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.



What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$?
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](https://mathoverflow.net/questions/19339/torsors-in-algebraic-geometry) 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.


