§ Stable homotopy theory
We like stable homotopy groups
because of the Freudenthal suspension theorem
which tells us that homotopy groups stabilise after many suspensions.
The basic idea seems to be something like a tensor-hom adjunction. We have
the loop spaces which are like and the suspension which
is like . The theory begins by considering the tensor-hom-adjunction
between these objects as fundamental. So curry stuff around to write things as
(S^1, A) -> B and
A -> (S^1 -> B), which is
Suspension(A) -> B and
A -> Loop(B).
This gives us the adjunction between suspension and looping.
- We then try to ask: how can one invert the suspension formally? One tries to do some sort of formal nonsense, by declaring that maps between and , but this doesn't work due to some sort of grading issue.
- Instead, one repaces a single object with a family of objects called as the spectrum. Then, we can invert the suspension by trying to invert maps between objects of the same index.