Subject: categories: Generalized Enrichment

From: Tom Leinster <[email-masked]>
Date: Fri, 29 Jan 1999 12:29:26 +0000 (GMT)

The following is now available, at http://www.dpmms.cam.ac.uk/~leinster/

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 the author, Burroni or Hermida.) 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 T_n-multicategory enriched in a T_{n+1}-multicategory, where
T_n is the monad expressing the pasting-together of n-opetopes.

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.

Tom Leinster

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