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I say this primarily as a warning to those (students in particular) who may think that a little bit of tweeking of the category or the universal property might yield better results. There are a lot of broken ideas along the way, some of which you will find surveyed in Isbell's paper. Breaking a correct idea like the universal property (by restricting its test object to a single space) is not going to help.

On the other hand, there are ways of enlarging the traditional category to make it cartesian closed. Equilogical Spaces and Filter Spaces by Pino Rosolini in Rendiconti del Circolo Matematico di Palermo 64 (2000) 157--175 gives an excellent survey of them, explaining how they are reflctive subcategories of presheaves on the traditional category. In particular, Scott had introduced equilogical spaces, defined as topological spaces equipped with formal equivalence relations; the theory is set out in full in Equilogical Spaces by Andrej Bauer, Lars Birkedal and Dana Scott.

My personal view is that these extensions are not topology but set theory with topological decoration. In this context, by "set theory" I mean, not the study of $\in$, but that of discrete spaces, whereas I believe (following Marshall Stone) that mathematical structures should be intrinsically topological. I have a reseach programme called Equideductive Topology that tries to look at such extensions without importing set theory.

I say this primarily as a warning to those (students in particular) who may think that a little bit of tweaking of the category or the universal property might yield better results. There are a lot of broken ideas along the way, some of which you will find surveyed in Isbell's paper. Breaking a correct idea like the universal property (by restricting its test object to a single space) is not going to help.

On the other hand, there are ways of enlarging the traditional category to make it cartesian closed. Equilogical Spaces and Filter Spaces by Pino Rosolini in Rendiconti del Circolo Matematico di Palermo 64 (2000) 157--175 gives an excellent survey of them, explaining how they are reflective subcategories of presheaves on the traditional category. In particular, Scott had introduced equilogical spaces, defined as topological spaces equipped with formal equivalence relations; the theory is set out in full in Equilogical Spaces by Andrej Bauer, Lars Birkedal and Dana Scott.

My personal view is that these extensions are not topology but set theory with topological decoration. In this context, by "set theory" I mean, not the study of $\in$, but that of discrete spaces, whereas I believe (following Marshall Stone) that mathematical structures should be intrinsically topological. I have a research programme called Equideductive Topology that tries to look at such extensions without importing set theory.

So this is the reason why local compactness of $X$ is necessary.

So I would be grateful if the homotopy theorists here would explain, without rehearsing the category theory, what the motivations were and are in their own subject for asking for "convenient" or cartesian closed categories.

[PS: Peter Johnstone has a neat argument involving preservation of injectivity, in the final chapter of his book Stone Spaces.]

So this is the reason why local compactness of $X$ is necessary.

So I would be grateful if the homotopy theorists here would explain, without rehearsing the category theory, what the motivations were and are in their own subject for asking for "convenient" or cartesian closed categories.

PS Thanks to Tyler Lawson for the comment below answering this question. Is there a slightly more detailed explanation of these methods, say of the length of a MO answer, or a survey paper?

In the context of an application of this kind, the next question is whether the cartesian closed categories that have been used (and mentioned above) are the most appropriate for the job. On the face of it, you're happy with "any old" CCC. But, when you look at the extra objects of this category, do the extensions of topological notions to them behave in the way that you would like? That is, according to whatever other intuitions of topology you have, such as developing results along the lines that Tyler mentions?

Many early applications of category theory imported the benefits of "set theory" by working in the Yoneda embedding (presheaves) or a smaller category of sheaves. Rosolini showed (in the paper cited above) how the CCC extensions of categories of topological spaces are subcategories of the Yoneda embedding. There is a close technical analogy in that both kinds of subcategory are reflective, but for sheaves the reflector (left adjoint to inclusion) preserves all finite limits, whereas in these CCCs it preserves products but not all equalisers or pullbacks.

My personal view is that these extensions are not topology but set theory with topological decoration. In this context, by "set theory" I mean, not the study of $\in$, but that of discrete spaces, whereas I believe (following Marshall Stone) that mathematical structures should be intrinsically topological. I have a reseach programme called Equideductive Topology that tries to look at such extensions without importing set theory.

Briefly, this works very nicely when $Y$ is locally compact, but not otherwise. Then the function space carries the compact-open topology.

Briefly, this works very nicely when $X$ is locally compact, but not otherwise. Then the function space carries the compact-open topology.