## A HOMOTOPY-THEORETIC UNIVERSAL PROPERTY OF LEINSTER’S OPERAD FOR WEAK ω-CATEGORIES (2008)

Citations: | 2 - 1 self |

### BibTeX

@MISC{Garner08ahomotopy-theoretic,

author = {Richard Garner},

title = { A HOMOTOPY-THEORETIC UNIVERSAL PROPERTY OF LEINSTER’S OPERAD FOR WEAK ω-CATEGORIES},

year = {2008}

}

### OpenURL

### Abstract

### Citations

327 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ... “weakened” version of that same structure. The picture is as follows: we begin with a category C equipped with a notion of higher-dimensionality coming from a model structure in the sense of Quillen =-=[14]-=-. We now contemplate some notion of algebraic theory on C: monads, operads, or Lawvere theories on C, for example. These algebraic theories themselves form a category Th(C), and by making use of vario... |

181 |
Model categories and their localizations
- Hirschhorn
(Show Context)
Citation Context .... Primary: 18D50, 55U35, 18D05. Supported by a Research Fellowship of St John’s College, Cambridge and a Marie Curie Intra-European Fellowship, Project No. 040802. 12 RICHARD GARNER Reedy category D =-=[9, 16]-=-, in which case we have canonical notions of both cell (the representable presheaves) and boundary (arising from the Reedy structure). Yet this rather appealing construction has a problem, which arise... |

143 |
Model categories, Mathematical surveys and monographs
- Hovey
- 1998
(Show Context)
Citation Context ...weak factorisation systems is the following result, first proved by Quillen in the finitary case [14, Chapter II, §3] and in the following transfinite form by Bousfield [4]. For a modern account, see =-=[10]-=-, for example. 2.1.1. Proposition (The small object argument). Let C be a locally presentable category, and let I be a set of maps in C. Define classes of maps I ↓ and I ↓↑ by I ↓ := { p ∈ C 2 j ⋔ p f... |

110 |
Monoidal globular categories as a natural environment for the theory of weak n-categories
- Batanin
- 1998
(Show Context)
Citation Context ...iversal cofibrant replacements, and then illustrate their utility by means of an example drawn from the study of weak ω-categories. More specifically, we consider Batanin’s theory of globular operads =-=[1]-=-, and by using the machinery outlined above, obtain a canonical and universal notion of cofibrant replacement for globular operads. We then show that applying this to the globular operad for strict ω-... |

45 |
A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so
- Kelly
(Show Context)
Citation Context ...ransformation Λ ′: idC2 ⇒ R ′. We now obtain the monad part R of the desired algebraic realisation as the free monad on the pointed endofunctor (R′,Λ ′). We may construct this using the techniques of =-=[12]-=-. To obtain the comonad part L we proceed as follows. The assignation f ↦→ λ ′ f underlies a functor L′ : C2 → C2 ; and a little manipulation shows that this functor in turn underlies a comonad L ′ on... |

38 |
Distributive laws, in: Seminar on Triples and Categorical Homology Theory
- Beck
(Show Context)
Citation Context ...es a monad on C 2 , and L = (L,Φ,Σ) a comonad; and that the natural transformation ∆: LR ⇒ RL: C 2 → C 2 with components Y πf idY Kf Y ρf Kf σf X ∆f = λρ f ρλ f Kρf πf Kf describes a distributive law =-=[3]-=- between L and R. Under these circumstances, we will say that (L,R) is an algebraic realisation of (L, R). The pairs (L,R) arising in this way are the natural weak factorisation systems of [8]. Though... |

32 | Operads in higher-dimensional category theory
- Leinster
- 2000
(Show Context)
Citation Context ... and universal notion of cofibrant replacement for globular operads. We then show that applying this to the globular operad for strict ω-categories yields precisely the operad singled out by Leinster =-=[13]-=- as the operad for weak ω-categories. 2. Weak factorisation and cofibrant replacement 2.1. Weak factorisation systems. A weak factorisation system [4] (L, R) on a category C is given by two classes L ... |

22 | Adjoint lifting theorems for categories of algebras - Johnstone - 1975 |

17 |
Homotopy theory of model categories. Unpublished manuscript, available electronically from http://www-math.mit.edu/~psh/#Reedy
- Reedy
(Show Context)
Citation Context .... Primary: 18D50, 55U35, 18D05. Supported by a Research Fellowship of St John’s College, Cambridge and a Marie Curie Intra-European Fellowship, Project No. 040802. 12 RICHARD GARNER Reedy category D =-=[9, 16]-=-, in which case we have canonical notions of both cell (the representable presheaves) and boundary (arising from the Reedy structure). Yet this rather appealing construction has a problem, which arise... |

14 | Quillen closed model structures for sheaves
- Crans
- 1995
(Show Context)
Citation Context ...echnique allows us to lift a cofibrantly generated weak factorisation system along a right adjoint functor. This process was first described in the general context of model categories by Sjoerd Crans =-=[6]-=-.A HOMOTOPY-THEORETIC UNIVERSAL PROPERTY... 9 2.4.2. Proposition. Let (L, R) be a cofibrantly generated w.f.s. on C, and let F ⊣ G: D → C with D locally presentable. Then there is a w.f.s. (L ′ , R ′... |

13 | Natural weak factorization systems
- Grandis, Tholen
- 2006
(Show Context)
Citation Context ...uction; but in practice, this would require some rather strange combinatorics of a nature entirely orthogonal to that of the mathematics one was trying to do. However, recent work of Grandis & Tholen =-=[8]-=- and the author [7] suggests a solution to this problem. Using the results of [7], we may equip any reasonable (which is to say, cofibrantly generated) weak factorisation system with a canonical and u... |

11 |
Bousfield, Constructions of factorization systems in categories
- K
- 1977
(Show Context)
Citation Context ...we need is a category C, a notion of algebraic structure, and a notion of higher-dimensionality; and for this last, we do not even need a full model structure on C. A single weak factorisation system =-=[4]-=- will do, and for a sufficiently well behaved (typically, locally presentable) C we may obtain this by using the small object argument of Quillen [14] and Bousfield [4]: for which it suffices to speci... |

6 | Algebras of higher operads as enriched categories. Applied Categorical Structures
- Batanin, Weber
- 2011
(Show Context)
Citation Context ...ised globular operads; this is the full subcategory of GOpd whose objects are those globular operads with O(⋆) a singleton. The restriction to normalised globular operads also plays a central role in =-=[2]-=-. 3.2.4. The candidate theory. The fourth and final ingredient we require for our machinery is a theory T ∈ NGOpd which we wish to weaken. We take this to be the terminal globular operad T given by T ... |

4 |
Understanding the small object argument. Applied categorical structures
- Garner
(Show Context)
Citation Context ...tice, this would require some rather strange combinatorics of a nature entirely orthogonal to that of the mathematics one was trying to do. However, recent work of Grandis & Tholen [8] and the author =-=[7]-=- suggests a solution to this problem. Using the results of [7], we may equip any reasonable (which is to say, cofibrantly generated) weak factorisation system with a canonical and universal notion of ... |

3 | Monad interleaving: a construction of the operad for Leinster’s weak ω-categories
- Cheng
- 2005
(Show Context)
Citation Context ...hus we will be done if we can show that the initial object (L,λ) of OWC lies in this reflective subcategory; in other words, if we can show that L is normalised. But this is known to be the case: see =-=[5]-=-, for example. □ 3.4.2. Remark. Note that the restriction to normalised globular operads is vital for the above proof to go through. Indeed, were we to take the universal cofibrant replacement for T i... |

3 | Cofibrance and Completion
- Radulescu-Banu
- 1999
(Show Context)
Citation Context ...nts underlie a cofibrant replacement comonad Q = (Q,ǫ,δ) which is the restriction and corestriction of L to the coslice under 0. The concept of a cofibrant replacement comonad was first considered by =-=[15]-=-, though it should be noted that the comonads constructed there do not coincide with the ones obtained from Proposition 2.3.1. Indeed, they are built using the small object argument, and so suffer fro... |

1 | factorization algebras, Journal of Pure and Applied Algebra 175 - Rosick´y, Tholen, et al. |