Higher fundamental functors for simplicial sets (Q2752243)

The homotopy theory for simplicial complexes developed by the author [\textit{M. Grandis}, Appl. Categ. Struct. 10, No.~2, 99-155 (2002; Zbl 0994.55011)] is extended to symmetric simplicial sets. (A symmetric simplicial set is a contravariant set-valued functor on the category of finite positive cardinals with all mappings as opposed to a simplicial set which is a contravariant set-valued functor on the category of finite positive ordinals with monotone maps.) This allows one to construct higher fundamental groupoids for symmetric simplicial sets as left adjoints of suitable nerve functors which implies preservation of all colimits, a strong, simple version of the Seifert-van Kampen property. A similar theory is developed for directed homotopy of simplicial sets giving rise to higher fundamental categories as left adjoints.