Where does the term "torsor" come from?

Question

Is there a heuristic reason why principal homogeneous spaces of a group (object) $G$ (in some categories) are called $G$-torsors? Does it have anything to do with the idea of "torsion", somehow? When and where did this piece of terminology originate?

Answer 1

There are three questions, but it's most sensible to answer them in reverse order.

(3) When and where did this piece of terminology originate?

Donu pointed out a reference to Giraud's thesis, and I tracked it down. Giraud's 1966 thesis was titled Cohomologie non abélienne de degré 2. His advisor was Grothendieck, so the SGA footnote is strong evidence that Giraud really did coin this terminology. In 1971, Giraud published a book with the same title. Chapter 3 is titled "Torseurs. Cohomologie de degré 1" and begins on page 107. He defines pseudo-torsors, then torsors in a topos, and $G$-torsors.

(2) Does it have anything to do with the idea of "torsion", somehow?

Giraud's Proposition 2.3.7 on page 146 relates torsors to torsion of bundles of groups. He also relates them to twist functors, on page 147 and 151, a motivation for the term suggested by Todd. This question is also related to:

(1) Is there a heuristic reason why principal homogeneous spaces of a group (object) $G$ (in some categories) are called $G$-torsors?

I think Fred's pointer to the essay of Baez is the best answer to this part of the question. Before giving Baez's heuristics, let me define $G$-torsor so that this answer is self contained. For a Grothendieck topology $T$, a scheme $X$, and a group scheme $G$ over $X$, a $G$-torsor is the same thing as a principal $G$-bundle, i.e., a scheme $P$ and a scheme morphism $f:P\to X$ with $G$-invariant action on $P$ and locally trivial with respect to $T$. Some people refer to $P$ as the $G$-torsor, e.g., Baez does when he writes that "the fiber of a principal $G$-bundle is a $G$-torsor." Baez spends most of his essay unpacking this definition. He gives examples (dates, and musical notes) where you can't add two elements of $X$ but can add an element of $G$ with an element of $X$.

The three heuristics Baez provides are below. They revolve around the idea that often we cannot really measure the thing we want to study, but rather can only measure differences, because to measure the thing itself we'd need to fix a convention about what represents zero:

• In Newtonian mechanics, energies form an $\mathbb{R}$-torsor, and we study the differences between energies, which lies in $\mathbb{R}$.
• In electromagnetism, voltages are an $\mathbb{R}$-torsor and voltage differences lie in $\mathbb{R}$.
• In quantum mechanics, relative phases lie in the unit group $U(1)$ of $\mathbb{C}$ but phases themselves are $U(1)$-torsors, again because you can't ask "what's the phase of this quantum state" without first fixing a convention.

He also gives other useful examples:

• The indefinite integral of some continuous function $f$ is an $\mathbb{R}$-torsor, namely $F + C$, where the $+C$ represents that we have a whole $\mathbb{R}$-worth of choices for the antiderivative, i.e., an action of $\mathbb{R}$ on the set of antiderivatives.
• Studying vectors in $\mathbb{R}^2$ without fixing a choice of origin leads you to the torsor way of thinking of an affine space, as a vector space where you've not had to specify the origin.
• Baez says "A torsor is like a group that has forgotten its identity" again on the theme of the examples above.
• The set of orthonormal frames at some point of an $n$-dimensional Riemannian manifold is an $O(n)$-torsor, not $O(n)$ itself, because you can rotate any frame by an element of $O(n)$. If you wanted to insist that a frame was the same as an element of $O(n)$, you'd have to pick an arbitrary choice of frame to make it so. Just like cosets of a group.
• Same example but with spinors on a manifold, and spin structure.

Lastly, I want to point out the connection between the word "torsion" in group theory and "twisting." In a wonderful answer a long time ago, Qiaochu Yuan explained that "torsion" in group theory came from "torsion" in integral homology, which had to do with twisting around polygons to make the edges fit together, in a simplicial decomposition. So, it's appropriate that "torsor" also fits with the story of (co)homology and twisting things together via a group action.

Answer 2

In the french school, un torseur sert à tordre, a torsor is used to twist. More precisely, let $\eta$ be an object in a topos, and $G=\operatorname{Aut}(\eta)$.

• If $\nu$ is a form of $\eta$ (another object of the topos locally isomorphic to $\eta$) then $T=\operatorname{Isom}(\eta,\nu)$ is a $G$-torsor.
• In the other direction, if $T$ is a $G$ torsor, then one can use descent along $T\rightarrow *$ (the terminal object) to get a form of $\eta$, that is $\nu =T\wedge^G \eta$, le tordu de $\eta$ par $T$.

So using twisting by torsors you get a one to one correspondence between forms of $\eta$ and $G$-torsors. This can be found in Giraud's book.

Nice example of forms: non degenerate quadratic forms over a (non algebraically closed) field. But you should pick up your favorite example.