Higher rank graphs, or $k$-graphs are $k$-dimensional analogues of directed graphs.

First introduced to provide combinatorial models for the higher-rank Cuntz-Krieger_algebras.

Given a nonnegative integer \(k\), a $K$-graph is a nonempty countable small category \(\Lambda\) equipped with a functor \(d:\Lambda \to \mathbb{N}^k\) satisfying the factorization property.

• Homology of $k$-graphs introduced in 2012 by Kumjian, Pask, and Sims in “Homology for higher-rank graphs and twisted \(C^\ast\text{-algebras.}\)”

$k$-graphs are defined using the language of cateogry theory. Here, a category \(C\) consists of two sets \(C^0\) and \(C^\ast\), two functors \(r,s: C^\ast \to C^0\), a partially defined product \((f,g) \mapsto fg\) from \[ \{ (f,g) \in C^\ast \times C^\ast : s(f) = r(g) \} \] to \(C^\ast\), and distinguised elements \(\{ \iota_{v \in C^\ast : v \in C^0} \}\) satisfying

1. \(r(fg) = r(f)\) and \(s(fg) = s(g)\);
2. \((fg)h = f(gh)\) when \(s(f) = r(g)\) and \(s(g) = r(h)\);
3. \(r(\iota_{v}) = v = s(\iota_{v})\) and $ι_vf = f,$ \(g\iota_{v = g}\) when \(r(f) = v\) and \(s(g) = v\).
4. \(C^0\) are the objects of the category
5. elements of \(C^*\) are morphisms
6. \(s(f)\) is the domain of \(f\) for \(f \in C^\ast\)
7. \(r(f)\) is the codomain of \(f\)
8. \((f,g) \mapsto fg\) is the composition
9. \(\iota_{v}\) is the identity morphism of the object onto \(v\)

If \(C\) and \(D\) are categories, a functor \(F:C \to D\) is a pair of maps \(F^0:C^0 \to D^0\) and \(F^\ast:C^\ast \to D^\ast\) which respect the domain and codomain maps and composition, and which satsify \(F^\ast (\iota_{v}) = \iota\)

If \(k = 1\) then the degree map is \(d: E^\ast \to \mathbb{N}\) hence it is the normal distance metric?

1. Notes on Homology for higher rank graphs and twisted C star algebras
2. Notes paper moves on k graphs preserving Morita equivalence
3. Research idea k-graph cohomology a groupoid invariant
4. RP k-graphs definitions and notation