## Iterated distributive laws (2007)

Citations: | 6 - 1 self |

### BibTeX

@TECHREPORT{Cheng07iterateddistributive,

author = {Eugenia Cheng},

title = {Iterated distributive laws},

institution = {},

year = {2007}

}

### OpenURL

### Abstract

We give a framework for combining n monads on the same category via distributive laws satisfying Yang-Baxter equations, extending the classical result of Barr and Wells which combines two monads via one distributive law. We show that this corresponds to iterating n-times the process of taking the 2-category of monads in a 2-category, extending the result of Street characterising distributive laws. We show that this framework can be used to construct the free strict n-category monad on n-dimensional globular sets; we first construct for each i a monad for composition along bounding i-cells, and then we show that the interchange laws define distributive laws between these monads, satisfying the necessary Yang-Baxter equations.

### Citations

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(Show Context)
Citation Context ...on suffice. This theorem can be proved directly by a diagram chase and induction, and this is what we do in Section 2. In Section 3 we look at a more abstract approach to monads provided by Street in =-=[13]-=-. Usually monads are defined by giving a category C, a functor T : C −→ C and natural transformations 1 η �� T �� µ 2 T 2 satisfying the usual unit and associativity axioms. Of course, we could make t... |

110 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
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(Show Context)
Citation Context ...tion of n-category [15], which has strict interchange but weak units. The results of the present work enable us to construct a comparison functor from Trimble n-categories to those defined by Batanin =-=[2]-=-. Finally we note that the proofs and constructions in this work are all completely straightforward (in that there are no surprises) though sometimes lengthy and afflicted with unavoidably complex not... |

107 |
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(Show Context)
Citation Context ...”. More precisely, we know that not every tricategory is triequivalent to a completely strict 3-category; the well-known coherence result is that every tricategory is triequivalent to a Gray-category =-=[5, 8]-=-. This has led to a generally accepted conjecture that every weak n-category should be equivalent to a semi-strict one, where “semi-strict” means that every thing is strict apart from interchange – in... |

96 | The algebra of oriented simplexes - Street - 1987 |

82 |
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(Show Context)
Citation Context ...lf a monad (on the category 2-Cat of 2-categories and 2-functors). Thus it can be expressed as a functor ∆ −→ Cat. Using the closed structure of 2-Cat with respect to the lax Gray tensor product (see =-=[6, 7]-=-), we see that Mnd n (B) ≃ [∆ ⊗n , B] and the Yang-Baxter equations for the monads correspond to Yang-Baxter equations arising from the relations on the generators defining the Gray tensor product. In... |

22 | A closed model structure for n-categories, internal Hom, n-stacks and generalized Seifert-Van Kampen - Simpson |

18 |
An algebraic theory of tricategories
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(Show Context)
Citation Context ...”. More precisely, we know that not every tricategory is triequivalent to a completely strict 3-category; the well-known coherence result is that every tricategory is triequivalent to a Gray-category =-=[5, 8]-=-. This has led to a generally accepted conjecture that every weak n-category should be equivalent to a semi-strict one, where “semi-strict” means that every thing is strict apart from interchange – in... |

17 |
Toposes, Triples, and Theories. Number 278 in Grundlehren der mathematischen Wissenschaften
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(Show Context)
Citation Context ... the monad structure for T”. The main theorem about distributive laws tells us about new monads that arise canonically as a result of the distributive law. 5�� � �� �� � Theorem 1.2 (Barr and Wells, =-=[1]-=-). The following are equivalent: • A distributive law of S over T. • A lifting of the monad T to a monad T ′ on S-Alg. • A lifting of the monad S to a monad S ′ on Kl(T). It follows that TS canonicall... |

10 |
Higher operads, higher categories. Number 298 in London Mathematical Society Lecture Note Series
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(Show Context)
Citation Context ...ow that the composite monad T0T2 · · · Tn−1 resulting from the distributive series of monads in question, is really the free strict n-category monad. We draw on abstract 27results from Appendix F of =-=[10]-=-, but writing down the definitions directly is not hard. The construction of the monad for i-composition proceeds in the following steps: 1. Construct free category monad on 1-GSet. 2. Use the enriche... |

8 |
Coherence for the tensor product of 2-categories, and braid groups. In Algebra, Topology and Category Theory, a collection in honor of Samuel Eilenberg
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(Show Context)
Citation Context ...lf a monad (on the category 2-Cat of 2-categories and 2-functors). Thus it can be expressed as a functor ∆ −→ Cat. Using the closed structure of 2-Cat with respect to the lax Gray tensor product (see =-=[6, 7]-=-), we see that Mnd n (B) ≃ [∆ ⊗n , B] and the Yang-Baxter equations for the monads correspond to Yang-Baxter equations arising from the relations on the generators defining the Gray tensor product. In... |

5 | Weak units and homotopy 3-types
- Joyal, Kock
- 2007
(Show Context)
Citation Context ... semi-strict one, where “semi-strict” means that every thing is strict apart from interchange – interchange thus being the only part that cannot be strictified. However, recent work of Joyal and Kock =-=[9]-=- has shown that a different type of “semi-strict 3-category” is also fruitful: one in which everything is strict apart from units. Joyal and Kock have shown that such 4� � �� � �� � �� �� � � � � �� ... |

4 | Limits in n-categories
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(Show Context)
Citation Context ...�� 3-categories do model homotopy 3-types, and do give rise to braided monoidal categories in the suitably degenerate case, two crucial things that fully strict 3-categories cannot do. Indeed Simpson =-=[12, 11]-=- conjectures that n-categories with weak units (and everything else strict) are enough to model n-types for all n. In the light of these recent results we consider the study of n-categories with stric... |

4 |
What are ‘fundamental n-groupoids
- Trimble
- 1999
(Show Context)
Citation Context ...tant, and the results of the present work will contribute towards that study. In a future work, and in [4] we will use the theory of iterated distributive laws to study Trimble’s notion of n-category =-=[15]-=-, which has strict interchange but weak units. The results of the present work enable us to construct a comparison functor from Trimble n-categories to those defined by Batanin [2]. Finally we note th... |

1 |
Operadic theories of n-category
- Cheng
- 2007
(Show Context)
Citation Context ...of these recent results we consider the study of n-categories with strict interchange to be important, and the results of the present work will contribute towards that study. In a future work, and in =-=[4]-=- we will use the theory of iterated distributive laws to study Trimble’s notion of n-category [15], which has strict interchange but weak units. The results of the present work enable us to construct ... |