What about the empty torsor?

Question

Let $G$ be a group. A $G$-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. 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 $G$.

Answer 1

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.

Answer 2

The empty set is known as a pseudo-torsor (the only one not a torsor if the base as here consists of single point). The reason why it is excluded is that torsor is really a relative notion, a map $X\to Y$ with $G$ acting acting freely on $X$ and $X\to Y$ being a quotient map. Hence the empty set is a torsor over the empty not over the one-point set as is considered in question. This condition may seem like nit-picking but becomes more relevant when the base is more general and particular in other categories. The condition that all fibres are non-empty implies that locally (we are here talking about the base category being a topos) all $G$-torsors are isomorphic which is important to capture the idea of a $G$-torsor being a twisted version of the group itself. It also goes back to the pre-example of a torsor, an affine space over a vector space which is like a vector space without the choice of an origin but where one wants to be able to choose, at will, an origin whenever needed.

Answer 3

A torsor is a map to the classifying stack BG. Of course in algebraic geometry this is (in one form or another) the definition of BG (and this answer is equivalent to the local nontriviality in the above answers), but in any case there's an underlying intuition based on topology of what this means, and there isn't room in this picture (as far as I can see) for the empty torsor. If you want there to exist a universal G-torsor (ie the point mapping to BG) it seems to me that you have to continue to exclude empty torsors. (Edit: or you could define BG as a simplicial scheme and get a marginally less tautologous identification of G-torsors with maps.)

Answer 4

In many circumstances, it is natural to define a torsor as a nonempty set $X$ with a $G$-action such that the map $G \times X \to X \times X$ is an isomorphism. (Here the map is given by the group action $G \times X \to X$ in one coordinate, and the projection $G \times X \to X$ onto $X$ in the other coordinate.) The advantage of this definition is that you never have to mention an element of $G$ or of $X$, so the definition works for a group object in any category.

If you remove the word "nonempty" from this definition, the empty set is a torsor for every group. It definitely comes up that one can prove all parts of this definition except the nonemptyness, and has to say things like "$X$, if nonempty, is a torsor for $G$". For example, in deformation theory, the first order deformations of a complex manifold are measured by the vector space $H^1(X, T_X)$ and the second order deformations extending a given first order deformation, if nonempty, are a torsor for $H^1(X, T_X)$.

As to whether it would make sense to broaden the definition in this way, I don't think it would. The empty set feels very different from a torsor, so one would just have to say "nonempty torsor" all the time and one would probably frequently forget to do so. But it is certainly a matter of taste to what extent definitions should capture the simplest concepts and to what extent they should capture the most useful ones.