# nLab pseudo-torsor

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

## Theorems

#### Bundles

bundles

fiber bundles in physics

# Contents

## 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.

## References

The term seems to be due to:

See also:

Last revised on May 24, 2021 at 23:12:41. See the history of this page for a list of all contributions to it.