nLab shear map

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:

• a free action if the shear map is a monomorphism,

• a transitive action if the shear map is an epimorphism,

• a regular action or $G$-torsor if the shear map is an isomorphism (or rather a pseudo-torsor if $A$ is not required to be inhabited).

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.

Related concepts

• group action

• torsor, pseudo torsor

• principal bundle

References

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

• Alexander Grothendieck, p. 312 (15 of 30) in: FGA Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire N. Bourbaki, 1960, exp. no190, p. 299-327 (numdam:SB_1958-1960__5__299_0, English translation: web version)

• Alexander Grothendieck, p. 105 (8 of 21) in: FGA Techniques de construction et théorèmes d’existence en géométrie algébrique III : préschémas quotients, Séminaire Bourbaki : années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf, English translation: web version)

• Alexander Grothendieck, 16.5.15 in: Éléments de géométrie algébrique : IV. Étude locale des schémas et des morphismes de schémas (Quatrième partie), Publications mathématiques de l’I.H.É.S., tome 32 (1967), p. 5-361 (numdam:PMIHES_1967__32__5_0)

• Alexander Grothendieck, p. 9 of Exemples et Complements (doi:10.1007/BFb0058666, pdf), which is p. 293 in:

  Alexander Grothendieck, Michèle Raynaud, Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics 224, Springer 1971 (arXiv:math/0206203, doi:10.1007/BFb0058656)