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.
> for chain complexes it took something like 4 years, and the satisfying motivation is much more complicated to explain than the definition.
You can’t just say that and not tell us what the motivation is! Is it on your blog somewhere?
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.