely determine one another, as algebras and coalgebras, yet they remain distinct, like brackets.

## 2018

### *Universal Probability-Free Prediction*

#### Vladimir Vovk and D.P.

We construct universal prediction systems in the spirit of Popperâ€™s falsifiability

and Kolmogorov complexity and randomness. These prediction systems do not depend on

any statistical assumptions (but under the IID assumption they dominate, to within the usual

accuracy, conformal prediction). Our constructions give rise to a theory of algorithmic com-

plexity and randomness of time containing analogues of several notions and results of the

classical theory of Kolmogorov complexity and randomness.

The COPA 2016 version of the same paper: *Universal Probability-Free Conformal Prediction*.

## 2015

### *Testing randomness by Matching Pennies*

In the game of Matching Pennies, Alice and Bob each hold a penny, and at every tick of the clock they simultaneously display the head or the tail sides of their coins. If they both display the same side, then Alice wins Bob’s penny; if they display different sides, then Bob wins Alice’s penny. To avoid giving the opponent a chance to win, both players seem to have nothing else to do but to randomly play heads and tails with equal frequencies. However, while not losing in this game is easy, not missing an opportunity to win is not. Randomizing your own moves can be made easy. Recognizing when the opponent’s moves are not random can be arbitrarily hard.

The notion of randomness is central in game theory, but it is usually taken for granted. The notion of outsmarting is not central in game theory, but it is central in the practice of gaming. We pursue the idea that these two notions can be usefully viewed as two sides of the same coin.

### *Smooth coalgebra: testing vector analysis*

#### with Bertfried Fauser

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to Banach spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton’s laws.

## 2014

### *Monoidal computer II: Normal complexity by string diagrams*

In Monoidal Computer I, we introduced a categorical model of computation where the formal reasoning about computability was supported by the simple and popular diagrammatic language of string diagrams. In the present paper, we refine and extend that model of computation to support a formal complexity theory as well. This formalization brings to the foreground the concept of normal complexity measures, which allow decompositions akin to Kleene’s normal form. Such measures turn out to be just those where evaluating the complexity of a program does not require substantially more resources than evaluating the program itself. The usual time and space complexity are thus normal measures, whereas the average and the randomized complexity measures are not. While the measures that are not normal provide important design time information about algorithms, and for theoretical analyses, normal measures can also be used at run time, as practical tools of computation, e.g. to set the bounds for hypothesis testing, inductive inference and algorithmic learning.

## 2013

### *Bicompletions of distance matrices*

In the practice of information extraction, the input data are usually arranged into *pattern matrices*, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The usual reason is that the matrix composition of linear algebra presents information as flowing in waves, whereas it sometimes flows in particles, which seek the shortest paths. This wave-particle duality in computation and information processing has been originally observed by Abramsky. In this paper we pursue a particle view of information, formalized in *distance spaces*, which generalize metric spaces, but are slightly less general than Lawvere’s generalized metric spaces. In this framework, the task of extracting the ‘principal components’ from a given matrix of data boils down to a *bicompletion*, in the sense of enriched category theory. We describe the bicompletion construction for distance matrices. The practical goal that motivates this research is to develop a method to estimate the hardness of attack constructions in security.

## 2012

*Monoidal computer I: Basic computability by string diagrams*

We present a new model of computation, described in terms of monoidal categories. It conforms the Church-Turing Thesis, and captures the same computable functions as the standard models. It provides a succinct categorical interface to most of them, free of their diverse implementation details, using the ideas and structures that in the meantime emerged from research in semantics of computation and programming. The salient feature of the language of monoidal categories is that it is supported by a sound and complete graphical formalism, string diagrams, which provide a concrete and intuitive interface for abstract reasoning about computation. The original motivation and the ultimate goal of this effort is to provide a convenient high level programming language for a theory of computational resources, such as one-way functions, and trapdoor functions, by adopting the methods for hiding the low level implementation details that emerged from practice. Only a modest first step towards this ambitious goal is presented.

## 2008

*A new description of orthogonal bases*

with Bob Coecke and Jamie Vicary

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative dagger-Frobenius monoid in the category FHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative dagger-Frobenius monoid is special. Hence orthogonal and orthonormal bases can be axiomatised in terms of composition of operations and tensor product only, without any explicit reference to the underlying vector spaces. This axiomatisation moreover admits an operational interpretation, as the comultiplication copies the basis vectors and the counit uniformly deletes them. That is, we rely on the distinct ability to clone and delete classical data as compared to quantum data to capture basis vectors. For this reason our result has important implications for categorical quantum mechanics.

## 1996

*Maps II: chasing diagrams in categorical proof theory*

In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositions-as-types and proofs-as-constructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logical-operations-as-adjunctions. It offers an answer to the notorious question of the equality of proofs. Diagram chasing comes about in algebra of proofs.

On the basis of these ideas, the present paper investigates proof theory of regular logic: the {\and,\exists}-fragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and single-valued relations. However, when enriched with proofs-as-arrows, this familiar concept must be supplied with an additional conversion rule, connecting the proof of the totality with the proof of the single-valuedness. We explain the logical meaning of this rule, and then determine precise conditions under which a regular fibration supports the principle of function comprehension (that each map corresponds to a unique function viz arrow in the base), thus lifting a basic theorem from regular categories (that Map(Rel(C) = C), which was recently relativized to factorisation systems. The obtained results open an interesting possibility of extending the P-set construction from triposes to non-posetal fibrations.

## 1995

*A categorical setting for the 4-Colour Theorem*

The 4-Colour Theorem has been proved in the late seventies , after more than a century of fruitless efforts. But the proof has provided very little new information about the map colouring itself. While trying to understand this phenomenon, we analyze colouring in terms of universal properties and adjoint functors.

It is well known that the 4-colouring of maps is equivalent to the 3-colouring of the edges of some graphs. We show that every slice of the category of 3-coloured graphs is a topos. The forgetful functor to the category of graphs is cotripleable; every loop-free graph is covered by a 3-coloured one in a universal way. In this context, the 4-Color Theorem becomes a statement about the existence of coalgebra structure on graphs.

In a sense, this approach seems complementary to the known combinatorial colouring procedures.

*On completeness and cocompleteness in and around small categories*

The simple connection of completeness and cocompleteness of lattices grows in categories into the Adjoint Functor Theorem. The connection of completeness and cocompleteness of Boolean algebras – even simpler – is similarly related to Pare’s Theorem for toposes. We explain these relations and then study the fibrational versionso of both theorems, for small complete categories. They can be interpreted as definability results in logic with proofs-as-constructions, and transferred into type theory.

*Maps I: relative to a factorisation system*

Originally, categorical calculus of relations was developed using the canonical factorisations in regular categories. More recently, relations restricted to a proper factorisation systems have been studied by several authors. In the present paper, we consider the general situation, where relations are induced by an arbitrary stable factorisation. This extension of the calculus of relations is necessary for categorical development of strongly constructivist logic, and of semantics of computation, where non-monic relations come about naturally.

In this setting we analyze the correspondence of *maps*, i.e. the total, single-valued relations, and *functions*, as given by the arrows of the underlying category.

## 1993

*Chu I: cofree equivalences, dualities and *-autonomous categories*

We study three comonads derived from the comma construction. The induced coalgebras correspond to the three concepts displayed in the title of the paper. The comonad that yields the cofree *-autonomous categories is, in essence, the Chu construction, which has recently awaken much interest in computer science. We describe its couniversal property. It is right adjoint to the inclusion of *-autonomous categories among autonomous categories, with lax structure-preserving morphisms. Moreover, this inclusion turns out to be comonadic: *-autonomous categories are exactly the Chu-coalgebras.

## 1992

*On the structure of paradoxes*

Paradox is a logical phenomenon. Usually, it is produced in type theory, over a type of “truth values”. A formula p, as a term p:Prop is constructed, such that p is equivalent to its negation – whereupon everything can be proved. We describe a general pattern which many constructions of the formula follow: for example, the well known arguments of Cantor, Russell, and Goedel. We generalize their common underlying structure, and prove that Reynolds’ construction of the type isomorphic to its second power type, contrary to his conjecture, cannot be extended to give a fixed point of every variable type derived from exponentiation. In particular, for contravariant types, such a fixed point would lead to a paradox.

## 1991

*Constructions and Predicates*

In this paper, Coquand and Huet’s Theory of Constructions is reinterpreted as a type theory of “sets” and “predicates”. Following some set-theoretical intuitions, this popular type system is modified at two points: (1) a simple new operation is added – to represent a constructive version of the comprehension principle; (2) a restriction on contexts is imposed – “sets” must not depend on “proofs” of “predicates”. The resulting theory is called Theory of Predicates. Sufficiently constructive arguments from naive set theory can be directly written down in it. On the other hand, modification (2) is relevant from a computational point of view, since it corresponds to a necessary condition of the modular approach to programming.

Our main result tells that, despite (2), Theory of Predicates is as expressive as Theory of Constructions: the type constructors preempted by (2) are recovered in an isomorphic form using (1). In fact, Theory of Constructions is equivalent to a quotient of Theory of Predicates.

## 1990

*Categorical Interpolation: Descent and the Beck-Chevalley Condition without Direct Images*

Fibred categories have been introduced by Grothendieck (1959, 1970) as the setting for his theory of descent. The present paper contains (in section 4) a characterisation of the effective descent morphisms under an arbitrary fibred category. This essentially geometric result complements a logical analysis of the Beck-Chevalley property (presented in section 1). This property was crucial in the well-known theorem on sufficient conditions for the descent under bifibrations, due to Benabou and Roubaud (1970) and Beck (unpublished). We describe the notion of *interpolant* as the common denominator of the concepts of descent and the Beck-Chevalley property.

*Predicates and Fibrations*

(thesis, 18MB scan)

The commonplace simplification of constructivist logic as a logic without the excluded middle conceals a more authentic idea of constructivism: that proofs are constructive functions, while the quantifiers should be sums and products. An exact realisation of this idea requires a very special mathematical setting. Expressed categorically, the idea is that there is a small category (of “propositions” and “proofs”), with all small sums and products (as “quantifiers”). In ordinary category theory (or over a Grothendieck topos) such a category must be a preorder, and there can be at most one “proof” leading from one “proposition” to another. In Hyland’s Effective Topos, however, a nondegenerate small complete category has been discovered recently. It can be regarded as the first mathematical model of logic with constructive proofs. On the other hand, a significant impluse to a formal development of such logics came from Computer Science, especially through the work of Coquand, Huet and their collaborators on the Calculus of Constructions.

In this thesis, we consider two mathematical formulations of constructivist logic: a type theoretical and a category theoretical. In the end, the former is completely interpreted in the latter. The purpose of such a connection is to yield a characterisation of type theoretic*structures* by categorical *properties*.