Related papers: A graphical calculus for semi-groupal categories
Properties of morphisms represented by so-called 'string diagrams' of monoidal categories (and their braided and symmetric derivatives), mainly their resistance in value to isotopic deformation, have made the usage of graphical calculi…
A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This…
A popular graphical calculus for monoidal categories makes computations tactile and intuitive. Complicated diagram chases can be expressed in a few pictures and discovered by playing with a shoelace. Joyal and Street's proof of the…
We propose a graphical language that accommodates two monoidal structures: a multiplicative one for pairing and an additional one for branching. In this colored PROP, whether wires in parallel are linked through the multiplicative structure…
Path calculus, or graphical linear algebra, is a string diagram calculus for the category of matrices over a base ring. It is the usual string diagram calculus for a symmetric monoidal category, where the monoidal product is the direct sum…
We describe how to perform surgeries on Joyal-Street style diagrams and thereby construct free Markov categories. We also show that the construction is functorial over ordered directed acyclic graphs.
This thesis provides an introduction to the various category theory ideas employed in topological quantum field theory. These theories are viewed as symmetric monoidal functors from topological cobordism categories into the category of…
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,…
This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and…
We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors…
Bimonoidal categories (also known as rig categories) are categories with two monoidal structures, one of which distributes over the other. We formally define sheet diagrams, a graphical calculus for bimonoidal categories that was informally…
We present here definitions and constructions basic for the theory of monoidal and tensor categories. We provide references to the original sources, whenever possible. Group-theoretical categories are used as examples
We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially…
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…
Morphisms in a monoidal category are usually interpreted as processes, and graphically depicted as square boxes. In practice, we are faced with the problem of interpreting what non-square boxes ought to represent in terms of the monoidal…
Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free…
We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of promonoidal category. Since the theory of promonoidal categories has not been developed…
String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to…
Motivated by the weighted Hurwitz product on sequences in an algebra, we produce a family of monoidal structures on the category of Joyal species. We suggest a family of tensor products for charades. We begin by seeing weighted derivational…
In this paper we present cartesian structure for symmetric Gray-monoidal double categories. To do this we first introduce locally cubical Gray categories, which are three-dimensional categorical structures analogous to classical, locally…