It seems to me that although homological localization (i.e. formally inverting $E$-homology equivalences for some $E$) is a reasonable thing to do to a spectrum, it's a pretty brutal thing to do to a space, particularly if that space is not simply connected. To put a finer point on it: shouldn't there be a better notion of localization for non-simply-connected spaces? Shouldn't it at least take into account local coefficients, for example?

As soon as one says "local coefficients" and has general cohomology theories in mind, one should start thinking in terms of parameterized homotopy theory. Recall that for $X$ a space, the $\infty$-category of *$X$-parameterized spectra* is just the functor category $Fun(X,Sp)$ where $Sp$ is spectra. For $E \in Sp$, let $Sp_E$ be the category of $E$-local spectra, and define the category of *$E$-local $X$-parameterized spectra* to be $Fun(X,Sp_E)$.

**Definition:** A map $f: Y \to X$ of spaces is a *strong $E$-local equivalence* if the induced functor $f_\ast: Fun(Y,Sp_E) \to Fun(X,Sp_E)$, between categories of parameterized $E$-local spectra, is an equivalence.

(Note that $f_\ast$ has a left adjoint $f^\ast$ which has a further left adjoint $f_!$; it's equivalent to ask for any one of these to be an equivalence).

I'm curious to what extent this notion agrees with just Bousfield localizing at $E$-homology equivalence. So, some test cases:

**Questions:**

Let $E = H \mathbb Q$. Is a rational equivalence between simply-connected spaces a strong $H\mathbb Q$-equivalence?

Let $E = H\mathbb Z_{(p)}$. Is a $p$-local equivalence between (sufficiently connected?) spaces a strong $H\mathbb Z_{(p)}$-equivalence?

Let $E = M(p)$. Is a $p$-complete equivalence between sufficiently connected spaces a strong $M(p)$-equivalence?

Let $E = K(n)$. Is a $K(n)_\ast$-equivalence between $(n+1)$-connected spaces a strong $K(n)$-equivalence?

**Guiding analogy:** My perspective is that a space is like a scheme which is etale over a point. A parameterized spectrum is like a quasicoherent sheaf over this scheme. So we're asking for an equivalence of localized categories of quasicoherent sheaves. From this perspective, an $E_\ast$-homology equivalence $f: Y \to X$ looks like pretty a crude thing -- it just means that $t_! f_! \mathbb S_Y \to t_! \mathbb S_X$ is an $E$-local equivalence, where $t: X \to pt$ is the unique map. In the algebraic geometry analogy, this is like saying that the map induces an $E$-local isomorphism of the cohomology-with-compact-support of the structure sheaves, which seems like a pretty weak condition.

So perhaps strong $E$-equivalences are truly a stronger notion than $E$-equivalences...