Unfortunately, no; it’s not a short explanation and I never worked up the motivation to write it up in full. The closest thing I have is this MathOverflow answer. The starting point is the Dold-Kan correspondence but the conceptual meat of the story is about spectra, which loosely speaking are “abelian oo-groups.” To really buy the story you have to first buy a story about natively caring about homotopy types which I also haven’t written up, but which also shows up in various MO answers of mine (among other places, e.g. the nLab).
Anyway, this is why it took so long; I had to learn and buy a bunch of homotopy theory before I was really satisfied.
Unfortunately, no; it’s not a short explanation and I never worked up the motivation to write it up in full. The closest thing I have is this MathOverflow answer. The starting point is the Dold-Kan correspondence but the conceptual meat of the story is about spectra, which loosely speaking are “abelian oo-groups.” To really buy the story you have to first buy a story about natively caring about homotopy types which I also haven’t written up, but which also shows up in various MO answers of mine (among other places, e.g. the nLab).
Anyway, this is why it took so long; I had to learn and buy a bunch of homotopy theory before I was really satisfied.