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Craig Westerland
Craig Westerland
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A compact Lie group $G$ is a loop space ($G \simeq \Omega BG$), and its integral homology is a Poincaré duality algebra (and in particular finite rank over $\mathbb{Z}$). One may relax these assumptions in the $E$-local setting. In particular, if we take $E =H\mathbb{F}_p$ to be mod $p$ homology, we can declare a loop space $G$ to be a $p$-compact group if it is $p$-local with $H_*(G, \mathbb{F}_p)$ finite rank over $\mathbb{F}_p$. Again, the $p$-localization of a compact Lie group is an example, but there are many exotic examples which arise. Notably, whenever $n$ is a divisor of $p-1$, the $p$-completion of $S^{2n-1}$ is a $p$-compact group (the Sullivan spheres), despite the fact that only $S^0$, $S^1$, and $S^3$ (and, if you're charitable, $S^7$) are groups or, for that matter, H-spaces.

A compact Lie group $G$ is a loop space ($G \simeq \Omega BG$), and its integral homology is a Poincaré duality algebra (and in particular finite rank over $\mathbb{Z}$). One may relax these assumptions in the $E$-local setting. In particular, if we take $E =H\mathbb{F}_p$ to be mod $p$ homology, we can declare a space $G$ to be a $p$-compact group if it is $p$-local with $H_*(G, \mathbb{F}_p)$ finite rank over $\mathbb{F}_p$. Again, the $p$-localization of a compact Lie group is an example, but there are many exotic examples which arise. Notably, whenever $n$ is a divisor of $p-1$, the $p$-completion of $S^{2n-1}$ is a $p$-compact group (the Sullivan spheres), despite the fact that only $S^0$, $S^1$, and $S^3$ (and, if you're charitable, $S^7$) are groups or, for that matter, H-spaces.

A compact Lie group $G$ is a loop space ($G \simeq \Omega BG$), and its integral homology is a Poincaré duality algebra (and in particular finite rank over $\mathbb{Z}$). One may relax these assumptions in the $E$-local setting. In particular, if we take $E =H\mathbb{F}_p$ to be mod $p$ homology, we can declare a loop space $G$ to be a $p$-compact group if it is $p$-local with $H_*(G, \mathbb{F}_p)$ finite rank over $\mathbb{F}_p$. Again, the $p$-localization of a compact Lie group is an example, but there are many exotic examples which arise. Notably, whenever $n$ is a divisor of $p-1$, the $p$-completion of $S^{2n-1}$ is a $p$-compact group (the Sullivan spheres), despite the fact that only $S^0$, $S^1$, and $S^3$ (and, if you're charitable, $S^7$) are groups or, for that matter, H-spaces.

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Craig Westerland
Craig Westerland
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This is perhaps orthogonal to your desire for an a priori reason to motivate localization. Indeed it's intrinsically a posteriori. I still think it's a good reason to care about localization that is deeper than the fact that the assignment $X \mapsto E_*(X)$ is a somewhat computable invariant.

If one has a category of things (like spaces) and some sort of notion of a structured gadget in that category, as long as your localization functor is reasonable (e.g., monoidal, if your gadget is defined operadically), then the image of any gadget in the localized category will remain the same sort of structured gadget. However, if one relaxes the requirements of your structured gadget so that they are only visible to the eyes of the localized category, one can in principle come up with a notion which admits exotic examples which are not the localization of a standard object.

This is pretty airy; let me explain via two examples. A spectrum $X$ is invertible if there is another spectrum $Y$ with $X \wedge Y \simeq S^0$. The collection of equivalence classes of invertible spectra forms the Picard group of the stable homotopy category, $Pic(S^0)$. As Hopkins-Mahowald-Sadofsky show, it's isomorphic to the integers, where $n$ corresponds to $S^n$ (with smash inverse $S^{-n}$).

This doesn't appear to change in the localization of the stable homotopy category with respect to any cohomology theory $E$, since, as you note, the local category is a subcategory of the original category. However, there is a problem, in that the smash product of two $E$-local spectra need not be $E$-local. So one defines a new monoidal structure on the $E$-local category by localizing after smashing:

$$X \otimes Y := L_E (X \wedge Y)$$

Now, one can ask: what is $Pic(L_E S^0)$, the set of isomorphism classes of invertible objects with respect to this new monoidal structure on the category of $E$-local spectra? Of course, the image of any spectra which are invertible with respect to $\wedge$ (i.e., spheres) are still invertible, so I get a map $Pic(S^0) \to Pic(L_E S^0)$. But generally the target group is much larger and more interesting. For instance, when $E$ is mod $2$ K-theory, many exotic examples can be constructed as Thom spectra over $\mathbb{R} P^\infty$. In general when $E$ is a Morava K-theory, computation of $Pic(L_{K(n)} S^0)$ is difficult, and a very active subject of investigation.

This highlights the fact: the notion of invertibility can be (in fact must be) modified in the $E$-local category, and yields a new and richer notion than the original. The same holds in the second example, $p$-compact groups.

A compact Lie group $G$ is a loop space ($G \simeq \Omega BG$), and its integral homology is a Poincaré duality algebra (and in particular finite rank over $\mathbb{Z}$). One may relax these assumptions in the $E$-local setting. In particular, if we take $E =H\mathbb{F}_p$ to be mod $p$ homology, we can declare a space $G$ to be a $p$-compact group if it is $p$-local with $H_*(G, \mathbb{F}_p)$ finite rank over $\mathbb{F}_p$. Again, the $p$-localization of a compact Lie group is an example, but there are many exotic examples which arise. Notably, whenever $n$ is a divisor of $p-1$, the $p$-completion of $S^{2n-1}$ is a $p$-compact group (the Sullivan spheres), despite the fact that only $S^0$, $S^1$, and $S^3$ (and, if you're charitable, $S^7$) are groups or, for that matter, H-spaces.

Lastly, I want to point out that these relaxed definitions are not "techniques in search of a problem." In fact, they create a home for existing constructions. For instance, the usual classification of compact Lie groups in terms of the action of their Weyl group on the lattice of the characters of their maximal torus extends to the $p$-compact case. In particular, when $p$ is odd there is a bijection between $p$-compact groups and $p$-adic reflection groups. So: if you were wondering what is the geometry associated to $p$-adic reflection groups, the answer is: $p$-compact groups.

Similarly, in the Picard group setting, there is a spectral sequence that computes $Pic(L_{K(n)} S^0)$ using classical computations of group cohomology of automorphism groups of Lubin-Tate formal groups, acting on the units in the ring of functions on the Lubin-Tate moduli space. So if you're interested in invertible sheaves on that moduli space, you're also interested in exotic elements of $Pic(L_{K(n)} S^0)$.