Weak complicial sets I. Basic homotopy theory

D. R B Verity*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

28 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1081-1149
Number of pages69
JournalAdvances in Mathematics
Issue number4
Publication statusPublished - 10 Nov 2008


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