Polygraph (mathematics)

In mathematics, and particularly in category theory, a polygraph is a generalisation of a directed graph. It is also known as a computad. They were introduced as "polygraphs" by Albert Burroni¹ and as "computads" by Ross Street.²

In the same way that a directed multigraph can freely generate a category, an n-computad is the "most general" structure which can generate a free n-category.³

In the context of a graph, each dimension is represented as a set of ${\displaystyle k}$-cells. Vertices would make up the 0-cells, edges connecting vertices would be 1-cells, and then each dimension higher connects groups of the dimension beneath it. For 2-cells and up, which connect edges themselves, a source or target may consist of multiple edges of the dimension below it, as long as each set of elements are composites, i.e., are paths connected tip-to-tail.³

A globular set can be seen as a specific instance of a polygraph. In a polygraph, a source or target of a ${\displaystyle k}$-cell may consist of an entire path of elements of (${\displaystyle k}$-1)-cells, but a globular set restricts this to singular elements of (${\displaystyle k}$-1)-cells.³⁴