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 well-pointed topological monoid of the homotopy type of a CW complex and $G$ a well-pointed 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 $Grightarrowmathrm{Map}_0(B_nG,BG)$ of the evaluation fiber sequence $mathrm{Map}_0(B_nG,BG)rightarrowmathrm{Map}(B_nG,BG)rightarrow BG$ is delooped. As other applications, we consider higher homotopy commutativity, $A_n$-types of gauge groups, $T_k^f$-spaces by Iwase--Mimura--Oda--Yoon and homotopy pullback of $A_n$-maps. In particular, we show that the $T_k^f$-space and the $C_k^f$-space are exactly the same concept and give some new examples of $T_k^f$-spaces.
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