Fibrations between mapping spaces

We study the behaviour of fibre maps under exponentiation, i.e. given a fibration p:E→B we ask for which spaces X is the induced map between mapping spaces p⁎:EX→BX also a fibration. If X is a locally compact space, the positive answer follows easily by the exponential law so in this paper we consider more general spaces and show that the preservation of fibrations is related to the local homotopy properties of the space X. For example, if p is a Dold fibration and X admits a deformation retraction on a compactly generated space, then the induced map p⁎ is also a Dold fibration. Similar results hold for Hurewicz fibrations with unique path-lifting and for covering spaces, and can be furthermore extended to spaces that admit some numerable cover, whose elements preserve fibration property.