Related papers: Notions of Computation as Monoids
A differential category is an additive symmetric monoidal category, that is, a symmetric monoidal category enriched over commutative monoids, with an algebra modality, axiomatizing smooth functions, and a deriving transformation on this…
Generalized operads, also called generalized multicategories and $T$-monoids, are defined as monads within a Kleisli bicategory. With or without emphasizing their monoidal nature, generalized operads have been considered by numerous authors…
We introduce a notion of signature whose sorts form a direct category, and study computads for such signatures. Algebras for such a signature are presheaves with an interpretation of every function symbol of the signature, and we describe…
We develop a symmetric monoidal closed category of games, incorporating sums and products, to model quantum computation at higher types. This model is expressive, capable of representing all unitary operators at base types. It is compatible…
We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which…
Computability logic is a formal theory of computational tasks and resources. Its formulas represent interactive computational problems, logical operators stand for operations on computational problems, and validity of a formula is…
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…
We develop further the theory of monoidal bicategories by introducing and studying bicategorical counterparts of the notions of a linear exponential comonad, as considered in the study of linear logic, and of a codereliction transformation,…
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…
We discuss a categorical version of the celebrated belief propagation algorithm. This provides a way to prove that some algorithms which are known or suspected to be analogous, are actually identical when formulated generically. It also…
This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…
Big data programming frameworks have become increasingly important for the development of applications for which performance and scalability are critical. In those complex frameworks, optimizing code by hand is hard and time-consuming,…
We describe a new logical data model, called the concept-oriented model (COM). It uses mathematical functions as first-class constructs for data representation and data processing as opposed to using exclusively sets in conventional…
We introduce contextads and the Ctx construction, unifying various structures and constructions in category theory dealing with context and contextful arrows -- comonads and their Kleisli construction, actegories and their Para…
A coherent mathematical overview of computation and its generalisations is described. This conceptual framework is sufficient to comfortably host a wide range of contemporary thinking on embodied computation and its models.
The theory of regular cost functions is a quantitative extension to the classical notion of regularity. A cost function associates to each input a non-negative integer value (or infinity), as opposed to languages which only associate to…
We use Lurie's symmetric monoidal envelope functor to give two new descriptions of $\infty$-operads: as certain symmetric monoidal $\infty$-categories whose underlying symmetric monoidal $\infty$-groupoids are free, and as certain symmetric…
Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. Although it can be treated purely as an algebraic subject, it is inherently topological in nature: the…
Monads are extensively used nowadays to abstractly model a wide range of computational effects such as nondeterminism, statefulness, and exceptions. It turns out that equipping a monad with a (uniform) iteration operator satisfying a set of…
Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal…