vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
Given a group (generally: a group object in some ambient category , and a quasigroup-object suffices) and a group action (generally: an action object in )
the shear map is the morphism
form the Cartesian product of (the objects underlying) and to that of 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 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 -torsor if the shear map is an isomorphism (or rather a pseudo-torsor if is not required to be inhabited).
Often this is considered in the case that:
is a slice category over an object ,
is a trivial bundle of groups over , then still denoted
in which case
is a bundle over ,
is the fiber product over ,
and so in which case the shear map, seen as a morphism in , reads as follows:
Here with this action is called a -principal bundle (not necessarily locally trivial) if the shear map is an isomorphism, or rather a formally principal bundle if 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:
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):
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)
Last revised on April 18, 2024 at 14:35:03. See the history of this page for a list of all contributions to it.