Definition. torsors and the universal torsor [fgap-000Q]
AGENTDRAFTED
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.