Let $G$ be a group. A [$G$-torsor](http://ncatlab.org/nlab/show/torsor) is a set $X$ together with an action of $G$ such that for all $x,y \in X$ there is exactly one $g \in G$ such that $gx=y$. This looks like a group which has forgotten its identity (no pun intended).


Usually it is assumed that $X$ is nonempty (or more generally an inhabited object according to the nlab article), but it seems to me that it makes perfect sense also to allow $X = \emptyset$ because of the equivalent definition using [Heaps](http://ncatlab.org/nlab/show/heap). This equivalence also suggests that $X = \emptyset$ should imply that $G$ is trivial. But this is not guaranteed by the definition given above.


**Question:** What is a natural definition of torsors which also includes the empty set with the action of the trivial group (the empty torsor)?


In the case of sets as above, we may just add that the action is faithful, i.e. the homomorphism $G \to \text{Sym}(X)$ is injective. But how can we give a definition for, say, group schemes acting on schemes, without making a nasty case distinction?


Also I would like to know if you agree with me that it is natural to include the empty torsor. It will be an initial object in the category of torsors and as I said, especially in the definition of a heap the assumption of being nonempty seems to be artificial according to universal algebra.


[added] Thank you for all the good answers. I agree that it's not natural to consider an empty torsor. A $G$-torsor should be something which is locally isomorphic to $G$.



What about the empty torsor?
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.


