## Generalized enrichment of categories (1999)

Venue: | Also Journal of Pure and Applied Algebra |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Leinster99generalizedenrichment,

author = {Tom Leinster},

title = {Generalized enrichment of categories},

journal = {Also Journal of Pure and Applied Algebra},

year = {1999},

pages = {391--406}

}

### OpenURL

### Abstract

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.

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Citation Context ...ally-arising T ′ -multicategory V such that V-enriched Tmulticategories are the structures called ‘relaxed multicategories’ by Borcherds in his definition of vertex algebras over a vertex group ([2], =-=[10]-=-, [11]), and called ‘pseudo-monoidal categories’ by Soibelman in his work on quantum affine algebras ([12], [13]). The general definition of enriched T-multicategory is very simple. Take a monad T on ... |

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Citation Context ...naturally-arising T ′ -multicategory V such that V-enriched Tmulticategories are the structures called ‘relaxed multicategories’ by Borcherds in his definition of vertex algebras over a vertex group (=-=[2]-=-, [10], [11]), and called ‘pseudo-monoidal categories’ by Soibelman in his work on quantum affine algebras ([12], [13]). The general definition of enriched T-multicategory is very simple. Take a monad... |

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Citation Context ...ticategory C consists of a diagram ✠� �� dom T(C0) C1 ❅ cod ❘ in E (a T-graph) together with functions defining ‘composition’ and ‘identity’; the full details can be found in Burroni [3] or Leinster (=-=[6]-=- or [8]). Thus when T is the identity monad on E = Set, a T-multicategory is simply a category. When T is the free-monoid monad on E = Set, a T-multicategory is a multicategory in the original sense o... |

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Citation Context ... multicategories’ by Borcherds in his definition of vertex algebras over a vertex group ([2], [10], [11]), and called ‘pseudo-monoidal categories’ by Soibelman in his work on quantum affine algebras (=-=[12]-=-, [13]). The general definition of enriched T-multicategory is very simple. Take a monad T on a category E, and let T ′ be the free T-multicategory monad, as above. Given an object A of E, we can form... |

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Citation Context ...categories’ by Borcherds in his definition of vertex algebras over a vertex group ([2], [10], [11]), and called ‘pseudo-monoidal categories’ by Soibelman in his work on quantum affine algebras ([12], =-=[13]-=-). The general definition of enriched T-multicategory is very simple. Take a monad T on a category E, and let T ′ be the free T-multicategory monad, as above. Given an object A of E, we can form I(A) ... |

1 | numbers such as math.CT/12345 or q-alg/67890 refer to the xxx preprint server, which can be found at http://xxx.lanl.gov (or one of its mirror sites, such as http://xxx.soton.ac.uk). For direct access to a paper, go to http://xxx.lanl.gov/abs/math.CT/1234 - Preprint - 1998 |