In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and particularly in
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. 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
Albert may refer to:
Companies
* Albert Computers, Inc., a computer manufacturer in the 1980s
* Albert Czech Republic, a supermarket chain in the Czech Republic
* Albert Heijn, a supermarket chain in the Netherlands
* Albert Market, a street mar ...
and as "computads" by
Ross Street.
[R. Street. Limits indexed by category-valued 2-functors. Journal of Pure and Applied Algebra, 8(2):149--181, 1976.]
In the same way that a
directed multigraph
In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called ''parallel edges''), that is, edges that have the same end nodes. Thus two vertices may be connected by more ...
can
freely
Freely is a British free-to-air IPTV service launched in 2024 by Everyone TV, a joint venture between the country's public broadcasters BBC, ITV, Channel 4 and 5. The service offers the ability to watch live television and on demand media from t ...
generate a
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
, an ''n''-computad is the "most general" structure which can generate a free
n-category.
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.
A
globular set
In category theory, a branch of mathematics, a globular set is a higher-dimensional generalization of a directed graph. Precisely, it is a sequence of sets X_0, X_1, X_2, \dots equipped with pairs of functions s_n, t_n: X_n \to X_ such that
* s_n ...
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.
References
Category theory
Directed graphs
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