As I mentioned in a previous post, I recently saw a talk by Rachel
Hardeman on the A-homotopy theory of graphs, and it really intrigued
me. In particular, it seemed to me that there was some nice structure
that could be abstractified: that of a “graded homotopy structure”, as
I’ve been calling it in my head. Rather than trying to type out
everything in `#math.CT:matrix.org`

,
I’ve decided to post it here, in the hope that I might be able to get
some answers.