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

Edit. (16/9/19) As one of my supervisors recently pointed out to me, the property of being an `n-homotopy’ is not transitive, and so this example is really a non-example. I’ll keep the post here for reference purposes, but the only useful/true bits are those quoted from [RH19].

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