## Operads In Higher-Dimensional Category Theory (2004)

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@TECHREPORT{Leinster04operadsin,

author = {Tom Leinster},

title = {Operads In Higher-Dimensional Category Theory},

institution = {},

year = {2004}

}

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### Abstract

The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak n-category. Included is a full explanation of why the proposed definition of n-category is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higher-dimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to n-categories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.

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Citation Context ...1), . . . of sets together with an `identity' element of C(1) and `composition' functions C(n) C(k 1 ) C(k n ) - C(k 1 + + k n ), obeying associativity and identity laws. (In the original definition, =-=[May1]-=-, the C(n)'s were not just sets but spaces with symmetric group action. Our operads never have symmetric group actions.) The simplest kind of multicategory---a plain multicategory---consists of a coll... |

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Citation Context ...m, it is variously described as `pseudo', `weak' and `strong', or not given a qualifier at all. In the context of maps between bicategories another word altogether is often used (`homomorphism'---see =-=[Ben]-=-). Not quite as severe a problem is the terminology for n-categories themselves: the version where things hold up to coherent isomorphism or equivalence is (almost) invariably called weak, and the ver... |

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Citation Context ...lgebra for this 2-monad. The definition of relaxed monoidal category in [Lei5, 4.4] implicitly uses this approach, but with lax algebras rather than weak algebras. For more on this point of view, see =-=[KS]-=- and [Pow]. We also use this approach in Appendix A. f. The notation (f n # #f 1 ) for the composite of a diagram A 0 f 1 - A 1 f 2 - fn - A n is sometimes inadequate in the case n = 0. When n = 0 the... |

125 |
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Citation Context ...s for a monoid or monad. Perhaps the coherence axioms for an unbiased lax functor are a little less obvious; however, they are the same shape as the axioms for a monad functor given in Street's paper =-=[Str1]-=-, and in any case seem to be quite canonical in some vague sense. Naturally, we would like to be able to compose lax functors. Given unbiased lax functors B (F,#) - B # (F # ,# # ) - B ## , define the... |

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Citation Context ...ulticategories are exactly plain multicategories enriched in V . 4. A definition of weak #-category In this section we present a definition of weak #-category, a variation on that given by Batanin in =-=[Bat]-=-. We start (4.1) by giving the definition in purely formal terms, which can be done very quickly. However, it is the explanation of why it is a reasonable definition that occupies most of the section ... |

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Citation Context ...y satisfactory way to weaken the classical definition of bicategory to get a lax version. Admittedly, one can drop the condition that the classical associativity and unit maps are isomorphisms (as in =-=[Borx1]-=-, after Definition 7.7.1); but somehow this does not seem quite right. OPERADS IN HIGHER-DIMENSIONAL CATEGORY THEORY 85 Another advertisement for the unbiased theory follows. To give it we need some p... |

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Citation Context ...he objects of the multicategory: thus an operad is a single-coloured operad. A two-object plain multicategory would be called an `operad of two colours', typically black and white. Baez and Dolan, in =-=[BD]-=-, use `operad' or `typed operad' for the same kind of purpose as we use `multicategory', and `untyped operad' where we use `operad'.) It is inherent that everything is small: when E = Set, for instanc... |

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Citation Context ...4.4] and [Lei7, 4.3] for more or less explicit references to the idea. Closely related issues have been considered in the study of 2-monads made by the (largely) Australian school: see, for instance, =-=[BKP]-=-, [Kel1] and [Pow]. The virtues of the main proof of this section (which is actually in Appendix A) are its directness, and that it uses an operad where a 2-monad might be used instead, which is more ... |

69 |
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Citation Context ...ieck fibration (category of elements) of the functor pd : G - Set, then Coll # = [G, Set]/pd # [Gr(pd), Set], and any category of the form [A, Set] (with A small) is locally finitely presentable: see =-=[Borx2]-=-, Example 5.2.2(b). OPERADS IN HIGHER-DIMENSIONAL CATEGORY THEORY 191 Hypotheses on U : CWC - Coll. We have to see that U is finitary and monadic. It is straightforward to calculate that U creates fil... |

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46 |
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Citation Context ...ding in elements got by contraction and then freely adding in elements got by operadic composition. However, we do not take this route here, instead relying on the following result from Kelly's paper =-=[Kel2]-=-: 190 TOM LEINSTER D.1.1. Theorem. Let D - C B ? P - A Q ? be a (strict) pullback diagram in CAT. If A is locally finitely presentable and each of P and Q is finitary and monadic, then the functor D -... |

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Citation Context ... a map between objects of E : on the one hand, genuine maps in E , and on the other, spans (i.e. 1-cells of Span(E , T )). A possible approach to formalizing this situation is via the `equipments' of =-=[CKVW]-=-. But this is not our approach: as explained in 3.6 and 3.7, fc-multicategories are the structures that capture exactly what we want. Any (E , T)-multicategory has an underlying (E , T )-graph, enabli... |

34 | 2000), Representable multicategories - Hermida |

33 |
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Citation Context ... of categories enriched in an fc-multicategory. This extends the usual theory of categories enriched in a monoidal category, as well as the less popular theory of categories enriched in a bicategory (=-=[BCSW]-=-, [CKW], [Wal]) and the evident but hardly-written-up theory of categories enriched in a plain multicategory. Categories enriched in an fc-multicategory are examined in each of [Lei4], [Lei5] and [Lei... |

33 | Categorical structures - Street - 1995 |

31 |
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Citation Context ...strong regularity: for in general, the monad yielded by any strongly regular theory is cartesian. This last result, and the notion of strong regularity, are due to Carboni and Johnstone. They show in =-=[CJ]-=- (Proposition 3.2 via Theorem 2.6) that a theory is strongly regular if and only if # andsare cartesian natural transformations and T preserves wide pullbacks. A wide pullback is by definition a limit... |

24 | Definitions: operads, algebras and modules
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Citation Context ...action of the nth symmetric group S n on C(n) for each n, satisfying certain axioms: in other words, an operad OPERADS IN HIGHER-DIMENSIONAL CATEGORY THEORY 99 in the usual sense of topologists (e.g. =-=[May2]-=-), except that the C(n)'s are sets rather than spaces or graded modules etc. A first attempt might be to take the free commutative monoid monad T on Set. But this is both misguided and doomed to failu... |

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18 |
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Citation Context ...3] for more or less explicit references to the idea. Closely related issues have been considered in the study of 2-monads made by the (largely) Australian school: see, for instance, [BKP], [Kel1] and =-=[Pow]-=-. The virtues of the main proof of this section (which is actually in Appendix A) are its directness, and that it uses an operad where a 2-monad might be used instead, which is more in the spirit of t... |

14 |
Vertex algebras
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Citation Context ...# # # # # . . S S S . . Q Q Q Q . # # # # # # # . The T-multicategories are a simpler version of Soibelman's pseudo-monoidal categories ([Soi]) or Borcherds' relaxed multilinear categories ([Borh]=-=-=-, [Sny1], [Sny2]); they omit the aspect of maps between trees. See the end of 3.8 for comments on the unsimplified version. OPERADS IN HIGHER-DIMENSIONAL CATEGORY THEORY 101 g. When E = Cat and T is t... |

14 |
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Citation Context ...dules construction. 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 =-=[CKW]-=- or [Kos]). 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 bi... |

14 | The role of Michael Batanin’s monoidal globular categories, in Higher Category Theory
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- 1998
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Citation Context ...semblance to the profunctors and natural transformations discussed in 3.7. Section 4 is A definition of weak #-category, based on the definition given by Batanin in [Bat] (and summarized by Street in =-=[Str3]-=-). I first wrote a version of this section in [Lei3]. At the time I thought I was writing an account of Batanin's definition, reshaped and very much simplified but with the same end result mathematica... |

13 | From coherent structures to universal properties - Hermida |

13 | Basic bicategories
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Citation Context ...ur terminology. The original definition of bicategory was made in Benabou's paper [Ben], along with the definition of lax functor (called `morphism' there). Other references for these definitions are =-=[Lei2]-=- and [Str2], which also include definitions of transformation and modification; but we will not need these further concepts here. We will typically denote 0-cells (or `objects') of a bicategory B by A... |

12 |
Weak n-categories: opetopic and multitopic foundations
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(Show Context)
Citation Context ...certain special T-operads. I have not investigated how well this works, but this idea seems to be related to the structures called `symmetric operads' at the beginning of [BD] and explored further in =-=[Che1-=-] and [Che2]. d. Let E = Set, and consider the monad --- + 1 of 2.1.3(e). A (Set, --- + 1)-graph is a diagram C 0 + 1 d C 1 c - C 0 of sets; this is like an ordinary (Set, id)-graph, except that some... |

12 |
On clubs and doctrines
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Citation Context ...d [Lei7, 4.3] for more or less explicit references to the idea. Closely related issues have been considered in the study of 2-monads made by the (largely) Australian school: see, for instance, [BKP], =-=[Kel1]-=- and [Pow]. The virtues of the main proof of this section (which is actually in Appendix A) are its directness, and that it uses an operad where a 2-monad might be used instead, which is more in the s... |

12 | On clubs and data-type constructors - Kelly - 1992 |

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9 |
Weak n-categories: comparing opetopic foundations
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Citation Context ...cial T-operads. I have not investigated how well this works, but this idea seems to be related to the structures called `symmetric operads' at the beginning of [BD] and explored further in [Che1] and =-=[Che2-=-]. d. Let E = Set, and consider the monad --- + 1 of 2.1.3(e). A (Set, --- + 1)-graph is a diagram C 0 + 1 d C 1 c - C 0 of sets; this is like an ordinary (Set, id)-graph, except that some arrows hav... |

9 | General operads and multicategories
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Citation Context ...[Lei9] is the place to look; otherwise, I hope that this will serve as a useful medium-length account. Much of what is here has appeared in preprints available electronically. The main references are =-=[Lei1]-=- and Sections I and II of [Lei3], and to a lesser extent [Lei5]. In many places I have added detail and rigour; indeed, much of the new writing is in the appendices. The first section, Bicategories, i... |

8 | Monads and interpolads in bicategories
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Citation Context ...struction. 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 [CKW] or =-=[Kos]-=-). 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... |

8 |
Generalized enrichment for categories and multicategories
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Citation Context ...erve as a useful medium-length account. Much of what is here has appeared in preprints available electronically. The main references are [Lei1] and Sections I and II of [Lei3], and to a lesser extent =-=[Lei5]-=-. In many places I have added detail and rigour; indeed, much of the new writing is in the appendices. The first section, Bicategories, is also largely new writing. However, the results it contains ar... |

6 | Homotopy algebras for operads - Leinster |

6 | Equivalence of Borcherds G-Vertex Algebras and Axiomatic Vertex Algebras, arXiv:math.QA/9904104. Anguelova: Centre de Recherches Mathematiques (CRM), Université de
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Citation Context ...# . . S S S . . Q Q Q Q . # # # # # # # . The T-multicategories are a simpler version of Soibelman's pseudo-monoidal categories ([Soi]) or Borcherds' relaxed multilinear categories ([Borh], [Sny1]=-=-=-, [Sny2]); they omit the aspect of maps between trees. See the end of 3.8 for comments on the unsimplified version. OPERADS IN HIGHER-DIMENSIONAL CATEGORY THEORY 101 g. When E = Cat and T is the free ... |

6 |
Sheaves and Cauchy-complete categories
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Citation Context ...enriched in an fc-multicategory. This extends the usual theory of categories enriched in a monoidal category, as well as the less popular theory of categories enriched in a bicategory ([BCSW], [CKW], =-=[Wal]-=-) and the evident but hardly-written-up theory of categories enriched in a plain multicategory. Categories enriched in an fc-multicategory are examined in each of [Lei4], [Lei5] and [Lei6]. The theory... |

5 |
Coherence for a closed functor
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Citation Context ...#(f 3 #f 2 #f 1 )) built up from coherence cells are equal. The form of the codomain is important, being F applied to a composite of 1-cells in B. In contrast, a counterexample in the introduction to =-=[Lew]-=- shows that there can be two distinct maps F1 - - F1#F1 built up from coherence cells. (The counterexample is stated in the context of classical bicategories---in fact, monoidal categories---but trans... |

5 | Meromorphic tensor categories
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Citation Context ...he graph structure is comprised of sets like C # # # # # # # # . . S S S . . Q Q Q Q . # # # # # # # . The T-multicategories are a simpler version of Soibelman's pseudo-monoidal categories ([Soi]) or Borcherds' relaxed multilinear categories ([Borh], [Sny1], [Sny2]); they omit the aspect of maps between trees. See the end of 3.8 for comments on the unsimplified version. OPERADS IN HIGHER-DIME... |

5 | A tensor product for Gray-categories - Crans - 1999 |

4 | Many variable functorial calculus I - Kelly - 1972 |

4 | Limits in n-categories - Simpson - 1997 |

4 | Equivalence between approaches to the theory of opetopes - Cheng - 2000 |