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Where the Idea-section mentions “$Q_{n+1}^{op}$” it should say what these symbols refers to. I added a link to quaternion group.
The grammar of the first sentence in the Definition-section seemed broken. I have tried to fix it, but please check if it is as intended.
In the Definition (here) I suppose you mean to say that the subcategory which $\Delta$ forms is (non-full and) wide (to make clear that there are no further objects)? I have made the edit, but please check.
Also I have replaced “every morphism is a product of” by “every morphism can be factored as”.
Added references to
{#Spalinski95}Jan Spaliński, Homotopy theory of dihedral and quaternionic sets, Topology 39 (1995) pp.35-52.
{#Balchin18}Scott Balchin, Homotopy of planar Lie group equivariant presheaves, J. Homotopy Relat. Struct. 13 (2018) pp.555-580. (doi)
as well as some minor terminological fine(r) tuning.
Mentioned that $\Delta Q$ is an Eilenberg-Zilber category.
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