Homological algebra and (semi)stable homotopy

✍ Scribed by Alex Heller

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 587 KB
• Volume: 115
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(96)00092-8

✦ Synopsis

If A is a complete and cocomplete abelian category, which we allow ourselves to conflate with the corresponding representable homotopy theory then the 2-functors HochA, taking the small category C to the homotopy category of chain complexes over AC and Hoch+A, with value the homotopy category of pasitiw chain compIexes, are both homotopy theories (in the sense of my monograph, A.M.S. Memoirs 383), the former being stable in the sense that the suspension hyperfunctor is an equivalence, while the latter is semistable. The hyperfunctors res: A 3 HochA and res+:A -+ Hoch+A which take an X in A" to a chain complex concentrated in degree 0 may be ch~ac~~zed as "resolvent". Then the two chain-complex theories associated to A are, respectively, the universal resolvent stabilization and semistabilization of A.

In other words, a "universal problem" of stabilization leads, for abelian categories, to the construction of chain complexes, just as a corresponding problem for topological spaces leads to the cons~ction of spectra.