# Could we have the simplicial definition of equivariant derived category of sheaves with arrow direction inversed?

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?