# nLab shear map

### Context

#### Bundles

bundles

fiber bundles in physics

# Contents

## Definition

Given a group $G$ (generally: a group object in some ambient category $\mathcal{C}$, and a quasigroup-object suffices) and a group action (generally: an action object in $\mathcal{C}$)

(1)$G \times A \overset{\rho}{\longrightarrow} A$

the shear map is the morphism

$\array{ G \times A &\overset{ (\rho,pr_2) }{\longrightarrow}& A \times A \\ (g,a) &\mapsto& \big( \rho(g)(a), a \big) \,. }$

form the Cartesian product of (the objects underlying) $G$ and $A$ to that of $A$ with itself, whose first component is the action morphism (1) and whose second component is the projection onto the second factor (or the other way around, equivalently).

The action $(A,\rho)$ is called:

Often this is considered in the case that:

1. $\mathcal{C}$ is a slice category over an object $X$,

2. $G$ is a trivial bundle of groups over $X$, then still denoted $G$

in which case

1. $A = (P \overset{p}{\longrightarrow} X)$ is a bundle over $X$,

2. $A \times A \,=\, P \times_X P$ is the fiber product over $X$,

3. $\rho$ is a fiber-wise action,

and so in which case the shear map, seen as a morphism in $\mathcal{C}$, reads as follows:

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

Here $P$ with this action is called a $G$-principal bundle (not necessarily locally trivial) if the shear map is an isomorphism, or rather a formally principal bundle if $P$ is allowed to be an empty bundle.

Notice that this condition (2) is equivalent to the condition that we have a pullback square as follows:

$\array{ G \times P &\overset{\rho}{\longrightarrow}& P \\ {}^{\mathllap{pr_2}} \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow {}^{\mathrlap{p}} \\ P &\underset{p}{\longrightarrow}& X \mathrlap{\,,} }$

because the shear map (2) is the universal comparison morphism induced from the commutativity of this square to the manifest fiber product pullback.

## References

Early explicit appearance of the shear map, alongside discussion of its isomorphy (pseudo-torsor-condition):