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In classical topology, it is natural to think in terms of function spaces yet the category of topological spaces is not cartesian closed. In 1967, Norman Steered proposed in effect to address that by working within coreflective subcategories which are cartesian closed. The essential idea is in Ronnie Brown's DPhil Thesis and the categorical underpinnings were folklore by the mid 1970s. Since then the more subtle framework of locales has come to prominence. In the first part of the talk I shall discuss the relatively neglected question of what happens for function spaces in that setting. There are many avenues to follow. In the second part of the talk I shall try to explain why what we are able to do is quite unsatisfactory from the point of view of homotopy type theory. That will lead to questions in topology or locale theory some of which I think extremely challenging.
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