What is classified by generalised Eilenberg MacLane spaces?

Question

Given an abelian group $A$, the Eilenberg MacLane spaces $K(A,n)$ represent the the nth cohomology group in $A$.

In a similar vein, given an arbitrary group $G$ and a space $X$, maps to the classifying space $X\to BG$ classify principal $G$-bundles on $X$.

In the literature I have encountered spaces $K(G,V,n)$, where $G$ is a group and $V$ is a finite dimensional $G$-representation, called generalised (or sometimes twisted) Eilenberg MacLane spaces. These spaces are determined up to homotopy by the property that

$\pi_{i}(K(G,V,n)) = \begin{cases} G,\ \ i=1,\\ V,\ \ i=n,\\ 0,\ \ \text{else}. \end{cases} $

My first question is, what do generalised Eilenberg MacLane spaces classify? Am I correct in thinking that they represent cohomology in the local system determined by $G$ and $V$?

My second question is, what does it mean to localise with respect to generalised Eilenberg MacLane spaces?

By this I am thinking of a Bousfield localisation on the model category of spaces in which a local equivalence is declared to be a map $f:X\to Y$ which induces a weak equivalence $$ f^{*}:\text{Map}(Y, K(G,V,n))\to \text{Map}(X,K(G,V,n)), $$ and a space $Z$ is local if any local equivalence $f:X\to Y$ induces a weak equivalence $$ f^{*}:\text{Map}(Y, Z)\to \text{Map}(X,Z). $$

Specifically, I would like to know what information is isolated by performing these localisations for a given group $G$ and representation $V$.

Answer 1

To answer your first question, take a look at the reference

Gitler, Samuel, Cohomology operations with local coefficients, Am. J. Math. 85, 156-188 (1963). ZBL0131.38006.

In particular, Theorem 7.18 in Chapter III does what you want.

To paraphrase, let $\mathcal{V}$ be the local system of groups on your Eilenberg--Mac Lane space $K(G,V,n)$ determined by the representation $G\to \operatorname{Aut}(V)$. Given pointed connected CW-complexes $X$ and $Y$ and a homomorphism $\alpha:\pi_1(X)\to \pi_1(Y)$ between their fundamental groups, let $[X,Y]_\alpha$ denote the set of pointed homotopy classes of pointed maps $f:X\to Y$ such that $f_*=\alpha:\pi_1(X)\to \pi_1(Y)$. Then there is a (natural in an appropriate sense) isomorphism $$ [X,K(G,V,n)]_\alpha \cong H^n(X;\alpha^*\mathcal{V}). $$ The isomorphism is given as in the untwisted case by pulling back an appropriately defined fundamental class $\iota\in H^n(K(G,V,n);\mathcal{V})$.