Moduli stack of principal bundles

In algebraic geometry, given a smooth projective curve X over a finite field ${\displaystyle \mathbf {F} _{q}}$ and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by ${\displaystyle \operatorname {Bun} _{G}(X)}$, is an algebraic stack given by: for any ${\displaystyle \mathbf {F} _{q}}$-algebra R,

${\displaystyle \operatorname {Bun} _{G}(X)(R)=}$ the category of principal G-bundles over the relative curve ${\displaystyle X\times _{\mathbf {F} _{q}}\operatorname {Spec} R}$.

In particular, the category of ${\displaystyle \mathbf {F} _{q}}$-points of ${\displaystyle \operatorname {Bun} _{G}(X)}$, that is, ${\displaystyle \operatorname {Bun} _{G}(X)(\mathbf {F} _{q})}$, is the category of G-bundles over X.

Similarly, ${\displaystyle \operatorname {Bun} _{G}(X)}$ can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define ${\displaystyle \operatorname {Bun} _{G}(X)}$ as the quotient stack of the space of holomorphic connections on X by the gauge group. Replacing the quotient stack (which is not a topological space) by a homotopy quotient (which is a topological space) gives the homotopy type of ${\displaystyle \operatorname {Bun} _{G}(X)}$.

In the finite field case, it is not common to define the homotopy type of ${\displaystyle \operatorname {Bun} _{G}(X)}$. But one can still define a (smooth) cohomology and homology of ${\displaystyle \operatorname {Bun} _{G}(X)}$.