Mapping spaces from 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 wellpointed topological monoid having the homotopy type of a CW complex and G' a well-pointed grouplike topological monoid. We prove that there is a natural weak equivalence between the pointed mapping space Map₀(B_nG, BG') and the space A_n(G, G') of all A_n-maps from G to G'. Moreover, if we suppose G = G', then an appropriate union of path-components of Map₀(B_nG, BG) is delooped. This fact has several applications. As the first application, we show that the evaluation fiber sequence Map₀(B_nG, BG) → Map(B_nG, BG) → BG extends to the right. As other applications, we investigate higher homotopy commutativity, A_n- types of gauge groups, T_k ^f-spaces and homotopy pullback of A_n-maps. The concepts of T_k ^f -space and Cf k -space were introduced by Iwase-Mimura-Oda-Yoon, which is a generalization of T_k-spaces by Aguadé. 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.