# 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.

## 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.

## References

The term seems to be due to: