TITLE: Loop space generalization
QUESTION [8 upvotes]: Let $X$ be a based connected space. The space of based continuous morphisms $Top_{\ast}(S^1,X)$ 
is the space of loops $\Omega X$. Since $S^1$ is homotopy equivalent to the Eilenberg-Mac Lane Space $K(\mathbf{Z},1)$ I was wandering if there is any "geometric interpretation" of the space of based continuous morphisms $Top_{\ast}(\mathbf{C}P^{\infty},X)$.  $\mathbf{C}P^{\infty}$ can be identified with the Eilenberg-Mac Lane Space $K(\mathbf{Z},2)$
More generally, what would be the "geometric interpretation" of the space of based continuous morphisms $Top_{\ast}(K(\mathbf{Z},n),X)$.

REPLY [2 votes]: [Edited after the comment below]
Well, I don't know exactly what you mean by "geometric interpretation", but 
for topological group $G$ the unpointed mapping spaces $Map(BG,X)$ are examples of homotopy fixed points. and the pointed mapping spaces $Map_*(BG,X)$ fits in the fibration
$$Map_*(BG,X) \rightarrow Map(BG,X) \rightarrow X$$ where the last map is
the evaluation map.
Now, $$Map(BG,X)\cong Map^G(EG,X)$$ where we provide $X$ with trivial action,
and this latter is nothing but the homotopy fixed points of $X$ with trivial $G$-action.
If $G$-action on $X$ is trivial, $X$ is just the fixed point set, so one can consider $Map_*(BG,X)$ as the space that measures the difference between the genuine fixed point set and the homotopy fixed point set.
As one can construct a model for $K(A,n-1)$ that has a structure of a topological group, you can take $G=K(A,n-1)$ in the above.