Kan extendable subcategories and fibrewise topology (Q6629786)

This paper uses pointwise Kan extensions to generate new subcategories out of old ones, investigating the properties of these newly produced categories and giving sufficient conditions for their cartesian closedness to hold.\N\NThe synopsis of the paper goes as follows:\N\N\begin{itemize}\N\item[\S 1] contains a brief discussion of reflective subcategories and their properties, establishing (Theorem 1.8) that a replete subcategory of \(\mathcal{C}\)\ containing \(\mathcal{W}\)\ as a codense subcategory is necessarily the reflective hull of \(\mathcal{W}\), and is therefore unique when it exists.\N\N\item[\S 2] recalls the basic definitions and facts about codensity monads and their idempotency.\N\N\item[\S 3] defines the notion of Kan extendable subcategories, studying their properties.\N\N\item[\S 4] uses the theory of Kan extendable subcategories to derive the Stone-Čech compactification and the Hewitt real-compactification procedures.\N\N\item[\S 5] establishes (Theorem 5.1) that subcategories of fiberwise topological spaces over \(B\)\ which abide by certain separation axioms are reflective subcategories of \textsf{Top}\(_{B}\).\N\N\item[\S 6] investigates the concept of fiberwise compact spaces, establishing (Theorem 6.15) that a fiberwise compact fiberwise Hausdorff space over a \textsf{T}\(_{1}\)\ base \(B\) is an exponential object of \textsf{Top}\(_{B}\).\N\N\item[\S 7] introduces the subcategories of \textsf{Top}\(_{B}\)\ of fiberwise weak Hausdorff spaces and fiberwise \(k\)-Hausdorff spaces, establishing (Theorem 7.7) that they are reflective in \textsf{Top}\(_{B}\).\N\N\item[\S 8] presents a sufficient condition for a subcategory of \textsf{Top}\(_{B}\) to be left Kan extendable (Theorem 8.2).\N\N\item[\S 9] states conditiions that ensure the cartesian closedness of the strong coreflective hull of a subcategory (Theorem 9.6).\N\N\item[\S 10] establishes the fiberwise Day's theorem (Theorem 10.2).\N\N\item[\S 11] uses Theorem 10.2 to establish (Theorem 11.2) that the category \textsf{kTop}\(_{B}\)\ of fiberwise compactly generated topological spaces over a \textsf{T}\(_{1}\)\ base \(B\)\ is cartesian closed.\N\N\item[\S 12] inspects properties of certain subcategories of \textsf{kTop}\(_{B}\).\N\N\item[\S 13] establishes (Proposition 13.4)\ the cartesian closedness of the category of fiberwise sequential spaces, provided the base \(B\)\ abides by the right separation axiom.\N\N\item[\S 14] establishes (Proposition 14.9)\ the cartesian closedness of the category of fiberwise Alexandroffl spaces, provided the base \(B\)\ abides by the right separation axiom.\N\N\item[Appendix A] investigates limits in a slice category.\N\N\item[Appendix B] is concerned with limits in a slice category of sets.\N\N\item[Appendix C] is concerned with limits in the category of fiberwise spaces.\N\end{itemize}