Question regarding notation in path homology paper

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In the paper, https://www.math.uni-bielefeld.de/~grigor/quivers.pdf
I have two questions regarding the definitions on Page $5$.

  1. What is $\sum c$ in Equation $3.1$ ? Is this just a new edge connecting the starting vertex of edge $a$ to end vertex of edge $b$ in path $[ab]$?
  2. What does $N$ stand for in Equations $3.2$, and later in $3.3$? Is this a chain-complex operator?

I understand that the section of the paper that I have pasted here is not self-contained. My apologies for that. enter image description here