Are generalized cohomology theories, spectra, and infinite loop spaces all the same thing up to homotopy?

Question

More specifically, John Baez mentions here that the following 3 things are equivalent (up to some technicalities).

1. the isomorphism classes of complex line bundles over $X$
2. the homotopy classes of maps from $X$ to $\mathbb{CP}^{\infty}$
3. the elements of the ordinary cohomology group $H^2(X)$

What are these technicalities, and what are some consequences of this equivalence?

Answer 1

The title and the body seem to be asking very different questions, so I'll answer them separately.

Title question: Not quite. While it is true that every generalized cohomology theory is represented by a spectrum and conversely that every spectrum represents a generalized cohomology theory, maps between spectra are richer than maps between generalized cohomology theories; see this MO question for a discussion. Infinite loop spaces are the same thing as connective spectra; this means spectra with no nonzero homotopy groups in negative degrees, and for example the K-theory spectra $KO$ and $KU$ are not connective.

Body question: The technicalities have to do with the failure of the first two things to be invariant under weak equivalences, or in other words with the possibility that $X$ has some point-set pathologies. I don't have details ready at hand, but I think that the usual definition of vector bundles doesn't behave the way you would expect on spaces that aren't paracompact (e.g. they don't give the "correct" version of K-theory). And homotopy classes of maps from $X$ to $\mathbb{CP}^{\infty}$ won't do the right thing if $X$ admits very few maps to spaces like $\mathbb{R}$. The correct version of #2 is to ask for maps from $X$ to $\mathbb{CP}^{\infty}$ in the weak homotopy category, or equivalently to ask for maps from a CW-replacement of $X$ to $\mathbb{CP}^{\infty}$. These technicalities all disappear if $X$ is, say, a manifold. (Maybe they all disappear if $X$ is a CW complex, even? It's the line bundle case I'm not sure about here.)

The consequences are vaguely that we have three different ways to think about the same thing, and having different ways to think about something is always useful.