One can possibly enlarge the simplex category to a one of the several interesting categories in homological algebra like cycle category of Connes and the skeletal category of finite sets. Such examples can be founded axiomatically as kind of extensions of the simplex category by a group with special properties. The resulting notion is a skew-simplicial group (synonym: crossed simplicial group): the presheaves on a skew-simplicial group are skew-simplicial sets of the corresponding kind. In addition to cyclic and symmetric simplicial sets other important applications are dihedral, quaternionic and hyperoctahedral sets/homology (cf. Loday MR89e:18024). The formalism has been found by Krasauskas and a bit later and independently also by Fiedorowicz and Loday.

Z. Fiedorowicz, Jean-Louis Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), no. 1, 57–87, MR91j:18018, doi

A connection to geometry of surfaces is sketched in

R. Krasauskas, Crossed simplicial groups of framed braids and mapping class groups of surfaces, Lith. Math. J. 36, 263–281 (1996) doi

A recent treatise on connection to geometry is

Tobias Dyckerhoff, Mikhail Kapranov, Crossed simplicial groups and structured surfaces, arxiv/1403.5799 In: Stacks and Categories in Geometry, Topology, and Algebra. Contemporary Mathematics 643, 37–110, Amer. Math. Soc. (2015) doi

They also play a major role in

S. Balchin, Homotopy of planar Lie group equivariant presheaves, J. Homotopy Relat. Struct. 13, 555–580 (2018) doi

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