Weak complicial sets I. Basic homotopy theory

By Dominic R. Verity. 2008

98 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/1c82402599bcc8bf2e2a55e66510d08e8405f609

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

In Neurology

Citations in the corpus

∞-Categories for the Working Mathematician (2018, background): “19 (iii) The 1-complicial sets of [21], equivalently the “naturally marked quasi-categories” of [13], define the objects of an ∞-cosmos 1-Comp in which the isofibrations, equivalences and trivial fibrations are the corresponding classes of the model structure from either of these sources.”
Fibrations and Yoneda's lemma in an ∞-cosmos (2017, background): “Specifically the axioms laid out above follow from standard results used in the construction of Joyal’s model structure on simplicial sets (see [8], [27], or I.”
On the Unicity of the Homotopy Theory of Higher Categories (2011, background): “le, be compared using our axioms. We therefore end with the following. 44 CLARK BARWICK AND CHRISTOPHER SCHOMMER-PRIES Conjecture 15.13. The ∞-categoryunderlyingVerity’s n-trivial weakcomplicial sets [37, 38] is a homotopy theory of (∞,n)-categories. The relative category consisting of Batanin’s ω-c”