## Higher dimensional algebra III: n-categories and the algebra of opetopes (1997)

Citations: | 74 - 6 self |

### BibTeX

@MISC{Baez97higherdimensional,

author = {John C. Baez and James Dolan},

title = {Higher dimensional algebra III: n-categories and the algebra of opetopes },

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘S-operads’, and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O + whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘n-dimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak n-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ω-categories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘n-coherent O-algebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal n-categories’, ‘stable n-categories’, ‘virtual n-functors ’ and ‘representable n-prestacks’. We also describe how n-coherent O-algebra objects may be defined in any (n + 1)-coherent O-algebra.

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Citation Context ...ry’ we always mean ‘weak n-category’, as defined in this paper. For more background on n-category theory and why it should be interesting, see our previous papers, which we refer to as HDA0 [3], HDA1 =-=[5]-=-, and HDA2 [2]. As in those papers, we use the ordering in which the composite of morphisms f: x → y and g: y → z is written as fg, but when dealing with operads we write the composite of a k-ary oper... |

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Citation Context ...rams. In fact, the bigon is the only basic shape of 2cell in the traditional globular approach to n-category theory; to achieve the effect of 2-cells with other shapes one resorts to pasting theorems =-=[11, 19, 25]-=-. In the opetopic approach the the basic shapes of cells are the opetopes, which may have any number of infaces but always exactly one outface. For example, we use a 2-cell shaped like the opetope in ... |

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Citation Context ...ary virtual n-functor from C1 × . . . × Ck to C ′, and write A: C1 × . . . × Ck ⇀ C ′ . Example 47. Virtual functors as saturated anafunctors. A virtual functor is essentially the same as what Makkai =-=[22]-=- calls a ‘saturated anafunctor’, which may be viewed as a special sort of distributor. A distributor A from the category C to the category D is a functor A: C op ×D → Set, and A is a saturated anafunc... |

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Citation Context ...fined by Eilenberg and MacLane [13] in their 1945 paper. In a category, composition of 1-morphisms is associative ‘on the nose’: (fg)h = f(gh). Weak 2-categories first appeared in the work of Bénabou =-=[8]-=- in 1967, under the name of ‘bicategories’. In a bicategory, composition of 1-morphisms is associative only up to an invertible 2-morphism, the ‘associator’: Af,g,h: (fg)h → f(gh). The associator allo... |

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Citation Context ...bras is the same as a homomorphism of S-operads. ⊓⊔ In fact, there is also a prof(C)-operad for C-operads for any small category C. This played an important role in an earlier version of our approach =-=[4]-=-, but for various reasons we now prefer in what follows to work only with operads having a set, rather than a category, of types. 3.2 The slice operad of an operad Definition 15. Given a S-operad O, l... |

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Citation Context ...hat the conditions α must satisfy to be an action are just the conditions given in Section 2. Our use of formal power series above appears already in the generating function approach to combinatorics =-=[12]-=- and its categorical interpretation in terms of ‘species’ by Joyal [20]. As shown in Figure 7, what is at work here is the analogy between ordinary set-theoretic linear algebra and categorified linear... |

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Citation Context ...into an operation of the traditional sort by arbitrarily choosing an extension of every horn. It is tempting, therefore, to develop a simplicial approach to weak n-categories. This was done by Street =-=[28]-=-, who actually dealt with weak ω-categories. Like Kan complexes, these are simplicial sets. However, only certain ‘admissible’ horns, 4having the correct sort of orientation, are required to have ext... |