Does any body know if comma categories have been defined in enriched contexts?
Lawvere's La Jolla paper, where general comma categories were introduced, showed how to construct them from pullbacks and a "cylinder" (or "arrow object") construction. John Gray (SLNM p. 254) showed that cylinder is a universal notion which a 2-category may or may not have. I pointed out [Fibrations and Yoneda's lemma in a 2-category, Lecture Notes in Math. 420 (1974) 104-133; MR53#585] that finite completeness for a 2-category should mean that it have pullbacks, a terminal object, and cylinders (a similar idea was in my PhD thesis for differential graded categories which are finitely complete when they admit pullbacks, a zero object and "suspension"). Finite completeness for 2-categories is further analysed in [Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8 (1976) 149-181; MR53#5695]. More generally, finite completeness for a V-category A (= a category with homs enriched in V) means that its underlying category has finite ordinary limits, which are preserved by representables A(a,-) into V, and that it admits cotensoring by the "finite" objects of V. There is some choice about what you mean by "finite" object in V however "finitely presentable" is often the right thing. Sometimes, as in the case of V = Cat, the finite objects are generated by a few finite objects - that is why "cylinder" plays the important role in 2-categories (it is the finite generating object, cotensor with which is cylinder). So why am I going on about finite limits in 2-categories? Well, Lawvere's construction shows that comma objects exist in any finitely complete 2-category. Comma objects are particular finite limits just like pullbacks. In particular, there is a 2-category V-Cat of V-categories, V-functors and V-natural transformations. It is certainly complete (as a 2-category) for any decent V. So, indeed, it is well known that comma objects (or comma V-categories) exist. They have their uses but NOT for the wonderful use that Lawvere put them to: Lawvere provided a formula for left (right) Kan extensions of ordinary functors which involves taking a colimit (limit) over a comma category. [Indeed, more is true; see my definition of "pointwise Kan extension" in "Fibrations and Yoneda's lemma in a 2-category".] However, this formula does not work even for additive categories (= categories enriched in the monoidal category of abelian groups). Regards, Ross http://www.mpce.mq.edu.au/~street/