## A folk model structure on omega-cat (2009)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Lafont09afolk,

author = {Yves Lafont and François Métayer and Krzysztof Worytkiewicz},

title = {A folk model structure on omega-cat},

year = {2009}

}

### OpenURL

### Abstract

The primary aim of this work is an intrinsic homotopy theory of strict ω-categories. We establish a model structure on ωCat, the category of strict ω-categories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while free objects are cofibrant. We further exhibit model structures of this type on n-categories for arbitrary n ∈ N, as specialisations of the ω-categorical one along right adjoints. In particular, known cases for n = 1 and n = 2 nicely fit into the scheme.

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Citation Context ...lobular sets preserving compositions and units. Thus, ω-categories and ω-functors build the category ωCat, which is our main object of study. The forgetful functor U : ωCat → Glob is finitary monadic =-=[2]-=- and Glob is a topos of presheaves on a small category: therefore ωCat is complete and cocomplete. On the other hand, the left adjoint to U takes a globular set to the free ω-category it generates. In... |

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Citation Context ...d Remark 2. If I ⊆ I ′ , then I−inj ⊇ I ′ −inj and J−inj = (J−cof)−inj. It is easy to see that I−cell ⊆ I−cof. The next proposition recalls standard formal properties of the classes just defined (see =-=[8]-=-). Proposition 3. I−inj as well as I−cof contain all identities. I−inj is closed under composition and pullback while I−cof is closed under retract, transfinite composition and pushout. We may now sta... |

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Citation Context ...presentable. Finally, we say that a cocomplete category is locally presentable if there is a regular cardinal witnessing this fact. Definition 1 is equivalent to the original one by Gabriel and Ulmer =-=[7]-=-. It proves especially powerful to establish factorisation results, when combined with the small object argument (Section 2.2). Let β be a regular cardinal. Recall that a β-colimit is a colimit of a f... |

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Citation Context ...ot (UMR 7126 du CNRS) - Laboratoire Preuves, Programmes et Systèmes ‡ Akademia Górniczo-Hutnicza Kraków - Katedra Informatyki 1�� �� � �� � �� � �� construction is based on a theorem by J.Smith (see =-=[3]-=-): under some fairly standard assumptions on the underlying category, conditions (S1) W has the 3 for 2 property and is stable under retracts; (S2) I−inj ⊆ W; (S3) I−cof ∩ W is closed under pushouts a... |

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Citation Context ...es for n = 1 and n = 2 nicely fit into the scheme. 1 Introduction hal-00395898, version 1 - 16 Jun 2009 1.1 Background and motivations The origin of the present work goes back to the following result =-=[1, 24]-=-: if a monoid M can be presented by a finite, confluent and terminating rewriting system, then its third homology group H3(M) is of finite type. The finiteness property extends in fact to all dimensio... |

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Citation Context ...es for n = 1 and n = 2 nicely fit into the scheme. 1 Introduction hal-00395898, version 1 - 16 Jun 2009 1.1 Background and motivations The origin of the present work goes back to the following result =-=[1, 24]-=-: if a monoid M can be presented by a finite, confluent and terminating rewriting system, then its third homology group H3(M) is of finite type. The finiteness property extends in fact to all dimensio... |

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Citation Context ...bjects support a structure of ω-category in a very direct way: this was the starting point of [19], which introduces a notion of resolution for ω-categories, based on computads [26, 21] or polygraphs =-=[6]-=-, the terminology we adopt here. Recall that a polygraph S consists of sets of cells of all dimensions, determining a freely generated ω-category S ∗ . A resolution of an ω-category C by a polygraph S... |

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Citation Context ...ly observes that both objects support a structure of ω-category in a very direct way: this was the starting point of [19], which introduces a notion of resolution for ω-categories, based on computads =-=[26, 21]-=- or polygraphs [6], the terminology we adopt here. Recall that a polygraph S consists of sets of cells of all dimensions, determining a freely generated ω-category S ∗ . A resolution of an ω-category ... |

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Citation Context ...ly observes that both objects support a structure of ω-category in a very direct way: this was the starting point of [19], which introduces a notion of resolution for ω-categories, based on computads =-=[26, 21]-=- or polygraphs [6], the terminology we adopt here. Recall that a polygraph S consists of sets of cells of all dimensions, determining a freely generated ω-category S ∗ . A resolution of an ω-category ... |

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Citation Context ...f a monoid M can be presented by a finite, confluent and terminating rewriting system, then its third homology group H3(M) is of finite type. The finiteness property extends in fact to all dimensions =-=[14]-=-, but the above theorem may also be refined in another direction: the same hypothesis implies that M has finite derivation type [25], a property of homotopical nature. We claim that these ideas are be... |

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Citation Context ...ite type. The finiteness property extends in fact to all dimensions [14], but the above theorem may also be refined in another direction: the same hypothesis implies that M has finite derivation type =-=[25]-=-, a property of homotopical nature. We claim that these ideas are better expressed in terms of ω-categories (see [10, 11, 17]). Thus we work in the category ωCat, whose objects are the strict ω-catego... |

17 |
Homotopical algebra, volume 43
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(Show Context)
Citation Context ...on 2Cat in a similar spirit (see [15, 16]). Incidentally, there is also a quite different, Thomason-like, model structure on 2Cat (see [27]). Its generalisation to ωCat remains an open problem. Since =-=[22]-=-, the notion of model structure has been gradually recognized as the appropriate abstract framework for developing homotopy theory in a category C: it consists in three classes of morphisms, weak equi... |

14 | A Quillen model structure for bicategories - Lack |

13 | Left-determined model categories and universal homotopy theories
- Rosick´y, Tholen
(Show Context)
Citation Context ...remark 5; − condition (S3) holds by corollary 1 and corollary 5; − condition (S4) holds by corollary 6. ⊳ Remark 7. By corollary 3, the model structure of theorem 3 is left-determined in the sense of =-=[23]-=-. hal-00395898, version 1 - 16 Jun 2009 5 Fibrant and cofibrant objects Recall that, given a model category C, an object X of C is fibrant if the unique morphism !X : X → 1 is a fibration. Dually, X i... |

12 | Understanding the small object argument
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(Show Context)
Citation Context ...ntable and let I be a set of morphisms of C. Then (I−cell, I−inj) is a functorial factorisation. Proof. The required factorisation is produced by the “small object argument”, due to Quillen (see also =-=[9]-=- for an extensive discussion): − For any f in C → , let Sf be the set of morphisms of C → with domain in I and codomain f, that is ♦ Sf = {s = (us, vs) ∈ C → | dom(s) = is ∈ I, cod(s) = f}. We get a f... |

9 |
A Quillen model structure for 2-categories
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(Show Context)
Citation Context ...nd Section 5 below). On the other hand, our model structure generalizes in a very precise sense the “folk” model structure on Cat (see [13]) as well a model structure on 2Cat in a similar spirit (see =-=[15, 16]-=-). Incidentally, there is also a quite different, Thomason-like, model structure on 2Cat (see [27]). Its generalisation to ωCat remains an open problem. Since [22], the notion of model structure has b... |

7 | The three dimensions of proofs
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(Show Context)
Citation Context ...n another direction: the same hypothesis implies that M has finite derivation type [25], a property of homotopical nature. We claim that these ideas are better expressed in terms of ω-categories (see =-=[10, 11, 17]-=-). Thus we work in the category ωCat, whose objects are the strict ω-categories and the morphisms are ω-functors (see Section 3). In fact, when considering the interplay between the monoid itself and ... |

6 |
Algebra and geometry of rewriting
- Lafont
(Show Context)
Citation Context ...n another direction: the same hypothesis implies that M has finite derivation type [25], a property of homotopical nature. We claim that these ideas are better expressed in terms of ω-categories (see =-=[10, 11, 17]-=-). Thus we work in the category ωCat, whose objects are the strict ω-categories and the morphisms are ω-functors (see Section 3). In fact, when considering the interplay between the monoid itself and ... |

5 | Two polygraphic presentations of Petri nets
- Guiraud
(Show Context)
Citation Context ...n another direction: the same hypothesis implies that M has finite derivation type [25], a property of homotopical nature. We claim that these ideas are better expressed in terms of ω-categories (see =-=[10, 11, 17]-=-). Thus we work in the category ωCat, whose objects are the strict ω-categories and the morphisms are ω-functors (see Section 3). In fact, when considering the interplay between the monoid itself and ... |

5 | A model structure à la Thomason on 2-Cat
- Worytkiewicz, Hess, et al.
(Show Context)
Citation Context ...olk” model structure on Cat (see [13]) as well a model structure on 2Cat in a similar spirit (see [15, 16]). Incidentally, there is also a quite different, Thomason-like, model structure on 2Cat (see =-=[27]-=-). Its generalisation to ωCat remains an open problem. Since [22], the notion of model structure has been gradually recognized as the appropriate abstract framework for developing homotopy theory in a... |

4 |
Higher Dimensional Word Problem, Category Theory and Computer
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(Show Context)
Citation Context ...ntroduced in [26], freely generating a 2-category hal-00395898, version 1 - 16 Jun 2009 S∗ 0 S∗ 1 τ0 σ0 This pattern has been extended to all dimensions, giving rise to n-computads [21] or polygraphs =-=[5, 6]-=-. More precisely, let nGlob (resp. nCat) denote the category of n-globular sets (resp. n-categories), we get a commutative diagram (n+1)Cat �� (n+1)Glob (4) Un nCat σ1 τ1 S ∗ 2 � nGlob where the horiz... |

4 |
Homotopical Algebra, volume 43 of Lecture Notes in Mathematics
- Quillen
- 1967
(Show Context)
Citation Context ...on 2Cat in a similar spirit (see [15, 16]). Incidentally, there is also a quite different, Thomason-like, model structure on 2Cat (see [27]). Its generalisation to ωCat remains an open problem. Since =-=[22]-=-, the notion of model structure has been gradually recognized as the appropriate abstract framework for developing homotopy theory in a category C: it consists in three classes of morphisms, weak equi... |

2 |
Resolutions by polygraphs. Theory and
- Métayer
(Show Context)
Citation Context ... itself and the space of computations attached to any presentation of it, one readily observes that both objects support a structure of ω-category in a very direct way: this was the starting point of =-=[19]-=-, which introduces a notion of resolution for ω-categories, based on computads [26, 21] or polygraphs [6], the terminology we adopt here. Recall that a polygraph S consists of sets of cells of all dim... |

2 |
Cofibrant objects among higher-dimensional categories
- Métayer
(Show Context)
Citation Context ...wise by using the universal property of the functors Ln. Y S∗ ��� v � � � �� Z u Thus freely generated ω-categories are cofibrant. The proof of the converse is much harder, and is the main purpose of =-=[20]-=-. The problem reduces to the fact that the full subcategory of ωCat whose objects are free on polygraphs is Cauchy complete, meaning that its idempotent morphisms split. p The results of [19] may be r... |

1 |
Sheafifiable homotopy model categories.II
- Beke
- 2001
(Show Context)
Citation Context ...tes a model structure on nCat, in which the weak equivalences are the n-functors f such that F (f) ∈ W, and (G(ik))k∈N is a family of generating cofibrations. The general situation is investigated in =-=[4]-=-, whose proposition 2.3 states sufficient conditions for the transport of a model structure along an adjunction. In our particular case, these conditions boil down to the following: hal-00395898, vers... |

1 |
Strong stacks and classifying spaces
- AMS
- 1999
(Show Context)
Citation Context ... 3 for 2 property if whenever h = g ◦ f and any two out of the three morphisms f, g, h belong to A, then so does the third. We now recall the basics of model structures, following the presentation of =-=[12]-=-. Definition 2. A model structure on a complete and cocomplete category C is given by three classes of morphisms, the class C of cofibrations, the class F of fibrations, and the class W of weak equiva... |

1 | Polygraphic resolutions and homology of monoids
- Lafont, Métayer
(Show Context)
Citation Context ...rs of X. The correspondence Γ turns out to be functorial and endowed with natural transformations from and to the identity functor. Reversible cylinders are in fact cylinders in the sense of [19] and =-=[18]-=-, satifying an additional reversibility condition. In the present work, “cylinder” means “reversible cylinder”, as the general case will not occur. Definition 9. By induction on n, we define the notio... |