V - Cat Is Locally Presentable or Locally Bounded If V Is So

We show, for a monoidal closed category V = (V0,⊗, I), that the category V -Cat of small V -categories is locally λ-presentable if V0 is so, and that it is locally λ-bounded if the closed category V is so, meaning that V0 is locally λ-bounded and that a side condition involving the monoidal structure is satisfied. Many important properties of a monoidal category V are inherited by the category V -Cat of small V -categories. For instance, if V is symmetric monoidal, V -Cat has a canonical symmetric monoidal structure, as was observed already in [4]. Much later [7, Remark 5.2], it was realized that if V is only braided monoidal then V -Cat still has a canonical monoidal structure, although it need not have a braiding unless the braiding on V is in fact a symmetry. Similarly, it is straightforward to show that V -Cat is monoidal closed when V is closed and complete, and that V -Cat is complete when V is so. All of these results are essentially routine; the less trivial fact that V -Cat is cocomplete when V is so was first proved in [11]. The properties of V or V -Cat that we consider here are of a less basic nature, being conditions on V which allow proofs by transfinite induction of the existence of various important adjoints. The best known of these conditions is local presentability [5], but there is also the notion of local boundedness [8], which is more general than local presentability, but also much more common, and sufficient for the central existence results of [8, Chapter 6], from which follow the basic results of the theory of enriched projective sketches. Recall that to be locally presentable is to be locally λ-presentable for some regular cardinal λ, and similarly that to be locally bounded is to be locally λ-bounded for some λ. It would be one thing to prove that V -Cat is locally presentable if V is so (in the sense that its underlying ordinary category V0 is so); here we prove the stronger result that V -Cat is locally λ-presentable if V0 is so, so that the passage from V to V -Cat does not require the regular cardinal λ to be changed. When it comes to local boundedness, we prove that V -Cat is locally λ-bounded when V is so “as a closed category”, meaning that V0 is locally λ-bounded and satisfies a side condition involving the monoidal structure. We recall the precise definitions of local λ-presentability and local λ-boundedness in Section 2, but the common aspect is that V0 is cocomplete and has a small set G of objects forming in some sense a generator of V0, with the representables V0(G,−) : V0 → Set preserving certain colimits: λ-filtered colimits in the locally λ-presentable case, and λ-filtered unions Both authors gratefully acknowledge the support of the Australian Research Council. Received by the editors 2001 November 12. Transmitted by R. J. Wood. Published on 2001 December 19. 2000 Mathematics Subject Classification: 18C35, 18D20, 18A32.