# What is classified by generalised Eilenberg MacLane spaces?

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 $$V$$.

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 $$\iota\in H^n(K(G,V,n);\mathcal{V})$$.