nLab
pseudo-torsor

Contents

Context

Representation theory

Bundles

Contents

Idea

In the definition of torsors and principal bundles one deals with group action objects G×PρPG \times P \overset{\rho}{\longrightarrow} P (generally over some base object XX) whose shear map is an isomorphism:

(1)G×P (ρ,pr 2) P× XP (g,p) (ρ(g)(p),p). \array{ G \times P & \underoverset {\simeq} { (\rho, pr_2) } {\longrightarrow} & P \times_X P \\ (g,p) &\mapsto& \big( \rho(g)(p), p \big) \,. }

Now if PP is inhabited (fiber-wise over XX), this implies a (fiber-wise) free and transitive (hence regular) action, which is typically what is understood to characterize torsors/principal bundles.

However, the condition alone that the shear map (1) be an isomorphism makes sense (and is then automatically satisfied) also for locally empty PP, meaning for fibers of PP being strict initial objects (internal to the ambient category). If that case is meant to be included, one speaks, following Grothendieck, alternatively of:

  1. a formally principal action (Grothendieck 60, p. 312 (15 of 30))

  2. a pseudo-torsor (Grothendieck 67, EGA IV.4, 16.5.15)

  3. a formally principal homogeneous action (ibid. & Grothendieck 71, p. 9 (293)).

Examples of references using this terminology: StacksProject, Moret-Bailly 13, slide 5, BGA 13, p. 73-74.

Examples

Properties

Proposition

Internal to some ambient category 𝒞\mathcal{C} with finite limits, let

Then the following are equivalent:

  1. PXP \to X is the GG-quotient coprojection;

  2. PXP \to X is an effective epimorphism.

Proof

The first condition is equivalent to

P× XPPX P \times_X P \rightrightarrows P \to X

being a coequalizer, the second to

P×GPX P \times G \rightrightarrows P \to X

being a coequalizer. But the pseudo-principality condition says that we have an isomorphism (the shear map)

P× XPP×G P \times_X P \simeq P \times G

which identifies these two diagrams.

References

The term seems to be due to:

See also:

Last revised on August 26, 2021 at 10:15:11. See the history of this page for a list of all contributions to it.