Generalizing the work of [1], we present the process of generating cartesian closed subcategories of (T,V)-Cat. From a fixed class C of objects, we form a new (reflective) subcategory whose objects are shown to be colimits of objects in C. For instance taking for C the class of compact Hausdorff objects of (T,V)-Cat, we construct the so called compactly generated (T,V)-categories; we recover results for topological spaces, namely the ones appearing in [1], and discuss some ongoing work.

[1] Martín Escardó, Jimmie Lawson and Alex Simpson. Comparing Cartesian closed categories of (core) compactly generated spaces. Topology Appl., 143(1-3):105-145, 2004.