Polygraphs are a higher-dimensional generalization of the notion of directed graph. Based on those as unifying concept, this monograph on polygraphs revisits the theory of rewriting in the context of strict higher categories, adopting the abstract point of view offered by homotopical algebra. The first half explores the theory of polygraphs in low dimensions and its applications to the computation of the coherence of algebraic structures. It is meant to be progressive, with little requirements on the background of the reader, apart from basic category theory, and is illustrated with algorithmic computations on algebraic structures. The second half introduces and studies the general notion of n-polygraph, dealing with the homotopy theory of those. It constructs the folk model structure on the category of strict higher categories and exhibits polygraphs as cofibrant objects. This allows extending to higher dimensional structures the coherence results developed in the first half.

翻译：多图模型是有向图概念的高维推广。本书以多图模型作为统一概念，在严格高阶范畴的背景下重新审视重写理论，并采用同调代数提供的抽象视角。前半部分探讨低维多图模型的理论及其在代数结构一致性计算中的应用，采用循序渐进的方式，除基本范畴论外对读者背景知识要求较少，并辅以代数结构上的算法计算示例。后半部分介绍并研究n-多图模型的一般概念，探讨其同伦理论，构建了严格高阶范畴范畴上的民俗模型结构，并展示多图模型作为余纤维对象。这使得前半部分发展的一致性结果得以推广至高维结构。