Related papers: A Linear Category of Polynomial Diagrams
A new proof for adjoint systems of linear equations is presented. The argument is built on the principles of Algorithmic Differentiation. Application to scalar multiplication sets the base line. Generalization yields adjoint inner vector,…
Linear Logic refines Intuitionnistic Logic by taking into account the resources used during the proof and program computation. In the past decades, it has been extended to various frameworks. The most famous are indexed linear logics which…
Categorial actions of braided tensor categories are defined and shown to be the right framework for a discussion of the categorial structure related to the group of braids in the cylinder. A Kauffman polynomial of links in the solid torus…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
To formalize calculations in linear algebra for the development of efficient algorithms and a framework suitable for functional programming languages and faster parallelized computations, we adopt an approach that treats elements of linear…
Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic…
This note explains how dependent sums and products are interpreted by adjoints of the base change functor in a locally cartesian closed category. An effort is made to unpack all the definitions so as to make the concepts more transparent to…
This paper exhibits a series of semantic characterisations of sublinear nondeterministic complexity classes. These results fall into the general domain of logic-based approaches to complexity theory and so-called implicit computational…
Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…
We construct a modular functor which takes its values in the monoidal bicategory of finite categories, left exact functors and natural transformations. The modular functor is defined on bordisms that are 2-framed. Accordingly we do not need…
Usually a name of the category is inherited from the name of objects. However more relevant for a category of objects and morphisms is an algebra of morphisms. Therefore we prefer to say a category of graphs if every morphism is a graph. In…
Let $\mathcal C$ be a category with finite colimits, writing its coproduct $+$, and let $(\mathcal D, \otimes)$ be a braided monoidal category. We describe a method of producing a symmetric monoidal category from a lax braided monoidal…
We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common…
Cartesian differential categories come equipped with a differential combinator that formalizes the derivative from multi-variable differential calculus, and also provide the categorical semantics of the differential $\lambda$-calculus. An…
The categorified theories known as "doctrines" specify a category equipped with extra structure, analogous to how ordinary theories specify a set with extra structure. We introduce a new framework for doctrines based on double category…
Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding…
We extend the recently introduced setting of coherent differentiation for taking into account not only differentiation, but also Taylor expansion in categories which are not necessarily (left)additive. The main idea consists in extending…
We consider tensor grammars, which are an example of \commutative" grammars, based on the classical (rather than intuitionistic) linear logic. They can be seen as a surface representation of abstract categorial grammars ACG in the sense…
Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more…
We show that contrary to common belief in the DisCoCat community, a monoidal category is all that is needed to define a categorical compositional model of natural language. This relies on a construction which freely adds adjoints to a…