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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 $G$-over an algebraic space/algebraic stack can be found in Definition 04TY

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

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 $\mathcal{C}=\mathcal{O}(X)$ 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 $\mathcal{O}(X)$. Instead, we have a fibered category $\mathcal{O}(X)\rightarrow \text{Top}$ and $G$ is the group object in the category $\text{Top}$. The Grothendieck topology on $\text{Top}$ gives a Grothendieck topology on $\mathcal{O}(X)$. This is the Grothendieck topology on $\mathcal{O}(X)$ we are assuming when defining (the usual) notion of sheaf on the topolgical space $X$ or the shaef on the site $\mathcal{O}(X)$.

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 $G$-over an algebraic space/algebraic stack can be found in Definition 04TY

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 $G$-over an algebraic space/algebraic stack can be found in Definition 04TY

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 $\mathcal{C}=\mathcal{O}(X)$ 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 $\mathcal{O}(X)$. Instead, we have a fibered category $\mathcal{O}(X)\rightarrow \text{Top}$ and $G$ is the group object in the category $\text{Top}$. The Grothendieck topology on $\text{Top}$ gives a Grothendieck topology on $\mathcal{O}(X)$. This is the Grothendieck topology on $\mathcal{O}(X)$ we are assuming when defining (the usual) notion of sheaf on the topolgical space $X$ or the shaef on the site $\mathcal{O}(X)$.

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.

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 $\mathcal{C}=\mathcal{O}(X)$ 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 $\mathcal{O}(X)$. Instead, we have a fibered category $\mathcal{O}(X)\rightarrow \text{Top}$ and $G$ is the group object in the category $\text{Top}$. The Grothendieck topology on $\text{Top}$ gives a Grothendieck topology on $\mathcal{O}(X)$. This is the Grothendieck topology on $\mathcal{O}(X)$ we are assuming when defining (the usual) notion of sheaf on the topolgical space $X$ or the shaef on the site $\mathcal{O}(X)$.

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 $G$-over an algebraic space/algebraic stack can be found in Definition 04TY