Generalized Enrichment of Categories

We define the phrase ‘category enriched in an fc-multicategory’ and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fc-multicategory extends the (more or less well-known) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fc-multicategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper is elementary and self-contained, we also explain why, from one point of view, fc-multicategories are the natural structures in which to enrich categories. Published as Journal of Pure and Applied Algebra 168 (2002), 391–406. 2000 Mathematics Subject Classification: 18D20, 18D05, 18D50, 18D10. A general question in category theory is: given some kind of categorical structure, what might it be enriched in? For instance, suppose we take braided monoidal categories. Then the question asks: what kind of thing must V be if we are to speak sensibly of V-enriched braided monoidal categories? (The usual answer is that V must be a symmetric monoidal category.) In another paper, [7], I have given an answer to the general question for a certain family of categorical structures (generalized multicategories). In particular, this theory gives an answer to the question ‘what kind of structure V can a category be enriched in’? The answer is: an ‘fc-multicategory’. Of course, the traditional answer to this question is that V is a monoidal category. But there is also a notion of a category enriched in a bicategory (see Walters [15]). And generalizing in a different direction, it is easy to see how Supported by the EPSRC and St John’s College, Cambridge