Polygraph (mathematics)

This article is about a notion of higher-dimensional directed graph. For the forensic instrument, see Polygraph. For the dual pen device that produces a simultaneous copy of an original while it is written in cursive writing, see Polygraph (duplicating device). For an author that can write on a variety of different subjects, see Polygraph (author).

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 Burroni[1] 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 -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 -cell may consist of an entire path of elements of (-1)-cells, but a globular set restricts this to singular elements of (-1)-cells.[3][4]

References

  1. ^ A. Burroni. Higher-dimensional word problems with applications to equational logic. TCS, 115(1):43--62, 1993.
  2. ^ R. Street. Limits indexed by category-valued 2-functors. Journal of Pure and Applied Algebra, 8(2):149--181, 1976.
  3. ^ a b c computad at the nLab
  4. ^ globular+set at the nLab


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