nLab
shear map

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Bundles

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Definition

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

(1)G×AρX G \times A \overset{\rho}{\longrightarrow} X

the shear map is the morphism

G×A (ρ,pr 2) A×A (g,a) (ρ(g)(a),a). \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) GG and AA to that of AA 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,ρ)(A,\rho) is called:

Often this is considered in the case that:

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

  2. GG is a trivial bundle of groups over XX, then still denoted GG

in which case

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

  2. A×A=P× XPA \times A \,=\, P \times_X P is the fiber product over XX,

  3. ρ\rho is a fiber-wise action,

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

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

Here PP with this action is called a GG-principal bundle (not necessarily locally trivial) if the shear map is an isomorphism, or rather a formally principal bundle if PP 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:

G×P ρ P pr 2 (pb) p P p X, \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):

Last revised on June 18, 2021 at 17:35:19. See the history of this page for a list of all contributions to it.