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2 Notation
They start with the standard category theory definition of $k$-graph.
How to model $k$-graphs using $k$-colored graphs
- Let \(G = (G^0, G^1, r, s)\) denote a directed graph with
- \(G^0\) set of vertices
- \(G^1\) set of edges
- \(r,s:G^1 \to G^0\) are the range and source map, respectively.
- Let \(n \geq 2\) then \(G^n\) is the set of paths of length \(n\)
- If \(\delta \in G^n\), write \(|\delta| := n\)
- Color the graph by assigning to each edge one of the standard basis vectors \(e_{i}\) of \(\mathbb{N}^k\) and leg \(G^{e_{i}}\) be the set of edges assigned to \(e_{i}\), so that $G1 = ∪i=1k Gei.$
- The {\it path category}, \(G^* = \cup_{n \in \mathbb{N}}G^n\), may
now be equippped with a degree functor \(d:G^* \to \mathbb{N}^k\)
- defined on vertices by \(d(v) = 0~\forall v \in G^0\), and on the edges by \(d(f) = e_{i}\) if \(f\) was assigned the basis vector \(e_{i}\).
Definitions
- For a finite path \(\lambda\) in an edge-colored directed graph \(G\), let \(\lambda_{i}\) denote the ith edge of \(\lambda\) (counting from the source of \(\lambda\)). The color order of \(\lambda\) is the $|λ|$-tuple $(d(λ1), d(λ2), \ldots, d(λ|λ|)).$