The question is related to this one.

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 $$ d_2^*\theta\circ d_0^*\theta=d_1^*\theta, \text{ and } s_0^*\theta=\text{id}_{\mathcal{F}}. $$

A morphism $\phi: (\mathcal{F},\theta)\to (\mathcal{G},\xi)$ is a sheaf morphism $\phi: \mathcal{F}\to \mathcal{G}$ that commute with $\theta$ and $\xi$. 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.

An equivariant sheaf $(\mathcal{I},\eta)$ is called injective if for any monomorphsim of equivariant sheaves $\phi: (\mathcal{F},\theta)\to (\mathcal{G},\xi)$ and any morphism of equivariant sheaves $\psi: (\mathcal{F},\theta)\to (\mathcal{I},\eta)$, $\psi$ could be extended to a morphism $\chi: (\mathcal{G},\xi)\to (\mathcal{I},\eta)$ such that $\chi\circ \phi=\psi$.

My question is: does $\text{Sh}_G(X)$ have enough injective objects? In other words, for any object $(\mathcal{F},\theta)\in \text{Sh}_G(X)$, does there exist an injective object $(\mathcal{I},\eta)$ together with a monomorphism $\phi: (\mathcal{F},\theta)\to (\mathcal{I},\eta)$?

I am not sure if this question is the same as the existence of enough injectives in the category of sheaves on stacks.


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