What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$?

Question

Let $X$ be a topological space and $G$ be a topological group. Let $\tilde{G}$ be the sheaf of groups defined by the sheaf of sections of the product $G$ bundle $ \pi_1:X\times G \rightarrow X $.

(1) Let $T\tilde{G}_X$ be the set of all isomorphism class of $\tilde{G}$-torsors over $X$.

(2) Let $H^1(X,\tilde{G})$ be the $\tilde{G}$ valued non-abelian cohomology of degree $1$ over the $X$.

(3) Let $BG_X$ be the set of all isomorphism class of Principal $G$ bundles over $X$.

Now it is well known that there exist one-one correspondence between (1) ,(2) and (3).

Now let we replace the topological space $X$ by a site $C$ then the definition of sheaf over a site is well known. Hence I am assuming that we can appropriately generalise $(1)$ and $(2)$ to the (1') and (2') as follows:(Though neither I have proved it personally nor I have seen anywhere where such generalisation is mentioned. )

(1') Let $T\tilde{G}_C$ be the set of all isomorphism class of $\tilde{G}$-torsors over $C$.

(2') Let $H^1(C,\tilde{G})$ be the $\tilde{G}$ valued non-abelian cohomology of degree $1$ over $C$.

Now my question is the following:

Is there any analogue of (3) if we replace the topological space $X$ by a site $C$?

I will also be very grateful if someone can suggest some literature in this direction.

Thank you.

Answer 1

You should read Section 4.5 of Olsson's book Algebraic Spaces and Stacks.

The notion of a site is a piece of category theory with no intrinsic geometry, so it doesn't really make sense to ask for a geometric description of torsors for a general site.

However, in the concrete geometric contexts where site theory is typically applied, you can generalize definition 3). It will always imply definitions 1)-2) (which are always equivalent, essentially by the definition of Čech cohomology), but the converse becomes a non-trivial question about descent.

Let's assume you have some category $\mathscr{S}$ of spaces and to each object $X$ of $\mathscr{S}$, you attach a site $\mathrm{Op}(X)$ consisting of a certain full subcategory of $\mathscr{S}/X$ (e.g. the site of open subsets of a topological space, the site of étale maps into a scheme/algebraic space/DM stack, etc). Let's also assume that for a morphism $f \colon X \rightarrow Y$, the pullback map $U \mapsto f^{-1} U := U \times_Y X$ defines a continuous morphism of sites $f \colon \mathrm{Op}(X) \rightarrow \mathrm{Op}(Y)$, i.e. that if $U$ is an object of $\mathrm{Op}(X)$, then $f^{-1} U$ is an object of $\mathrm{Op}(Y)$ and that covers pull back to covers. (There are interesting contexts where this is not true, e.g. the crystalline or lisse-étale sites; in such cases, you need to be extremely careful!)

Moreover, assume that to any map of spaces $f \colon X \rightarrow Y$, the presheaf $h_X$ on $\mathrm{Op}(Y)$ defined by $U \mapsto \mathrm{Mor}_Y(U, X)$ is a sheaf, where $\mathrm{Mor}$ is the set of morphisms in $\mathscr{S}$. We say that $X$ represents the sheaf $h_X$ (note that $X$ might not be unique; the Yoneda lemma would only apply if $X$ is an object of $\mathrm{Op}(X)$).

If you want to state this abstractly, we're requiring that we have a fibered category over $\mathscr{S}$ with fiber $X \mapsto \mathrm{Op}(X)$, that this is a full subcategory of the natural fibration $X \mapsto \mathscr{S}/X$ with the same notion of pullbacks, and that this fibration satisfies the stack/descent condition for morphisms.

If $\mathcal{G}$ is a sheaf of groups on $\mathrm{Op}(X)$, the equivalent definitions 1) and 2) give a notion of when a sheaf $\mathcal{P}$ on $\mathrm{Op}(X)$ is a $\mathcal{G}$-torsor.

On the other hand, if $G$ is a group object in $\mathscr{S}/X$, we can make the following geometric notion of a $G$-torsor: a $G$-torsor is a map $P \rightarrow X$ in $\mathscr{S}$ with an action of $G$ given by a map $\rho \colon G \times_X P \rightarrow P$ (which is compatible with multiplication on $G$ in the sense that the evident diagrams commute) such that:

1. The map $(1, \rho) \colon G \times_X P \rightarrow P \times_X P$ is an isomorphism.
2. There is a covering $\{U_\alpha\}$ in $\mathrm{Op}(X)$ such that the map $P \times_X U_\alpha \rightarrow U_\alpha$ has a section (pulling back the isomorphism from point 1. along this section then gives an isomorphism $G \times_X U_\alpha \rightarrow P \times_X U_\alpha$).

Now, the sheaf $h_G$ is a sheaf of groups on $\mathrm{Op}(X)$, and the sheaf $h_P$ is a $h_G$-torsor in the sheaf-theoretic sense. Now, it makes sense to ask the following question:

If $G$ is a group object in $\mathscr{S}/X$ and $\mathscr{P}$ is an $h_G$-torsor, is there some $G$-torsor $P$ in $\mathscr{S}/X$ such that $\mathscr{P} = h_P$?

This is now a question of descent in $\mathscr{S}$.

Namely, since $\mathscr{P}$ is an $h_G$-torsor, we may find a covering $\{U_\alpha\}$ in $\mathrm{Op}(X)$ such that for each $\alpha$, we may choose a trivialization $h_G|_{U_\alpha} \simeq \mathscr{P}|_{U_\alpha}$. Therefore, $\mathscr{P}|_{U_\alpha}$ is represented by the trivial geometric $G|_{U_\alpha}$-torsor $P_\alpha = G|_{U_\alpha} \rightarrow U_\alpha$, with $G|_{U_\alpha}$-action given by left multiplication. The descent data for $\mathscr{P}$ gives us a Čech cocycle $(g_{\alpha \beta})$ with $g_{\alpha \beta} \in h_G(U_{\alpha, \beta}) = \mathrm{Mor}_X(U_{\alpha, \beta}, G)$, where $U_{\alpha, \beta} = U_\alpha \times_X U_\beta$. This is the same thing as a $G|_{U_{\alpha, \beta}}$-equivariant isomorphism $P_\alpha|_{U_{\alpha, \beta}} \rightarrow P_\beta|_{U_{\alpha, \beta}}$ in $\mathscr{S}/U_{\alpha, \beta}$. In particular, these isomorphisms satisfy the triple overlap condition because $(g_{\alpha \beta})$ is a cocycle.

If this descent datum is effective, then there is an object $P$ of $\mathscr{S}/X$ representing $\mathscr{P}$. This will always be a geometric $G$-torsor (note that this doesn't immediately follow from the Yoneda lemma, since $G$ and $P$ may not be objects of $\mathrm{Op}(X)$):

The action maps $\rho_\alpha \colon G|_{U_\alpha} \times_X P_\alpha \rightarrow P_\alpha$ glue to a map $\rho \colon G \times_X P \rightarrow P$: apply the fact that representable presheaves are sheaves to the open cover $\{G|_{U_\alpha} \times_X P_\alpha\}$ of $G \times P$. Moreover, the same argument shows that the map $(1, \rho) \colon G \times_X P \rightarrow P \times_X P$ is an isomorphism.

When you're dealing with topological spaces and open subsets, descent is always effective (in abstract terminology, the fibration $X \rightarrow (\mathscr{S}/X)$ is a stack). This is often not true in algebraic geometry!

For example, let's take $\mathscr{S}$ to be the category of schemes with the fppf topology. If $G \rightarrow X$ is affine, then we know that fppf descent is effective, and thus any sheaf-theoretic torsor $\mathscr{P}$ is represented by a geometric torsor $P \rightarrow X$. This is also true (it's a hard result of Raynaud) if $X$ is Dedekind and $G \rightarrow X$ is an abelian scheme, but it can fail in general. See this MO question and section III.4 of Milne's book Etale Cohomology.

It's a hard theorem of Artin (using the full force of his deformation-theoretic representation criteria for algebraic spaces) that fppf descent is effective for algebraic spaces, so we can in fact represent all sheaf-theoretic torsors for a group algebraic space $G$ by geometric torsors which are algebraic spaces. (See Tag 04SJ in the Stacks Project).

Edit Since you mention it in the question, I should add that this whole conversation should carry over essentially verbatim in a higher-categorical context (for example, you could replace the group $G$ by $BG$, and then talk about $G$-gerbes instead of torsors and look at cohomology in degree $2$). I'm not an expert in these things, but certainly Lurie discusses the matter comprehensively in Higher Topos Theory.

Answer 2

This is not a complete answer, too long for a comment.

If we start with an arbitrary site $\mathcal{C}$ and if we want to define the notion of a $G$-torsor over $\mathcal{C}$, then $G$ is not expected to be a group object in the category $\mathcal{C}$.

Observe that when $\mathcal{C}=\underline{X}$ for a topological space $X$, when you define $G$-torsor over the topological space $X$, the candidate $G$ was not a group object in the category $\underline{X}$. Instead, we have a fibered category $\underline{X}\rightarrow \text{Top}$ and $G$ is the group object in the category $\text{Top}$. The Grothendieck topology on $\text{Top}$ gives a Grothendieck topology on $\underline{X}$. This is the Grothendieck topology on $\underline{X}$ we are assuming when defining (the usual) notion of sheaf on the topolgical space $X$ or the shaef on the site $\underline{X}$.

So, if you want to imitate the notion of $G$-torsor to an arbitrary site $\mathcal{C}$, it is only reasonable to expect that there is a (fibered category ??) functor $\mathcal{C}\rightarrow \mathcal{D}$ for some $\mathcal{D}$ and $G$ is a group object in the category $\mathcal{D}$. Further, the Grothendieck topology that we have fixed on $\mathcal{C}$ is expected to come from a Grothendieck topology on $\mathcal{D}$.

For example, consider the case when $\mathcal{D}=\text{Man}$, the category of manifolds. Fix a Grothendieck topology on $\mathcal{D}$, say open cover topology. Let $\mathcal{C}$ be a differentiable stack; that is, $\mathcal{C}$ is a fibered category with the functor $\mathcal{C}\rightarrow \text{Man}$, satisfying certain special properties. Then, $\mathcal{C}$ can be made as a site. A cover $\{U_\alpha\rightarrow U\}$ is a cover for an object $U$ of $\mathcal{C}$ if, its image $\{F(U_\alpha)\rightarrow F(U)\}$ is a cover for the object $F(U)$ of $\text{Man}$. Then, fixing a group object in $\text{Man}$, that is a Lie group, we have the notion of a principal $G$-bundle over the site $\mathcal{C}$.

So, when $\mathcal{C}$ a category of special type, equipped with a functor $\mathcal{C}\rightarrow \text{Man}$ or $\mathcal{C}\rightarrow (\text{Sch}/S)$ and for a group object $G$ of $\text{Man}$ or $\text{Sch}/S$, we can define the notion of principal $G$ budnle over $\mathcal{C}$. It is not clear how one can define for sites which are not of this type.

The other two notions of $G$-torsors and $H^1(\mathcal{C},G)$ also require same assumptions as above.

References:

1. The notion of principal bundle over an algebraic stack; that is a special kind of fibered category $\mathcal{C}\rightarrow \text{Sch}/S$ can be found in section $1.2$ of Root stacks, principal bundles and connections.
2. The notion of principal bundle over an differentiable stack; that is a special kind of fibered category $\mathcal{C}\rightarrow \text{Man}$ can be found in section $4$ of Differentiable stacks and gerbes.
3. The notion of $G$-torsor over an algebraic space/algebraic stack can be found in Definition 04TY