# Mapping spaces form projective spaces

We denote the $n$-th projective space of a topological monoid $G$ by $B_nG$ and the classifying space by $BG$. Let $G$ be a topological monoid of pointed homotopy type of a CW complex and $G'$ a grouplike topological monoid. We prove the weak equivalence between the pointed mapping space $\mathrm{Map}_0(B_nG,BG)$ and the space of all $A_n$-maps from $G$ to $G'$. This fact has several applications. As the first application, we show that the connecting map $G\rightarrow\mathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $\mathrm{Map}_0(B_nG,BG)\rightarrow\mathrm{Map}(B_nG,BG)\rightarrow BG$ is delooped. As other applications, we consider higher homotopy commutativity, $A_n$-types of gauge groups and $T_k^f$-spaces by Iwase-Mimura-Oda-Yoon.