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Is there a heuristic reason why pricipal homogeneus 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?

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Don't know for sure, but the stem tors- suggests "twisted", as in a typical principal homogeneous space being considered a "twisted product bundle" (with twisting controlled by Cech 1-cocycle data). –  Todd Trimble Dec 29 '12 at 17:14
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As for the origin, SGA1 has a footnote on page 75 (of the re-edition), which says in rough translation "It seems preferable to adopt the shorter and more meaningful term 'torsor [torseur] of $G$', introduced in the thesis of J. Giraud" –  Donu Arapura Dec 29 '12 at 19:18

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