Timeline for Relation between different definitions of homotopy
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 11, 2016 at 17:54 | comment | added | Tyrone | Quillen was actually heavily influenced by Kan's work if I recall and cites it as motivation. I just thought I would use it as an opportunity to advertise an open area of modern mathematics and some beautiful proofs if the original poster would like to delve a little deeper. And yes, I just noticed the initial of the second author of "Abstract Homotopy and Simple Homotopy Theory". | |
| Nov 11, 2016 at 16:47 | comment | added | Tim Porter | Actually the notion of cylinder functor predates Quillen by a few years and is in work of Dan Kan. (Quillen abstracted further of course.) This was then explored by Heiner Kamps before being picked up by Hans Baues. | |
| Nov 11, 2016 at 16:45 | comment | added | Tim Porter | Tyrone: the relationship is much older than Quillen and is very clearly explored in several of the really classical texts such as Mac Lane's Homology or Spanier's Algebraic Topology, without the clutter that model category theory surrounds it with. (The clutter is very useful for pushing things further but is not needed for the explicit question of the link between the two notions of homotopy.) My advice to the original questioner would be to keep the source simple (and possibly fairly classical) to start with. | |
| Nov 11, 2016 at 15:36 | comment | added | Tyrone | A good place to start would be trace the relationship back to Quillen's abstraction of homotopy in his book 'Homotopical Algebra'. There are many more modern treatments of his notion of a Model Category. See for instance Baues's "Abstract Homotopy" or Riehl's "Categorical Homotopy Theory". The proof that the category or R-modules (every small abelian category embeds in the category of abelian groups by Freyd-Heron-Lubkin) can be found in Quillen or in "Homotopy Theories and Model Categories" by Dwyer & Spalinski. | |
| Nov 10, 2016 at 17:43 | history | answered | Tim Porter | CC BY-SA 3.0 |