Let ▫$SF(n)$▫ be the usual monoid of orientation- and base point-preserving self-equivalences of the ▫$n$▫-sphere ▫${\mathbb{S}}^n$▫. If ▫$Y$▫ is a (right) ▫$SF(n)$▫-space, one can construct a ...classifying space ▫$B(Y, SF(n), \ast)=B_n$▫ for ▫${\mathbb{S}^n$▫-fibrations with ▫$Y$▫-structure, by making use of the two-sided bar construction. Let ▫$k: B_n \to BSF(n)▫$ be the forgetful map. A ▫$Y$▫-structure on a spherical fibration corresponds to a lifting of the classifying map into ▫$B_n$▫ Let ▫$K_i =K(\mathbb Z_, i)$▫ be the Eilenberg-Mac Lane space of type ▫$(\mathbb Z_2,i)$▫ In this paper we study families of structures on a given spherical fibration. In particular, we construct a universal family of ▫$Y$▫-structures, where ▫$Y=W_n$▫ is a space homotopy equivalent to ▫$\Pi_{i \ge 1} K_i$▫ Applying results due to Booth, Heath, Morgan and Piccinini, we prove that the universal family is a spherical fibration over the space map ▫${\B_n,B_n }\times B_n$▫. Furthermore, we point out the significance of this space for secondary characteristic classes. Finally, we calculate the cohomology of ▫$B_n$▫.
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