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Let $X$ be a topological space and $G$ be a topological group acting on $X$ from the left. We consider the simplicial space $[G\backslash X]_{\cdot}$ where $$ [G\backslash X]_n=\underbrace{G\times \ldots \times G}_{n\text{ copies of }G\text{'s}}\times X $$ with structural maps $$ d_0(g_1,\ldots, g_n,x)=(g_2,\ldots,g_n,g_1^{-1}x); $$ $$ d_i(g_1,\ldots, g_n,x)=(g_1,\ldots, g_ig_{i+1},\ldots, g_n,x), ~1\leq i\leq n-1; $$ $$ d_n(g_1,\ldots, g_n,x)=(g_1,\ldots, g_{n-1},x); $$ and $$ s_0(g_1,\ldots, g_n,x)=(e,g_1,\ldots, g_n,x); $$ $$ s_i(g_1,\ldots, g_n,x)=(g_1,\ldots, g_i,e , g_{i+1},\ldots, g_nx),~1\leq i\leq n-1; $$ $$ s_n(g_1,\ldots, g_n,x)=(g_1,\ldots, g_n,e,x). $$

According to the Notes by Zhiwei Yun, a simplicial sheaf on $[G\backslash X]_{\cdot}$ is a collection of sheaves $\mathcal{F}_n$ on each $[G\backslash X]_n$ and a collection of structure morphisms $\theta_h: h^*\mathcal{F}_n\to \mathcal{F}_m$ for each structure map $h: [G\backslash X]_m\to [G\backslash X]_n$ satisfies $$ \theta_{h^{\prime}h}=\theta_h\circ h^*(\theta_{h^{\prime}}). $$ We call $\mathcal{F}$ Cartesian if $\theta_h$ is an isomorphism for each $h$.

We consider the bounded derived category of simplicial sheaves $D^b([G\backslash X]_{\cdot})$ on $[G\backslash X]_{\cdot}$. Then the equivariant derived category of sheaves $D^b_{eq}([G\backslash X]_{\cdot})$ is defined to be the full subcategory of $D^b([G\backslash X]_{\cdot})$ consisting of complexes of simplicial sheaves such that their cohomologies are Cartisian simplicial sheaves.

Now my question is: could we define the simplicial sheaf to be a collection of sheaves $\mathcal{F}_n$ on each $[G\backslash X]_n$ and a collection of structure morphisms $\theta_h:\mathcal{F}_m \to h^*\mathcal{F}_n$ for each structure map $h: [G\backslash X]_m\to [G\backslash X]_n$ satisfies $$ \theta_{h^{\prime}h}= h^*(\theta_{h^{\prime}})\circ \theta_h? $$ If we defined the equivariant derived category as the full subcategory with Cartesian cohomologies, then is it equivalent to the previous definition?

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