  This article presents a novel approach to construct a model category
structure designed to model the homotopy theory of spaces equipped with an
action by the group $C_2$, where morphisms are considered to be isovariant. Our
methodology centers on simplicial techniques. We replace the conventional
simplex category $\Delta$ with a modified category $C_2 \Delta$ and then delve
into the study of presheaves of sets on $C_2 \Delta$. To establish the model
category structure we employ Cisinski's methods for model structures in
categories of presheaves. In particular, we use an analogous idea to the one
employed by Cisinski and Moerdjik in the construction of a model category
structure for Dendroidal Sets (arXiv:0902.1954). Our approach distinguishes
itself from prior work, such as Yeakel's (arXiv:1907.12135) which primarily
focuses on a more topological context and, brings a new perspective to the
study of isovariant homotopy theory for $C_2$-spaces.
