V-Cat is nice when V is
Steve Lack – 7 March 2001
This talk described joint work with Max Kelly.
I proved that V is a symmetric monoidal closed category which is locally k-presentable (in the sense of Gabriel-Ulmer), for some regular cardinal k, then the category V-Cat of (small) V-categories is also locally k-presentable.
I also proved that if V is locally k-bounded as a closed category (in the sense of Max) then V-Cat is locally k-bounded; the definition of local boundedness is similar to that of local presentability, but rather than arbitrary k-filtered colimits uses only k-filtered unions of subobjects with respect to a given proper factorization system. Examples of symmetric monoidal closed categories which are locally bounded but not locally presentable include the various topological closed categories, such as the category of compactly-generated topological spaces.
There is a preprint, which should be available soon ...
[note added 22/2/02: it is now published.]