Timeline for Equivalence of homotopy categories and model structure theory

4 events

when toggle format	what		by	license	comment
Mar 26, 2014 at 15:38	comment	added	Elden Elmanto		@KotelKanim - perhaps a simpler example of the application of model categories would be what Quillen first used it for. Namely he answered: what is the cohomology of commutative rings? Concretely, he wanted to extend an exact sequence of abelian groups involving 3 rings (highly non-abelian objects) into a long exact sequence reminiscent of the usual one you get from derived functors cohomology in the sense of abelian categories. In this sense, model categories is a way to make sense of "abelian" ideas (e.g. derived functors, cohomology) in non-abelian settings!
Feb 9, 2014 at 19:24	comment	added	Peter May		Actually, I first became truly convinced of the force of model category theory while writing EKMM. I had long wanted to prove that the periodic K-theory spectra are E_{\infty} ring spectra, knowing by infinite loop space theory that their connective covers are such. Thinking model theoretically, this became so easy it was like a joke (in fact, I burst out laughing in the shower when I noticed it).
Feb 9, 2014 at 16:00	comment	added	KotelKanim		I understand that it is not the point. Thank you for the explanation though.
Feb 9, 2014 at 14:30	history	answered	Peter May	CC BY-SA 3.0