mathlib documentation

data.bundle

Bundle #

Basic data structure to implement fiber bundles, vector bundles (maybe fibrations?), etc. This file should contain all possible results that do not involve any topology. We provide a type synonym of Σ x, E x as bundle.total_space E, to be able to endow it with a topology which is not the disjoint union topology sigma.topological_space. In general, the constructions of fiber bundles we will make will be of this form.

References #

source
def bundle.total_space {B : Type u_1} (E : B → Type u_2) :
Type (max u_1 u_2)

total_space E is the total space of the bundle Σ x, E x. This type synonym is used to avoid conflicts with general sigma types.

Equations
source
@[protected, instance]
def bundle.total_space.inhabited {B : Type u_1} (E : B → Type u_2) [inhabited B] [inhabited (E default)] :
inhabited (bundle.total_space E)
Equations
source
@[simp]
def bundle.proj {B : Type u_1} (E : B → Type u_2) :
bundle.total_space E → B

bundle.proj E is the canonical projection total_space E → B on the base space.

Equations
source
@[simp]
def bundle.total_space_mk {B : Type u_1} (E : B → Type u_2) (b : B) (a : E b) :
bundle.total_space E

Constructor for the total space of a topological_fiber_bundle_core.

Equations
source
@[protected, instance]
def bundle.total_space.has_coe_t {B : Type u_1} (E : B → Type u_2) {x : B} :
has_coe_t (E x) (bundle.total_space E)
Equations
source
@[simp]
theorem bundle.coe_fst {B : Type u_1} (E : B → Type u_2) (x : B) (v : E x) :
↑v.fst = x
source
theorem bundle.to_total_space_coe {B : Type u_1} (E : B → Type u_2) {x : B} (v : E x) :
↑v = ⟨x, v⟩
source
def bundle.trivial (B : Type u_1) (F : Type u_2) :
B → Type u_2

bundle.trivial B F is the trivial bundle over B of fiber F.

Equations
source
@[protected, instance]
def bundle.trivial.inhabited {B : Type u_1} {F : Type u_2} [inhabited F] {b : B} :
inhabited (bundle.trivial B F b)
Equations
source
def bundle.trivial.proj_snd (B : Type u_1) (F : Type u_2) :
bundle.total_space (bundle.trivial B F) → F

The trivial bundle, unlike other bundles, has a canonical projection on the fiber.

Equations
source
@[simp]
theorem bundle.coe_snd_map_apply {B : Type u_1} (E : B → Type u_2) [Π (x : B), add_comm_monoid (E x)] (x : B) (v w : E x) :
↑(v + w).snd = ↑v.snd + ↑w.snd
source
@[simp]
theorem bundle.coe_snd_map_smul {B : Type u_1} (E : B → Type u_2) [Π (x : B), add_comm_monoid (E x)] (R : Type u_3) [semiring R] [Π (x : B), module R (E x)] (x : B) (r : R) (v : E x) :
↑(r • v).snd = r • ↑v.snd
source
@[protected, instance]
def bundle.trivial.add_comm_monoid {B : Type u_1} {F : Type u_3} (b : B) [add_comm_monoid F] :
add_comm_monoid (bundle.trivial B F b)
Equations
source
@[protected, instance]
def bundle.trivial.add_comm_group {B : Type u_1} {F : Type u_3} (b : B) [add_comm_group F] :
add_comm_group (bundle.trivial B F b)
Equations
source
@[protected, instance]
def bundle.trivial.module {B : Type u_1} {F : Type u_3} {R : Type u_4} [semiring R] (b : B) [add_comm_monoid F] [module R F] :
module R (bundle.trivial B F b)
Equations