Après avoir exposé l'importance des dualités géométrico-algébriques dans l'histoire des mathématiques, la thèse propose de rassembler bon nombre d'entre elle sous une approche unifiée abstraite, la dualité d'Isbell. La dualité d'Isbell est formellement définie comme une adjonction entre un préfaisceau et un copréfaisceau, et permet de définir un nouveau paradigme de constructivité baptisé P3. En mathématique, nous montrons que cette dualité est présente en géométrie algébrique, en géométrie algébrique dérivée, en topologie algébrique et en analyse fonctionnelle. En logique contemporaine, nous montrons qu'elle peut être rendue explicite dans la géométrie de l'interaction de Girard. Nous montrons ensuite comment les sciences appliquées peuvent faire usage de la dualité d'Isbell, en permettant de renouveler significativement les théories. En sciences physiques, nous montrons qu'elle ouvre une perspective dans la théorie quantique des champs, vers la dualisation des représentations de Heisenberg et de Schrödinger. En sciences économiques et sociales, nous montrons qu'elle permet de renouveler la théorie de l'équilibre générale et la théorie de la valeur. En sciences de l'apprentissage, nous montrons qu'il est possible de reconsidérer la théorie de l'enquête de Dewey en termes de dualité espace-action, pour finalement dégager une dualité d'Isbell. Nous concluons en ouvrant un débat sur la notion bachelardienne d'obstacle épistémologique, pour montrer comment P3 peut avoir des difficultés à s'imposer, et en consacrant quelques développements ontologiques sur la nature kantienne et post-hégélienne de la thèse. / After exposing the importance of geometric-algebraic dualities in the history of mathematics, the thesis proposes to bring together many of them under an unified abstract approach, the Isbell duality. The Isbell duality is formally defined as an adjunction between a presheaf and a copresheaf, and allows to define a new paradigm of constructivity called P3. In mathematics, we show that this duality is present in algebraic geometry, derived algebraic geometry, algebraic topology and functional analysis. In contemporary logic, we show that Isbell duality can be made explicit in the geometry of interaction of Girard. We then show how applied sciences can make use of Isbell duality, allowing to significantly renew theories. In physical sciences, we show that it opens a perspective in quantum field theory, towards the dualization of Heisenberg and Schrödinger representations. In economic and social sciences, we show that it allows to renew the general equilibrium theory and the theory of value. In learning sciences, we show that it is possible to reconsider Dewey's theory of inquiry in terms of space-action duality, ultimately to reveal an Isbell duality. We conclude by opening a debate on the Bachelardian notion of epistemological obstacle, showing how P3 can have difficulties to establish itself as reference constructive paradigm, and by devoting some ontological developments to the Kantian and post-Hegelian nature of the thesis.

This thesis consists of four papers about the intersection between semigroup theory, category theory and representation theory. We say that a representation of a semigroup by a matrix semigroup is effective if it is injective and define the effective dimension of a semigroup S as the minimal n such that S has an effective representation by square matrices of size n. A multisemigroup is a generalization of a semigroup where the multiplication is set-valued, but still associative. A 2-category consists of objects, 1-morphisms and 2-morphisms. A finitary 2-category has finite dimensional vector spaces as objects and linear maps as morphisms. This setting permits the notion of indecomposable 1-morphisms, which turn out to form a multisemigroup. Paper I computes the effective dimension Hecke-Kiselman monoids of type A. Hecke-Kiselman monoids are defined by generators and relations, where the generators are vertices and the relations depend on arrows in a given quiver. Paper II computes the effective dimension of path semigroups and truncated path semigroups. A path semigroup is defined as the set of all paths in a quiver, with concatenation as multiplication. It is said to be truncated if we introduce the relation that all paths of length N are zero. Paper III defines the notion of a multisemigroup with multiplicities and discusses how it better captures the structure of a 2-category, compared to a multisemigroup (without multiplicities). Paper IV gives an example of a family of 2-categories in which the multisemigroup with multiplicities is not a semigroup, but where the multiplicities are either 0 or 1. We describe these multisemigroups combinatorially.

We investigate categorical models of the unit-free multiplicative and multiplicative-additive fragments of linear logic by representing derivations as particular structures known as dinatural transformations. Suitable categories are considered to satisfy a property known as full completeness if all such entities are the interpretation of a correct derivation. It is demonstrated that certain Hyland-Schalk double glueings [HS03] are capable of transforming large numbers of degenerate models into more accurate ones. Compact closed categories with finite biproducts possess enough structure that their morphisms can be described as forms of linear arrays. We introduce the notion of an extended tensor (or ‘extensor’) over arbitrary semirings, and show that they uniquely describe arrows between objects generated freely from the tensor unit in such categories. It is made evident that the concept may be extended yet further to provide meaningful decompositions of more general arrows. We demonstrate how the calculus of extensors makes it possible to examine the combinatorics of certain double glueing constructions. From this we show that the Hyland-Tan version [Tan97], when applied to compact closed categories satisfying a far weaker version of full completeness, produces genuine fully complete models of unit-free multiplicative linear logic. Research towards the development of a full completeness result for the multiplicative-additive fragment is detailed. The proofs work for categories of finite arrays over certain semirings under both the Hyland-Tan and Schalk [Sch04] constructions. We offer a possible route to finishing this proof. An interpretation of these results with respect to linear logic proof theory is provided, and possible further research paths and generalisations are discussed.

The concept of a recursive function has been extensively studied using traditional tools of computability theory. However, with the development of category-theoretic methods it has become possible to study recursion in a more general (abstract) sense. The particular model this thesis is structured around is known as a Turing category. The structure within a Turing category models the notion of partiality as well as recursive computation, and equips us with the tools of category theory to study these concepts. The goal of this work is to build a formal language description of this computation model. Specifically, to use the Coq proof assistant to formulate informal definitions, propositions and proofs pertaining to Turing categories in the underlying formal language of Coq, the Calculus of Co-inductive Constructions (CIC). Furthermore, we have instantiated the more general Turing category formalism with a CIC description of the category which models the language of partial recursive functions exactly.

Dans cette thèse, on définit et on étudie les notions de structure de Poisson et coïsotrope sur un champ dérivé, dans le contexte de la géométrie algébrique dérivée. On considère deux présentations différentes de structure de Poisson : la première est purement algébrique, alors que la deuxième est plus géométrique. On montre que les deux approches sont en fait équivalentes. On introduit aussi la notion de structure coïsotrope sur un morphisme de champs dérivés, encore une fois en présentant deux définitions équivalentes : la première est basée sur une généralisation appropriée de l'opérade Swiss-Cheese de Voronov, tandis que la deuxième est formulée en termes de champs de multivecteurs rélatifs. En particulier, on montre que le morphisme identité admet une unique structure coïsotrope ; cela produit une application d'oubli des structures de Poisson n-décalées aux structures de Poisson (n-1)-décalées. On montre aussi que l'intersection de deux morphismes coïsotropes dans un champ de Poisson n-décalée est naturellement equipée d'une structure de Poisson (n-1)-décalée canonique. En outre, on fournit une équivalence entre l'espace de structures coïsotropes non-dégénérées et l'espace des structures Lagrangiennes en géométrie dérivée, introduites dans les travaux de Pantev-Toën-Vaquié-Vezzosi. / In this thesis, we define and study Poisson and coisotropic structures on derived stacks in the framework of derived algebraic geometry. We consider two possible presentations of Poisson structures of different flavour: the first one is purely algebraic, while the second is more geometric. We show that the two approaches are in fact equivalent. We also introduce the notion of coisotropic structure on a morphism between derived stacks, once again presenting two equivalent definitions: one of them involves an appropriate generalization of the Swiss Cheese operad of Voronov, while the other is expressed in terms of relative polyvector fields. In particular, we show that the identity morphism carries a unique coisotropic structure; in turn, this gives rise to a non-trivial forgetful map from n-shifted Poisson structures to (n-1)-shifted Poisson structures. We also prove that the intersection of two coisotropic morphisms inside a n-shifted Poisson stack is naturally equipped with a canonical (n-1)-shifted Poisson structure. Moreover, we provide an equivalence between the space of non-degenerate coisotropic structures and the space of Lagrangian structures in derived geometry, as introduced in the work of Pantev-Toën-Vaquié-Vezzosi.

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Diese Arbeit beschäftigt sich mit der Speziestheorie aus Mochizukis Beweis(versuch) der abc-Vermutung. Es wird ein Standpunkt eingeführt, der Parallelen zwischen der Kategorientheorie und der Speziestheorie aufzeigt und es werden so die Besonderheiten der Speziestheorie herausgearbeitet. In der Speziestheorie möchte man Konstruktionen ausführen, welche von keinen Auswahlen abhängen. Dieses Problem wird in einem allgemeinen Kontext für universelle Morphismen gelöst. An Beispielen wird die in der Arbeit behandelte Theorie erklärt.

The main goal of this thesis was to implement a functional programming (FP) library named Funk that extends C# with support for concepts present in functional programming languages, such as F# and Scala. Funk utilizes many functional programming concepts, including immutability, pattern matching, and various types of monads, together with stronger typing. Introduction of these concepts into C# helps in avoiding many runtime errors and boilerplate code, and it also lets developers write C# code in a declarative rather than in an imperative way, making the day-to-day software development easier and less error-prone. Additionally, the thesis analyzes and compares Funk with existing functional programming libraries such as Language-ext and FuncSharp. Finally, it analyzes the new features of C# 8, which include nullable reference types and pattern matching and compares them with the functionalities of the Funk library.

In this work we introduce some category-theoretical concepts and techniques to study probability distributions on metric spaces and ordered metric spaces. In Chapter 1 we give an overview of the concept of a probability monad, first defined by Giry. Probability monads can be interpreted as a categorical tool to talk about random elements of a space X. We can consider these random elements as formal convex combinations, or mixtures, of elements of X. Spaces where the convex combinations can be actually evaluated are called algebras of the probability monad. In Chapter 2 we define a probability monad on the category of complete metric spaces and 1-Lipschitz maps called the Kantorovich monad, extending a previous construction due to van Breugel. This monad assigns to each complete metric space X its Wasserstein space PX. It is well-known that finitely supported probability measures with rational coefficients, or empirical distributions of finite sequences, are dense in the Wasserstein space. This density property can be translated into categorical language as a colimit of a diagram involving certain powers of X. The monad structure of P, and in particular the integration map, is uniquely determined by this universal property. We prove that the algebras of the Kantorovich monad are exactly the closed convex subsets of Banach spaces. In Chapter 3 we extend the Kantorovich monad of Chapter 2 to metric spaces equipped with a partial order. The order is inherited by the Wasserstein space, and is called the stochastic order. Differently from most approaches in the literature, we define a compatibility condition of the order with the metric itself, rather then with the topology it induces. We call the spaces with this property L-ordered spaces. On L-ordered spaces, the stochastic order induced on the Wasserstein spaces satisfies itself a form of Kantorovich duality. The Kantorovich monad can be extended to the category of L-ordered metric spaces. We prove that its algebras are the closed convex subsets of ordered Banach spaces, i.e. Banach spaces equipped with a closed cone. The category of L-ordered metric spaces can be considered a 2-category, in which we can describe concave and convex maps categorically as the lax and oplax morphisms of algebras. In Chapter 4 we develop a new categorical formalism to describe operations evaluated partially. We prove that partial evaluations for the Kantorovich monad, or partial expectations, define a closed partial order on the Wasserstein space PA over every algebra A, and that the resulting ordered space is itself an algebra. We prove that, for the Kantorovich monad, these partial expectations correspond to conditional expectations in distribution. Finally, we study the relation between these partial evaluation orders and convex functions. We prove a general duality theorem extending the well-known duality between convex functions and conditional expectations to general ordered Banach spaces.

In this dissertation we define an analogue, in the setting of infinity categories, of the classical notion of an enriched operad. We introduce six different models of enriched infinity operads. In particular, we generalize the operator category approach of Clark Barwick to the enriched setting as well as Moerdijk-Weiss' notion of dendroidal sets. The main part of the thesis consists of the comparison between different approaches to enriched operads.