Contents

bundles

# 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.

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