The significance of filtered colimits in homotopy theory

Question

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.

1. 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:

2. 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.

Answer 1

I think it's easier to understand if you look at it the other way around. Singular homology preserves filtered colimits (exercise: prove it), but it does not preserve other types of colimits in general (exercise: find a counterexample, a very simply one in fact; if you're stuck, have a look here). So then the theorem shows how useful it is:

Every homotopy type is a filtered colimit of finite CW complexes.

By this theorem, it means that when you want to understand homology, it's sufficient to know what it does with finite CW complexes, and then, because you know it preserves filtered colimits, you will also know what it does to every homotopy type.

And now it also makes sense why we restrict our attention to generalized homology theories that only preserve filtered colimits (but not necessarily general ones): otherwise, singular homology wouldn't even be an example of a generalized homology theory, so it's not quite clear what we would be generalizing here...

And now the adjective "finitary" makes sense: a homology theory is said to be finitary if it preserves filtered colimits, and then by the first theorem it is indeed determined by its value on finite CW complexes.

Answer 2

For your first claim: every weak homotopy type can be represented by some CW complex $X$. This is one of Whitehead's most famous theorems. But $X$ is given as the union of its finite-dimensional skeleta $X^n$, and such a nested union is a particular example of a filtered colimit.

The reason to restrict to finitary homology theories, equivalently, those which commute with filtered colimits, is to get the best possible representability theorem. In cohomology theory Brown's original representability theorem says that every extraordinary cohomology theory is representable by a spectrum.

From a cohomology theory represented by a spectrum $X$ we can get a homology theory defined on finite-dimensional CW complexes by Spanier-Whitehead duality, and so a finitary homology theory by the claim from the first paragraph, and this process is reversible (this is morally the reason for the theorem of Whitehead you cite.) But there is no Spanier-Whitehead duality for arbitrary spaces, so there's no way to use Brown representability to get a spectrum representing a non-finitary homology theory. And indeed not all extraordinary homology theories are representable!

Answer 3

I think the most basic things you can say here have nothing much to do with homotopy theory in particular. It's a general feature of many familiar categories that every object is a filtered colimit of "small" objects which are easier to understand: for example, every group is a filtered colimit of finitely generated groups, and every commutative ring is a filtered colimit of finitely generated (in particular Noetherian!) rings. If you look at a functor on such a category which commutes with filtered colimits, then to understand what it does in general you only need to understand what it does to "small" objects, and this is a useful thing to be able to do.

The condition that a functor commutes with filtered colimits is particularly interesting when applied to representable functors $\text{Hom}(X, -)$, where it usually corresponds to some interesting "finiteness" or "compactness" condition on $X$. See compact object for some details and examples.