If you have any category that admits finite products, you have a notion of $G$-pseudotorsor, namely an object $X$ equipped with a map $act: G \times X \to X$, such that

1. $act \circ (id_G, act) = act \circ (m_G,id_X): G \times G \times X \to X$ and $act \circ (e_G, id_X) = id_X: X \to X$ (i.e., it is an action).
2. $(act, p_2): G \times X \to X \times X$ is an isomorphism.

An initial object in your category is a pseudotorsor under any group object. I would say that the notion of pseudotorsor is a natural one, but I don't think the name "torsor" should be used. (See EGA IV 16.5.15)

The notion of torsor requires your category to admit a notion of objects being locally isomorphic - in particular, if $X$ is a $G$-torsor, it should be locally isomorphic to $G$, in a $G$-equivariant way. There are equivalent definitions, where you demand the local existence of a section, or you demand that the object be nonempty/non-initial over all opens, and although these conditions are useful in a bootstrapping sense, I don't think they look as natural (Warning: the non-emptiness condition is not global, and this makes a difference over non-connected bases). The isomorphism condition gives the category of torsors a groupoid structure (and base change yields a category fibered in groupoids). The existence of an initial pseudotorsor ensures that the category whose objects are $G$-pseudotorsors and whose morphisms are $G$-equivariant morphisms is not a groupoid, so you have to make a choice of which nice property you need when choosing between pseudotorsors and torsors.

When you work with $G$ as a set-theoretic group, or even as a group object in sets over some other set, the notion of local isomorphism collapses, and you demand that $X$ is in fact $G$-equivariantly isomorphic to $G$. The language of torsors in sets is useful when you want to make sure your constructions are canonical, but I think torsors become more interesting when you can glue locally trivial objects in nontrivial ways.

It looks like you also want a notion of pseudotorsor with a faithful $G$-action. If I'm not mistaken, you can define faithfulness by finality of the equalizer of $G \times X \overset{act}{\underset{p_2}{\rightrightarrows}} X$. This is equivalent to being a torsor if and only if $G$ is nontrivial.

If you have any category that admits finite products, you have a notion of $G$-pseudotorsor, namely an object $X$ equipped with a map $act: G \times X \to X$, such that

1. $act \circ (id_G, act) = act \circ (m_G,id_X): G \times G \times X \to X$ and $act \circ (e_G, id_X) = id_X: X \to X$ (i.e., it is an action).
2. $(act, p_2): G \times X \to X \times X$ is an isomorphism.

An initial object in your category is a pseudotorsor under any group object. I would say that the notion of pseudotorsor is a natural one, but I don't think the name "torsor" should be used. (See EGA IV 16.5.15)

The notion of torsor requires your category to admit a notion of objects being locally isomorphic - in particular, if $X$ is a $G$-torsor, it should be locally isomorphic to $G$, in a $G$-equivariant way. There are equivalent definitions, where you demand the local existence of a section, or you demand that the object be nonempty (non-initial) over all nonempty opens (non-initial objects), and although these conditions are useful in a bootstrapping sense, I don't think they look as natural (Warning: the non-emptiness condition is not global, and this makes a difference over non-connected bases). The isomorphism condition gives the category of torsors a groupoid structure (and base change yields a category fibered in groupoids). The existence of an initial pseudotorsor ensures that the category whose objects are $G$-pseudotorsors and whose morphisms are $G$-equivariant morphisms is not a groupoid, so you have to make a choice of which nice property you need when choosing between pseudotorsors and torsors.

When you work with $G$ as a set-theoretic group, or even as a group object in sets over some other set, the notion of local isomorphism collapses, and you demand that $X$ is in fact $G$-equivariantly isomorphic to $G$. The language of torsors in sets is useful when you want to make sure your constructions are canonical, but I think torsors become more interesting when you can glue locally trivial objects in nontrivial ways. Basic examples include the orientation double cover of a manifold as a $\mathbb{Z}/2\mathbb{Z}$-torsor, whose global trivializations are in bijection with choices of orientation, and the spectrum of a Galois extension of fields as a torsor under the Galois group, where trivializations exist étale-locally, but not Zariski-locally.

It looks like you also want a notion of pseudotorsor with a faithful $G$-action. If I'm not mistaken, you can define faithfulness by finality of the equalizer of $G \times X \overset{act}{\underset{p_2}{\rightrightarrows}} X$. This is equivalent to being a torsor if and only if $G$ is nontrivial.

If you have any category that admits finite products, you have a notion of $G$-pseudotorsor, namely an object $X$ equipped with a map $act: G \times X \to X$, such that

1. $act \circ (id_G, act) = act \circ (m_G,id_X): G \times G \times X \to X$ and $act \circ (e_G, id_X) = id_X: X \to X$ (i.e., it is an action).
2. $(act, p_2): G \times X \to X \times X$ is an isomorphism.

You could also say that the action is pseudo-transitive if that last map is surjective (but I don't think this name is standard). An initial object in your category is a pseudotorsor under any group object. I would say that the notion of pseudotorsor is a natural one, but I don't think the name "torsor" should be used.

The notion of torsor requires your category to admit a notion of objects being locally isomorphic - in particular, if $X$ is a $G$-torsor, it should be locally isomorphic to $G$, in a $G$-equivariant way. There are equivalent definitions, where you demand the existence of a local section, or you demand that the object be locally nonempty, and although they are useful in a bootstrapping sense, I don't think they look as natural. The isomorphism condition gives the category of torsors a groupoid structure (and base change yields a category fibered in groupoids). The existence of an initial pseudotorsor ensures that the category whose objects are $G$-pseudotorsors and whose morphisms are $G$-equivariant morphisms is not a groupoid, so you have to make a choice of which nice property you need when choosing between pseudotorsors and torsors.

When you work with $G$ as a set-theoretic group, or even as a group object in sets over some other set, the notion of local isomorphism collapses, and you demand that $X$ is in fact $G$-equivariantly isomorphic to $G$. The language of torsors in sets is useful when you want to make sure your constructions are canonical, but I think torsors become more interesting when you can glue locally trivial objects in nontrivial ways.

It looks like you also want a notion of pseudotorsor with a faithful $G$-action. If I'm not mistaken, you can define faithfulness by finality of the equalizer of $G \times X \overset{act}{\underset{p_2}{\rightrightarrows}} X$. This is equivalent to being a torsor if and only if $G$ is nontrivial.

If you have any category that admits finite products, you have a notion of $G$-pseudotorsor, namely an object $X$ equipped with a map $act: G \times X \to X$, such that

1. $act \circ (id_G, act) = act \circ (m_G,id_X): G \times G \times X \to X$ and $act \circ (e_G, id_X) = id_X: X \to X$ (i.e., it is an action).
2. $(act, p_2): G \times X \to X \times X$ is an isomorphism.

An initial object in your category is a pseudotorsor under any group object. I would say that the notion of pseudotorsor is a natural one, but I don't think the name "torsor" should be used. (See EGA IV 16.5.15)

The notion of torsor requires your category to admit a notion of objects being locally isomorphic - in particular, if $X$ is a $G$-torsor, it should be locally isomorphic to $G$, in a $G$-equivariant way. There are equivalent definitions, where you demand the local existence of a section, or you demand that the object be nonempty/non-initial over all opens, and although these conditions are useful in a bootstrapping sense, I don't think they look as natural (Warning: the non-emptiness condition is not global, and this makes a difference over non-connected bases). The isomorphism condition gives the category of torsors a groupoid structure (and base change yields a category fibered in groupoids). The existence of an initial pseudotorsor ensures that the category whose objects are $G$-pseudotorsors and whose morphisms are $G$-equivariant morphisms is not a groupoid, so you have to make a choice of which nice property you need when choosing between pseudotorsors and torsors.

When you work with $G$ as a set-theoretic group, or even as a group object in sets over some other set, the notion of local isomorphism collapses, and you demand that $X$ is in fact $G$-equivariantly isomorphic to $G$. The language of torsors in sets is useful when you want to make sure your constructions are canonical, but I think torsors become more interesting when you can glue locally trivial objects in nontrivial ways.

It looks like you also want a notion of pseudotorsor with a faithful $G$-action. If I'm not mistaken, you can define faithfulness by finality of the equalizer of $G \times X \overset{act}{\underset{p_2}{\rightrightarrows}} X$. This is equivalent to being a torsor if and only if $G$ is nontrivial.