## Homomorphisms of higher categories (2008)

Venue: | U.U.D.M. REPORT 2008:47 |

Citations: | 5 - 0 self |

### BibTeX

@MISC{Garner08homomorphismsof,

author = {Richard Garner},

title = { Homomorphisms of higher categories},

year = {2008}

}

### OpenURL

### Abstract

We describe a construction that to each algebraically specified notion of higher-dimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricategories; and here we do not obtain the trihomomorphisms defined by Gordon, Power and Street, but rather something which is equivalent in a suitable sense. The second application is to Batanin’s weak ω-categories.

### Citations

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110 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
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(Show Context)
Citation Context ...be appropriate for homological algebra is rather infrequently satisfied in the case of higher categories. We may try and rectify this by moving from symmetric operads to the higher operads of Batanin =-=[2]-=-; but here a different problem arises, namely that the tensor product of bimodules over a globular operad is ill-defined, for the reason that, in the category whose monoids are globular operads, the t... |

107 |
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(Show Context)
Citation Context ... The second is that the homomorphisms it yields may be composed in a strictly associative and unital fashion. Even in dimension three this is an unusual feature, not shared by the trihomomorphisms of =-=[10]-=-: which, as shown in [9], form only a bicategory. This last observation may lead us to question whether our notion of homomorphism is sufficiently weak. In order to show that it is, we devote Section ... |

73 |
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(Show Context)
Citation Context ...t homomorphisms—the maps we described earlier as preserving the categorical structure “on-the-nose”. In the case of bicategories, this relationship is described by the two-dimensional monad theory of =-=[5]-=-. We write CatGph for the 2-category of Cat-enriched graphs—whose objects are given by a set X together with a functor X ×X → Cat—and T for the 2-monad thereupon whose algebras are small bicategories.... |

69 | Model categories, volume 63 of Mathematical Surveys and Monographs - Hovey - 1999 |

58 |
Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads, Journal of pure and applied algebra 89
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(Show Context)
Citation Context ...nce cells of various dimensions—is given in terms of a finitary monad. Both the existence and the finitariness of this monad are a consequence of it admitting a presentation in the sense described by =-=[15]-=-; and though the details are of little importance here, the conclusion is, namely that the adjunction (2) Tricats K ⊥ V where Tricats denotes the category of tricategories and strict homomorphisms, is... |

43 |
Aspects of topoi
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(Show Context)
Citation Context ...gher-dimensional categories. Let us at once say that we concern ourselves exclusively with those notions of higher-dimensional category which are essentially-algebraic in the sense described by Freyd =-=[7]-=-; for which composition and its associated coherence are realised by specified operations subject to equational laws. Of course any species of essentially-algebraic structure has a concomitant notion ... |

38 | Higher-dimensional word problems with applications to equational logic
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(Show Context)
Citation Context ...Computads were introduced in [22] as a tool for presenting free higher-dimensional categories, and have been studied extensively in the context of strict ω-categories under the name of polygraph: see =-=[6, 20]-=-. For the weak ω-categories under consideration here, we have the following notion of computad due to Batanin [1]. In the definition, we make use of the functors En := ω-Cats(2n,–): ω-Cats → Set and B... |

34 |
Limits indexed by category-valued 2-functors
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(Show Context)
Citation Context ... ω-categories and homomorphisms is the co-Kleisli category of the comonad Q on ω-Cats. There is, in fact, another description of the comonad Q, one given using computads. Computads were introduced in =-=[22]-=- as a tool for presenting free higher-dimensional categories, and have been studied extensively in the context of strict ω-categories under the name of polygraph: see [6, 20]. For the weak ω-categorie... |

32 | Operads in higher-dimensional category theory
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(Show Context)
Citation Context ...nary composites are supplied. Again, from the latter we can derive the former; but again, in a non-canonical way that is determined only up to isomorphism. In recognition of this similarity, we adopt =-=[18]-=-’s terminology here, referring to the homomorphisms of Definition 3 as unbiased homomorphisms, and to those of [12, Section 3.3] as biased homomorphisms. Our goal in the remainder of this section is t... |

18 |
An algebraic theory of tricategories
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- 2006
(Show Context)
Citation Context ... = [˙e(β ⊗ α)] ; • If γ = 1f : f ⇒ f for some f : x → y, then we take ργ = Uf; • If γ = lf, rf or afgh, then we take ργ = Lf, Rf or Afgh respectively; • If γ = l � f : f ⇒ Iy ⊗ f—where we recall from =-=[12]-=- that such a 2-cell participates in a specified adjoint equivalence (ηf,ǫf) with lf—then we obtain ργ as follows. First we define a 3-cell ˜ηf : 1f ⇛ [l˙ef] ◦ [l� ˙ef ] as the composite (5) 1f Uf [η˙e... |

14 |
A Quillen model structure for bicategories
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(Show Context)
Citation Context ... OF HIGHER CATEGORIES 3 In this paper, we adopt a quite different means of generating a suitable comonad on the category of strict homomorphisms, one informed by categorical homotopy theory. Lack, in =-=[16]-=-, establishes that the comonad on Bicats generated by the adjunction in (1) gives a notion of cofibrant replacement for a certain Quillen model structure on Bicats; whose generating cofibrations are t... |

13 | Natural weak factorization systems
- Grandis, Tholen
- 2006
(Show Context)
Citation Context ...(Q,ǫ,∆) on C whose counit morphisms ǫY : QY → Y provide cofibrant replacements for each Y ∈ C. Such comonads were first studied in [21], but are best understood with reference to the natural wfs’s of =-=[11]-=-; which are wfs’s equipped with a cofibrant replacement comonad on each coslice and an acyclic fibrant replacement monad on each slice, subject to compatibility axioms. Recall that a locally finitely ... |

10 | Computads for finitary monads on globular sets, Higher category theory
- Batanin
- 1997
(Show Context)
Citation Context ...extensively in the context of strict ω-categories under the name of polygraph: see [6, 20]. For the weak ω-categories under consideration here, we have the following notion of computad due to Batanin =-=[1]-=-. In the definition, we make use of the functors En := ω-Cats(2n,–): ω-Cats → Set and Bn := ω-Cats(∂n,–): ω-Cats → Set and the natural transformation ρn := ω-Cats(ιn,–): En ⇒ Bn. 4.2. Definition. For ... |

7 | Co-rings over operads characterize morphisms
- Hess, Parent, et al.
(Show Context)
Citation Context ...blem of finding a suitable notion of homomorphism in terms of that of finding a suitable comonad on the category of strict homomorphisms. One technique for solving this latter problem is suggested in =-=[13]-=-. For this we must suppose the category HCats to be presentable as the category of algebras for a symmetric operad O on a suitable base category V; and may then consider co-rings over the operad O—the... |

6 | Algebras of higher operads as enriched categories. Applied Categorical Structures
- Batanin, Weber
- 2011
(Show Context)
Citation Context ...onad morphism κ: K → T, where T is the monad for strict ω-categories, and where to call κ cartesian is to assert that all of its naturality squares are pullbacks. By Lemma 6.8 and Proposition 6.11 of =-=[3]-=-, any given monad K admits at most one such augmentation κ, so that for a monad to be a globular operad is a property, not extra structure. 3 Since the identity monad on ˆ G is a globular operad, it i... |

5 |
2-nerves for bicategories
- Lack, Paoli
(Show Context)
Citation Context ... subclass of the tritransformations, those whose 1- and 2-cell components are all identity maps: these are the tricategorical icons 2 of [9], themselves a generalisation of the bicategorical icons of =-=[17]-=-. Since the 1- and 2-dimensional data for a tricategorical icon is trivial, it may be specified purely in terms of a collection of 3-cells satisfying axioms; and it is this which allows us to equip th... |

4 |
Understanding the small object argument. Applied categorical structures
- Garner
(Show Context)
Citation Context ...f the basic n-dimensional boundaries into the basic n-dimensional cells. For the general case, we can run this argument backwards: given a Quillen model structure on HCats, we can—by the machinery of =-=[8]-=-— use it to generate a “cofibrant replacement comonad”, and so obtain a notion of homomorphism. In fact, to generate a cofibrant replacement comonad we do not need a full model structure on HCats, but... |

3 | Cofibrance and Completion
- Radulescu-Banu
- 1999
(Show Context)
Citation Context ... cofibrant replacement comonad for a wfs on C, we mean a comonad Q = (Q,ǫ,∆) on C whose counit morphisms ǫY : QY → Y provide cofibrant replacements for each Y ∈ C. Such comonads were first studied in =-=[21]-=-, but are best understood with reference to the natural wfs’s of [11]; which are wfs’s equipped with a cofibrant replacement comonad on each coslice and an acyclic fibrant replacement monad on each sl... |

2 |
Cofibrant objects among higher-dimensional categories
- Métayer
(Show Context)
Citation Context ...Computads were introduced in [22] as a tool for presenting free higher-dimensional categories, and have been studied extensively in the context of strict ω-categories under the name of polygraph: see =-=[6, 20]-=-. For the weak ω-categories under consideration here, we have the following notion of computad due to Batanin [1]. In the definition, we make use of the functors En := ω-Cats(2n,–): ω-Cats → Set and B... |

1 |
The low-dimensional structures tricategories form
- Garner, Gurski
- 2008
(Show Context)
Citation Context ...omomorphisms it yields may be composed in a strictly associative and unital fashion. Even in dimension three this is an unusual feature, not shared by the trihomomorphisms of [10]: which, as shown in =-=[9]-=-, form only a bicategory. This last observation may lead us to question whether our notion of homomorphism is sufficiently weak. In order to show that it is, we devote Section 3 to a demonstration tha... |

1 | Are operads algebraic theories? The
- Leinster
(Show Context)
Citation Context ...nction. Because ω-Cats is finitarily monadic over the locally finitely presentable category ˆ G, it is itself locally 3 Note that this is by contrast with the situation for plain operads, as noted in =-=[19]-=-. K ⊥ V ˆ GHOMOMORPHISMS OF HIGHER CATEGORIES 23 finitely presentable; and so in order to apply the machinery of Section 2, it remains only to distinguish in ω-Cats a set of maps describing the basic... |