A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into the subcategory of "local" ones that "see all $E$-homology isomorphisms as equivalences". I'm looking for a motivation for this construction that will make sense to a category theorist.

In particular, I'm *not* looking for a motivation by applications. Homological localization certainly has many applications, but what I'm hoping for is a "philosophical" argument for why one would *expect* it to be useful, before even having it in hand.

I'm also not looking for an argument that specializes to localization or completion with respect to primes (which can be described as localization with respect to $\mathbb{Z}_{(p)}$-homology and $\mathbb{Z}/p$-homology respectively) and then says something about the behavior on nilpotent spaces. I feel like I have a pretty good motivation for localization and completion of nilpotent spaces, by analogy with the importance of the analogous constructions in algebra and the fact that a space or spectrum can be considered a sort of "generalized algebraic object" put together out of its homotopy groups (e.g. this is well-described in *More concise algebraic topology*). But homological localization is only one way to generalize these constructions to non-nilpotent spaces; why is it a good one, and why should we think about doing it for other homology theories as well?

The best I've been able to come up with so far is the following fairly obvious remark: "Homology (and cohomology) are easier to compute than homotopy, so it's natural to restrict attention to those aspects of a space that can be detected by some homology theory." But this doesn't satisfy me, because there is another much more straightforward functor which "restricts attention to those aspects of a space that are detected by $E$-homology", namely "$E$-homology". (Or, if you prefer to land somewhere homotopical, the free $E$-module spectrum.) Why should we think of instead formally inverting the $E$-homology isomorphisms?

In particular, are there analogous constructions in ("cowardly old") algebra where we get something useful by formally inverting the maps inverted by some other functor? Note that for localization at primes, formally inverting the maps in question is *equivalent* to tensoring with the localized base ring, so it doesn't argue for why to use the former rather than the latter — while for completion at primes, the naive completion functors, at least, are *not* reflections into subcategories at all!

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