Discussion of exponential objects in slice categories of compactly generated topological spaces (towards local cartesian closure, see at parameterized homotopy theory):
Peter I. Booth, The Exponential Law of Maps I, Proceedings of the London Mathematical Society s3-20 1 (1970) 179-192 doi:10.1112/plms/s3-20.1.179
Peter I. Booth, The exponential law of maps. II, Mathematische Zeitschrift 121 (1971) 311–319 doi:10.1007/BF01109977
Peter I. Booth, Ronnie Brown, Spaces of partial maps, fibred mapping spaces and the compact-open topology, General Topology and its Applications 8 2 (1978) 181-195 doi:10.1016/0016-660X(78)90049-1
Peter I. Booth, Ronnie Brown, On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps, General Topology and its Applications 8 2 (1978) 165-179 doi:10.1016/0016-660X(78)90048-X
On adjoint strings of functors:
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