## 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.

### Citations

365 | Gromov-Witten classes, quantum cohomology, and enumerative geometry - Manin - 1994 |

141 | Introduction to bicategories - Bénabou - 1967 |

125 |
The formal theory of monads
- Street
- 1972
(Show Context)
Citation Context ...n M’. A monoid in M is also the same thing as a multicategory map 1 ✲ M, where 1 is the terminal multicategory. 3. If B is a bicategory then an object of Mon(B) is a monad in B in the sense of Street =-=[14]-=-: that is, it’s an object X of B together with a 1-cell t : X ✲ X and 2-cells µ : t◦t ✲ t, η : 1 ✲ t satisfying the usual monad axioms. There are no maps (X, t, µ, η) ✲ (X ′, t ′, µ′, η ′) in Mon(B) u... |

110 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
- Batanin
- 1998
(Show Context)
Citation Context ... original sense of Lambek [5]. When T is the free (strict) ∞-category monad on the category E of globular sets (‘∞-graphs’), a T-multicategory C with C0 = 1 is a higher operad in the sense of Batanin =-=[1]-=-. The example which concerns us here is when T is the free category monad fc on the category E of C0 2directed graphs. A T-multicategory is then an fc-multicategory in the sense of the following expl... |

33 | Variation through enrichment - Betti, Carboni, et al. |

33 | Representable multicategories - Hermida - 2000 |

32 | Operads in higher-dimensional category theory - Leinster - 2000 |

29 |
Deductive systems and categories i
- Lambek
- 1968
(Show Context)
Citation Context ... 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 of Lambek =-=[5]-=-. When T is the free (strict) ∞-category monad on the category E of globular sets (‘∞-graphs’), a T-multicategory C with C0 = 1 is a higher operad in the sense of Batanin [1]. The example which concer... |

24 | Definitions: operads, algebras and modules
- May
- 1997
(Show Context)
Citation Context ...that any symmetric monoidal category S gives rise to a T ′ - multicategory V, and a one-object V-enriched T-multicategory is then exactly what topologists call a (non-symmetric) operad in S (see e.g. =-=[9]-=-). Secondly, there is a certain naturally-arising T ′ -multicategory V such that V-enriched Tmulticategories are the structures called ‘relaxed multicategories’ by Borcherds in his definition of verte... |

19 | On weak higher dimensional categories - Hermida, Makkai, et al. - 1997 |

14 |
An axiomatics for bicategories of modules
- Carboni, Kasangian, et al.
- 1987
(Show Context)
Citation Context ...r. 3 Bimodules Bimodules have traditionally been discussed in the context of bicategories. Thus given a bicategory B, one constructs a new bicategory Bim(B) whose 1-cells are bimodules in B (see e.g. =-=[4]-=-). The drawback is that this is only possible when B has certain properties concerning the existence and behaviour of local reflexive coequalizers. Here we extend the Bim construction from bicategorie... |

14 | Vertex algebras - Borcherds - 1998 |

11 | Quantum cohomology of a product - Manin - 1996 |

9 | On the Breen–Baez–Dolan stabilization hypothesis for Tamsamani’s weak n-categories, available as arXiv:math/9810058 - Simpson |

8 |
Generalized enrichment for categories and multicategories
- Leinster
- 1999
(Show Context)
Citation Context ...uestion 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 ... |

6 | Equivalence of Borcherds G-Vertex Algebras and Axiomatic Vertex Algebras, arXiv:math.QA/9904104. Anguelova: Centre de Recherches Mathematiques (CRM), Université de
- Snydal
(Show Context)
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 ... |

6 |
Sheaves and Cauchy-complete categories
- Walters
- 1981
(Show Context)
Citation Context ...e 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 1one might speak of a category enriched in an ordinary multicategory (‘ch... |

5 | Meromorphic tensor categories - Soibelman - 1997 |

4 | Categorification - Baez, Dolan - 1998 |

2 |
Relaxed multi category structure of a global category of rings and modules
- Snydal
- 1999
(Show Context)
Citation Context ...rising 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 a cate... |

1 |
Vertex algebras, e-print q-alg/9706008
- Borcherds
- 1997
(Show Context)
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... |

1 |
General operads and multicategories, e-print math.CT/9810053
- Leinster
- 1997
(Show Context)
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... |

1 |
Meromorphic tensor categories, e-print q-alg/9709030
- Soibelman
- 1997
(Show Context)
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... |

1 | The meromorphic braided category arising in quantum affine algebras
- Soibelman
- 1999
(Show Context)
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 |