6
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.