A set (or sheaf, scheme, etc.) with a simply transitive action of a group G: there is a free transitive G-action but no distinguished identity point, so the torsor is a 'principal homogeneous space' that becomes isomorphic to G once a basepoint is chosen.

Principle

A torsor for a group G has locally (for a chosen topology) the structure of G but globally may be nontrivial; torsors over a base are classified by H^1 of the base with coefficients in G (Čech or sheaf cohomology), and twisting by torsors produces nontrivial forms of objects.

Demonstration

A nonzero vector space V over a field k is a torsor under its general linear group GL(V) once a point is fixed; more geometrically, a principal G-bundle over a scheme that is locally trivial in the étale (or Zariski) topology is a G-torsor, and line bundles are torsors under G_m.

Misapplication

Treating a torsor as a group by assuming the existence of a canonical identity or composing torsor elements without reference to the group action; confusing torsor cohomology classes with ordinary group cohomology without the correct coefficient interpretation.

Consequence

Torsors encode obstruction and twisting data: they classify forms, parameterize principal bundles, and translate between geometric objects and cohomology classes, providing a concrete incarnation of nontrivial descent and gluing.

Reversal

The reversal views groups as torsors with chosen identity: equipping a torsor with a distinguished point recovers the group structure; conversely, forgetting the identity of a group yields a torsor—so torsor versus group is a choice of origin.

Boundary

Requires a specified acting group and a simply transitive action; objects with nontransitive or nonfree actions are excluded, and torsor theory typically presumes a topology for local triviality (Zariski, étale, analytic) depending on context.

Semantic Tension

Tension occurs between torsors, principal bundles, and gerbes: torsors are 0‑dimensional cocycles classified by H^1, while gerbes and higher torsors live in higher cohomology; also one must distinguish bitorsors (actions on both sides) from ordinary torsors.

Synthesis

A torsor is a group-shaped object without a chosen origin: locally identical to the acting group but globally possibly twisted, it is the basic geometric representative of a first cohomology class that encodes descent, forms, and principal bundle data.