Weak complicial sets I. Basic homotopy theory

• Autores: D. R. B. Verity
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 219, Nº 4, 2008, págs. 1081-1149
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.06.003
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• Resumen

  • This paper develops the foundations of a simplicial theory of weak ?-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) ?-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 ?-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 ?-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.