8.9 A general statement of the encode-decode method
We have used the encode-decode method to characterize the path spaces of various types, including coproducts (\crefthm:path-coprod), natural numbers (\crefthm:path-nat), truncations (\crefthm:path-truncation), the circle (\crefcor:omega-s1), suspensions (\autorefthm:freudenthal), and pushouts (\crefthm:van-Kampen). Variants of this technique are used in the proofs of many of the other theorems mentioned in the introduction to this chapter, such as a direct proof of , the Blakers–Massey theorem, and the construction of Eilenberg–Mac Lane spaces. While it is tempting to try to abstract the method into a lemma, this is difficult because slightly different variants are needed for different problems. For example, different variations on the same method can be used to characterize a loop space (as in \crefthm:path-coprod,cor:omega-s1) or a whole path space (as in \crefthm:path-nat), to give a complete characterization of a loop space (e.g. ) or only to characterize some truncation of it (e.g. van Kampen), and to calculate homotopy groups or to prove that a map is -connected (e.g. Freudenthal and Blakers–Massey).
However, we can state lemmas for specific variants of the method. The proofs of these lemmas are almost trivial; the main point is to clarify the method by stating them in generality. The simplest case is using an encode-decode method to characterize the loop space of a type, as in \crefthm:path-coprod and \crefcor:omega-s1.
Lemma 8.9.1 (Encode-decode for Loop Spaces).
If a fiberwise equivalence between and is desired, it suffices to strengthen condition (iii) to
However, to calculate a loop space (e.g. ), this stronger assumption is not necessary.
Another variation, which comes up often when calculating homotopy groups, characterizes the truncation of a loop space:
Lemma 8.9.2 (Encode-decode for Truncations of Loop Spaces).
|Title||8.9 A general statement of the encode-decode method|