) is a countable category
, called the degree map, which satisfy the following factorization property: if
μ , ν ∈
λ = μ ν
An immediate consequence of the factorization property is that morphisms in a
-graph can be factored in multiple ways: there are also unique
μ ν = λ =
A 1-graph is just the path category of a directed graph.
In this case the degree map takes a path to its length.
-graphs can be considered higher-dimensional analogs of directed graphs.
-colored directed graph together with additional information to record the factorization property.
-graph is referred to as its skeleton.
-graphs can have the same skeleton but different factorization rules.
Kumjian and Pask originally introduced
-graphs as a generalization of a construction of Robertson and Steger.
[1] By considering representations of
-graphs as bounded operators on Hilbert space, they have since become a tool for constructing interesting C*-algebras whose structure reflects the factorization rules.
Some compact quantum groups like
[2] There is also a close relationship between
-graphs and strict factorization systems in category theory.
-graphs is borrowed extensively from the corresponding notation for categories:
as being edges in a directed graph of a color indexed by
To be more precise, the skeleton of a
is a k-colored directed graph
, range and source maps inherited from
The extra information about factorization can be encoded in a complete and associative collection of commuting squares.
, there must exist unique
A different choice of commuting squares can yield a distinct
Just as a graph C*-algebra can be associated to a directed graph, a universal C*-algebra can be associated to a
is the universal C*-algebra generated by a Cuntz–Krieger