### Abstract

This paper develops the foundations of a simplicial theory of weak ω-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ω-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict ω-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to study the weak ω-category theory of these structures in a series of companion papers. In particular, we establish their basic homotopy theory by constructing a Quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets. As a simple corollary of this work we provide an independent construction of Joyal's model structure on simplicial sets for which the fibrant objects are the quasi-categories.

Language | English |
---|---|

Pages | 1081-1149 |

Number of pages | 69 |

Journal | Advances in Mathematics |

Volume | 219 |

Issue number | 4 |

DOIs | |

Publication status | Published - 10 Nov 2008 |

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*Advances in Mathematics*,

*219*(4), 1081-1149. https://doi.org/10.1016/j.aim.2008.06.003

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*Advances in Mathematics*, vol. 219, no. 4, pp. 1081-1149. https://doi.org/10.1016/j.aim.2008.06.003

**Weak complicial sets I. Basic homotopy theory.** / Verity, D. R B.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Weak complicial sets I. Basic homotopy theory

AU - Verity, D. R B

PY - 2008/11/10

Y1 - 2008/11/10

N2 - This paper develops the foundations of a simplicial theory of weak ω-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ω-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict ω-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to study the weak ω-category theory of these structures in a series of companion papers. In particular, we establish their basic homotopy theory by constructing a Quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets. As a simple corollary of this work we provide an independent construction of Joyal's model structure on simplicial sets for which the fibrant objects are the quasi-categories.

AB - This paper develops the foundations of a simplicial theory of weak ω-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) ω-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict ω-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to study the weak ω-category theory of these structures in a series of companion papers. In particular, we establish their basic homotopy theory by constructing a Quillen model structure on the category of stratified simplicial sets whose fibrant objects are the weak complicial sets. As a simple corollary of this work we provide an independent construction of Joyal's model structure on simplicial sets for which the fibrant objects are the quasi-categories.

UR - http://www.scopus.com/inward/record.url?scp=51349167155&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2008.06.003

DO - 10.1016/j.aim.2008.06.003

M3 - Article

VL - 219

SP - 1081

EP - 1149

JO - Advances in Mathematics

T2 - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 4

ER -