Free category

In mathematics, the free category or path category generated by a directed graph or quiver is the category that results from freely concatenating arrows together, whenever the target of one arrow is the source of the next.

More precisely, the objects of the category are the vertices of the quiver, and the morphisms are paths between objects. Here, a path is defined as a finite sequence

where is a vertex of the quiver, is an edge of the quiver, and n ranges over the non-negative integers. For every vertex of the quiver, there is an "empty path" which constitutes the identity morphisms of the category.

The composition operation is concatenation of paths. Given paths

their composition is

.[1][2]

Note that the result of the composition starts with the right operand of the composition, and ends with its left operand.

  1. ^ Awodey, Steve (2010). Category theory (2nd ed.). Oxford: Oxford University Press. pp. 20–24. ISBN 978-0199237180. OCLC 740446073.
  2. ^ Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 49–51. ISBN 1441931236. OCLC 851741862.