nLab pseudo-torsor

Idea

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

(1)$$\array{ G \times P & \underoverset {\simeq} { (\rho, pr_2) } {\longrightarrow} & P \times_X P \\ (g,p) &\mapsto& \big( \rho(g)(p), p \big) \,. }$$

Now if $P$ is inhabited (fiber-wise over $X$), 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 $P$, meaning for fibers of $P$ 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

• equivariant principal bundles

  Even if a $G$-equivariant principal bundle is an actual torsor (in topological G-spaces sliced over a given base, SS 2021, Def. 2.1.3) its fixed loci will generally only be (similarly internal) pseudo-torsors. (For more see SS 2021, Rem. 2.18).

Properties

Proposition

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

• $G \,\in\, Grp(\mathcal{C})$ be a group object,
• $P \,\in\, G Act(\mathcal{C})$ an action object,
• $(P \to X) \,\in\, G PsTor(\mathcal{C}_{/X})$ a formally principal bundle.

Then the following are equivalent:

1. $P \to X$ is the $G$-quotient coprojection;

2. $P \to X$ is an effective epimorphism.

Proof

The first condition is equivalent to

$$P \times_X P \rightrightarrows P \to X$$

being a coequalizer, the second to

$$P \times G \rightrightarrows P \to X$$

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

$$P \times_X P \simeq P \times G$$

which identifies these two diagrams.

Related concepts

• empty heap

• invertible semigroup

• associative quasigroup

References

The term seems to be due to:

• Alexander Grothendieck, p. 312 (15 of 30) in: Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire N. Bourbaki exp. no190 (1960) 299-327 [numdam:SB_1958-1960__5__299_0]

• Alexander Grothendieck, 16.5.15 in: Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas (Quatrième partie), Publications mathématiques de l’I.H.É.S., tome 32 (1967), p. 5-361 (numdam:PMIHES_1967__32__5_0)

• Alexander Grothendieck, p. 9 of Exemples et Complements (doi:10.1007/BFb0058666, pdf), which is p. 293 in:

  Alexander Grothendieck, Michèle Raynaud, Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics 224, Springer 1971 (arXiv:math/0206203, doi:10.1007/BFb0058656)

See also:

• The Stacks Project, Pseudo G-Torsor (Definition 03AH, Definition 0498)

• Laurent Moret-Bailly, slide 5 of: Principal Bundles over Valued Fields, Oberwolfach 2013 (pdf, pdf)

• Alessandra Bertapelle, Cristian D. Gonzalez-Aviles, The Greenberg functor revisited, European Journal of Mathematics volume 4, pages 1340–1389 (2018) (arXiv:1311.0051, doi:10.1007/s40879-017-0210-0)