Exponentiable Streams and Prestreams

Inspired by a construction of Escardó, Lawson, and Simpson, we give a general construction of C-generated objects in a topological construct. When C consists of exponentiable objects, the resulting category is Cartesian-closed. This generalizes the familiar construction of compactly-generated spaces, and we apply this to Krishnan’s categories of streams and prestreams, as well as to Haucourt streams. For that, we need to identify the exponentiable objects in these categories: for prestreams, we show that these are the preordered core-compact topological spaces, and for streams, these are the core-compact streams.