Generalized Enrichment for Categories and Multicategories

In this paper we answer the question: ‘what kind of a structure can a general multicategory be enriched in?’ (Here ‘general multicategory’ is used in the sense of [Lei1], [Bur] or [Her].) The answer is, in a sense to be made precise, that a multicategory of one type can be enriched in a multicategory of the type one level up. In particular, we will be able to speak of a Tn-multicategory enriched in a Tn+1-multicategory, where Tn is the monad expressing the pasting-together of n-opetopes, as constructed in [Lei2]. The answer for general multicategories reduces to something surprising in the case of ordinary categories: a category may be enriched in an ‘fc-multicategory’, a very general kind of 2-dimensional structure encompassing monoidal categories, plain multicategories, bicategories and double categories. It turns out that fc-multicategories also provide a natural setting for the bimodules construction. We also explore enrichment for some multicategories other than just categories. An extended application is given: the relaxed multicategories of Borcherds and Soibelman are explained in terms of enrichment.