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Relational properties describe multiple runs of one or more programs. They characterize many useful notions of security, program refinement, and equivalence for programs with diverse computational effects, and they have received much…
Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann, we define canonical computability models over categories and appropriate Set-valued functors…
Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively…
The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. The titular category has nice formal properties: it is bicomplete and is a symmetric monoidal category, with…
Recognizing analogies, synonyms, antonyms, and associations appear to be four distinct tasks, requiring distinct NLP algorithms. In the past, the four tasks have been treated independently, using a wide variety of algorithms. These four…
Adjoint functors and projectivization in representation theory of partially ordered sets are used to generalize the algorithms of differentiation by a maximal and by a minimal point. Conceptual explanations are given for the combinatorial…
This paper presents an abstraction of Hoare logic to traced symmetric monoidal categories, a very general framework for the theory of systems. Our abstraction is based on a traced monoidal functor from an arbitrary traced monoidal category…
A type theory is presented that combines (intuitionistic) linear types with type dependency, thus properly generalising both intuitionistic dependent type theory and full linear logic. A syntax and complete categorical semantics are…
We propose another interpretation of well-known derivatives computations from regular expressions, due to Brzozowski, Antimirov or Lombardy and Sakarovitch, in order to abstract the underlying data structures (e.g. sets or linear…
We explain two related constructions on the data of two monoidal symmetric closed categories $\mathscr{A}$ and $\mathscr{E}$ and monoidal functors $F: \mathscr{E}\to \mathscr{A}$ and $G: \mathscr{A}\to \mathscr{E}$. In a first part, we…
In this paper we define a pair of faithful functors that map isomorphic and isotopic finite-dimensional algebras over finite fields to isomorphic graphs. These functors reduce the cost of computation that is usually required to determine…
It is well established that equational algebraic theories, and the monads they generate, can be used to encode computational effects. An important insight of Power and Shkaravska is that comodels of an algebraic theory T -- i.e., models in…
Monadic programming presents a significant challenge for many programmers. In light of category theory, we offer a new perspective on the use of monads in functional programming. This perspective is clarified through numerous examples coded…
We explore the relationship between polynomial functors and (rooted) trees. In the first part we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category…
We explore some connections between vectors of integers and integer partitions seen as bi-infinite words. This methodology enables us on the one hand to obtain enumerations connecting products of hook lengths and vectors of integers. This…
The article explores function terms within uniform theories. It examines the uniformity of these theories through an algebraic lens. The paper compares the uniformity of terms and predicates within axiom schemas. It demonstrates the…
Taking inspiration from the monadicity of complete atomic Boolean algebras, we prove that profinite modal algebras are monadic over Set. While analyzing the monadic functor, we recover the universal model construction - a construction…
We characterize the equational theories and Lawvere theories that correspond to the categories of analytic and polynomial monads on Set, and hence also the categories of the symmetric and rigid operads in Set. We show that the category of…
When formalizing mathematics in (generalized predicative) constructive type theories, or more practically in proof assistants such as Coq or Agda, one is often using setoids (types with explicit equivalence relations). In this note we…
In this work, we study 'reduction monads', which are essentially the same as monads relative to the free functor from sets into multigraphs. Reduction monads account for two aspects of the lambda calculus: on the one hand, in the monadic…