Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.

Let $G$ be a topological group which act on $X$ continuously from the left. 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 $$ s_0(g_1,\ldots, g_n,x)=(e,g_1,\ldots, g_n,x); $$ $$ s_i(g_1,\ldots, g_n,x)=(g_1,\ldots, g_i,e , g_{i+1},\ldots, g_nx),~1\leq i\leq n-1; $$ $$ s_n(g_1,\ldots, g_n,x)=(g_1,\ldots, g_n,e,x). $$

A $G$-equivariant sheaf on $X$ is a pair $(\mathcal{F},\theta)$, where $\mathcal{F}\in \text{Sh}(X)$ and $\theta$ is an isomorphism $$ \theta: d_0^*\mathcal{F}\overset{\sim}{\to} d_1^*\mathcal{F}, $$ satisfying the cocycle condition $$ d_2^*\theta\circ d_0^*\theta=d_1^*\theta, \text{ and } s_0^*\theta=\text{id}_{\mathcal{F}}. $$ We denote the category of $G$-equivariant sheaves on $X$ by $\text{Sh}_G(X)$. It is clear that $\text{Sh}_G(X)$ is an abelian category and the forgetful functor For$: \text{Sh}_G(X)\to \text{Sh}(X)$ is exact.

I am interested in the injective objects in $\text{Sh}_G(X)$. We know that in general, $\text{For}: \text{Sh}_G(X)\to \text{Sh}(X)$ does not preserve injective objects, unless $G$ is discrete.

Here are my questions

In the case that $G$ is discrete, $\mathcal{F}$ is injective as a sheaf is a necessary but not sufficient condition that $(\mathcal{F},\theta)$ is an injective object in $\text{Sh}_G(X)$. Then what are additional requirements to make sure that $(\mathcal{F},\theta)$ is injective?

In then general case that $G$ is not discrete, what are injective objects in $\text{Sh}_G(X)$?


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