I have been allowed to attend some preparatory lectures for a seminar on the Goodwillie Calculus of Functors. I found in my notes from one of the lectures two statements which I would like to ask about.
The first one is probably straightforward and I'm guessing is related to Whitehead-type theorems. Still, I would still like a detailed explanation of what it means.
- Every homotopy type is a filtered colimit of finite CW complexes.
The second statement is a lot more problematic because I don't understand any of the context. Here it is:
- We want to look at (extraordinary) homology theories $h_\ast :\mathsf{Top}\rightarrow \mathsf{grAb}$ which commute with filtered colimits.
My question is why do we want to study homology theories which commute with filtered colimits? So that we may reduce to (finite) CW complexes? Is there anything else?
This statement is preceded in my notes by the following theorem of Whitehead:
Theorem. For any extraordinary homology theory which is finitary ($\overset?=$ determined by values on finite CW complexes) there exists a spectrum $E\in \mathsf{Sp}$ such that $h_\ast (X)=\pi_\ast (E\wedge X)$ where $\pi _\ast$ are stable homotopy groups and $\wedge $ is the smash product.
Now I don't yet know anything about either spectra no stable homotopy, so I can't make out much of this theorem myself.