Stable Homotopy Seminar, 10: Fiber and Cofiber Sequences of Spectra
Channel: Paul VanKoughnett
Date: 23/04/2021

(Note: some internet issues forced me to do the second half of this on the blackboard, so the quality's a bit worse.)


We generalize the notion of fiber & cofiber sequence to arbitrary pointed model categories, using the notion of homotopy limit and colimit defined last time. Cofiber sequences are naturally related to suspensions (and fiber sequences are naturally related to loopspaces): any cofiber sequence X → Y → Z can be continued, X → Y → Z → ΣX → ΣY → .... These induce long exact sequences in sets of homotopy classes, again generalizing something we've already learned in spaces. Additionally, for any X, ΣΧ is a cogroup object on the homotopy category that naturally coacts on any Z in a cofiber sequence X → Y → Z, and gives [ΣX, W] a group structure for any W. In particular, we can immediately show that sets of homotopy classes of maps in a stable model category like Spectra are naturally abelian groups.


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This is part of the stable homotopy learning seminar organized by Paul VanKoughnett. For notes and more, visit https://www.math.purdue.edu/~pvankoug/spectra/index.html