Using Rewriting Systems to Compute Kan Extensions and Induced Actions of Categories

Rewriting Procedures Generalise to Kan Extensions of Actions of Categories

Kan extensions provide a natural general framework for a variety of combinatorial problems. We have developed rewriting procedures for Kan extensions (over the category of sets) and this enables one program to address a wide range of problems. Thus it is possible to use the same framework (and therefore program) to enumerate monoid or group (or category of groupoid) elements, to enumerate cosets or congruence classes on monoids, calculate equivariant equivalence relations, induced actions of groups, monoids or categories and even more. This extended abstract is an outline of "Using Rewriting S…

Grobner Basis Techniques for Computing Actions of K-Categories

This paper involves categories and computer science. The paper is motivated by a question which arises from two pieces of research. Firstly, the work of Brown and Heyworth which extends rewriting techniques to enable the computation of left Kan extensions over the category of sets. It is well known that left Kan extensions can be defined over categories other than Sets. Secondly, the `folklore' that rewriting theory is a special case of noncommutative Groebner basis theory. It is therefore natural to ask whether Groebner bases can provide a method for computing Kan extensions beyond the specia…

Homotopies in Multiway (Non-Deterministic) Rewriting Systems as $n$-Fold Categories

We investigate algebraic and compositional properties of abstract multiway rewriting systems, which are archetypical structures underlying the formalism of the Wolfram model. We demonstrate the existence of higher homotopies in this class of rewriting systems, where homotopical maps are induced by the inclusion of appropriate rewriting rules taken from an abstract rulial space of all possible such rules. Furthermore, we show that a multiway rewriting system with homotopies up to order $n$ may naturally be formalized as an $n$-fold category, such that (upon inclusion of appropriate inverse morp…

表格单体：一种重写理论方法

The tableaux monoid: a rewriting-theoretic approach

本文从重写理论视角研究表格单体，构造了有限二次收敛表示并证明其双自动性，进而扩展为相干表示，提供规范范式并产生同调与算法推论。

黏合范畴中的规则代数

Rule Algebras for Adhesive Categories

本文证明双推重写在黏合范畴中沿重叠的规则顺序组合具有自然结合性，并定义了规则代数，在具有初始对象时得到单位代数，为随机DPO重写系统中统计矩的计算提供了规范表示。

On a simplicial monoid whose underlying simplicial set is not a quasi-category

It is well known that the underlying simplicial set of any simplicial group is a Kan complex. Roughly speaking, Kan complex is an infinite-dimensional analogue of groupoid, and the relation between groupoids and categories resembles that between groups and monoids. Thus one may ask if the underlying simplicial set of each simplicial monoid is a quasi-category. In this short note, we construct a simplicial monoid whose underlying simplicial set is not a quasi-category.