Definition. torsors and the universal torsor [fgap-000Q]
Definition. torsors and the universal torsor [fgap-000Q]
Let \(G\) be a group object in a topos \(\mathcal {E}\). A left \(G\)-object \(T\) is a torsor when \(T\to 1\) is an epimorphism and \[ (\mu ,\pi _2):G\times T\longrightarrow T\times T, \qquad (g,t)\longmapsto (g\mathbin {\cdot }t,t) \] is an isomorphism. The second condition is the internal form of freeness and transitivity. See [maclane1992sheaves, sec. VIII.2, pp. 429--430].
For an ordinary group \(G\), write \[ \mathsf {B}G=\mathsf {Set}^{BG^{\mathrm {op}}} \] for the topos of right \(G\)-sets, regarded as presheafs on the one-object category \(BG\). Its universal torsor \(U_G\) has underlying right \(G\)-set \(G\) with regular right multiplication. The constant group object \(\underline {G}\), whose right \(G\)-action is trivial, acts on \(U_G\) by left multiplication. This supplies the torsor action; the left and right actions commute.
For \(C_2\), the regular left action is a torsor in \(\mathsf {Set}\): \[ C_2\times C_2\longrightarrow C_2\times C_2, \qquad (g,h)\longmapsto (gh,h) \] is a bijection. The trivial action on a 2-point set \(D\), despite \(D\to 1\) being onto, is not a torsor. Its corresponding map sends \((g,d)\) to \((d,d)\), so it is neither injective nor surjective. Inhabitation and cardinality alone do not supply a torsor.