aarjaneiro/planetmath huggingface dataset:
2-groupoid

2-groupoid

Definition 0.1

.

A

2-groupoid

is a

2-category

whose

morphisms

are all

invertible

, that is, ones such that,
each

1

-arrow (morphism) is invertible with respect to the morphism

composition

.

Remark 0.1

.

Note, however that

ω

-

groupoid

has a distinct meaning from that of

ω

-

category

).

An important reason for studying

2

–categories, and especially

2

-groupoids, is to use them as coefficient objects for non-Abelian Cohomology theories. Thus, some

double groupoids

defined over

Hausdorff spaces

that are

non-Abelian

(or

non-commutative

) are relevant to non-Abelian

Algebraic Topology

(NAAT) and

http://planetphysics.org/?op=getobj&from=lec&id=61

NAQAT (or NA-QAT).

One needs to distinguish between a 2-groupoid and a double-groupoid as the two

concepts

are very different. Interestingly, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) have true two-dimensional geometric

representations

with special properties that allow

generalizations

of important

theorems

in algebraic topology and

higher dimensional algebra

, such as the generalized

van Kampen theorem

with significant

consequences

that cannot be obtained through

Abelian

means.

Furthermore, whereas the definition of an

n

-groupoid is a straightforward generalization of a 2-groupoid, the notion of a

multiple

groupoid

is not at all an obvious generalization or

extension

of the concept of double groupoid.

Title

2-groupoid

Canonical name

2groupoid

Date of creation

2013-03-22 19:21:09

Last modified on

2013-03-22 19:21:09

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

17

Author

bci1 (20947)

Entry type

Definition

Classification

msc 55Q35

Classification

msc 55Q05

Classification

msc 20L05

Classification

msc 18D05

Classification

msc 18-00

Synonym

2-category with invertible morphisms

Defines

2-groupoid

Defines

HDA

Defines

higher dimensional algebra

Defines

(m-1) arrows

Defines

Generated on Sat Feb 10 11:43:48 2018 by

LaTeXML
contractible

contractible

A

topological space

is said to be

contractible

if it is

homotopy equivalent

to a point. Equivalently, the space is contractible if a constant map is

homotopic

to the identity map. A contractible space has a trivial

fundamental group

.

Title

contractible

Canonical name

Contractible

Date of creation

2013-03-22 12:13:24

Last modified on

2013-03-22 12:13:24

Owner

RevBobo (4)

Last modified by

RevBobo (4)

Numerical id

9

Author

RevBobo (4)

Entry type

Definition

Classification

msc 55Q52

Related topic

HomotopyOfMaps

Related topic

HomotopyEquivalence

Related topic

HomotopyWithAContractibleDomain

Generated on Sat Feb 10 11:44:02 2018 by

LaTeXML
classification of covering spaces

classification of covering spaces

Let

X

be a

connected

,

locally path connected

and

semilocally simply
connected

space. Assume furthermore that

X

has a

basepoint

*

.

A

covering

p

:

E

→

X

is called

based

if

E

is endowed with a
basepoint

e

and

p

⁢

(

e

)

=

*

. Two based coverings

p

i

:

E

i

→

X

,

i

=

1

,

2

are called

equivalent

if there is a basepoint preserving equivalence

T

:

E

1

→

E

2

that
covers the

identity

, i.e.

T

is a

homeomorphism

and the following diagram
commutes

Generated on Sat Feb 10 11:44:14 2018 by

LaTeXML
covering space

covering space

Let

X

and

E

be

topological spaces

and suppose there is a surjective

continuous map

p

:

E

→

X

which satisfies the following condition: for each

x

∈

X

, there is an open neighborhood

U

of

x

such that

•

p

-

1

⁢

(

U

)

is a disjoint union of

open sets

E

i

⊂

E

, and

•

each

E

i

is mapped homeomorphically onto

U

via

p

.

Then

E

is called a

covering space

,

p

is called a

covering map

, the

E

i

’s are

sheets

of the

covering

of

U

and for each

x

∈

X

,

p

-

1

⁢

(

x

)

is the

fiber

of

p

above

x

. The open set

U

is said to be

evenly covered

.
If

E

is

simply connected

, it is called the

universal covering space

.

From this we can derive that

p

is a

local homeomorphism

, so that any local property

E

has is inherited by

X

(local

connectedness

, local path connectedness etc.). Covering spaces
are foundational in the study of the

fundamental group

of a topological space; in particular, there is a correspondence (

http://planetmath.org/ClassificationOfCoveringSpaces

) between

connected

coverings of

X

and subgroups of the group of

deck transformations

of its universal covering space which is exactly analogous to the

fundamental theorem of Galois theory

.

Covering maps are especially important in the study of Riemann surfaces; in this context, one sometimes discusses a generalized notion of covering map called a “

ramified

covering”; this allows one to replace a

discrete set

of the local homeomorphisms by maps that locally look like

z

↦

z

n

in the

complex plane

near zero. Covering maps are also generalized in

algebraic geometry

; there the corresponding notion is that of étale morphism.

Note that this is a completely separate usage of the word “cover” than we encounter in “

open cover

”; confusion usually does not arise.

Title

covering space

Canonical name

CoveringSpace

Date of creation

2013-03-22 11:59:30

Last modified on

2013-03-22 11:59:30

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

21

Author

rmilson (146)

Entry type

Definition

Classification

msc 55R05

Synonym

covering map

Related topic

Cover

Related topic

Site

Related topic

EtaleMorphism

Generated on Sat Feb 10 11:44:16 2018 by

LaTeXML
deck transformation

deck transformation

Let

p

:

E

→

X

be a covering map. A

deck transformation

or

covering transformation

is a map

D

:

E

→

E

such that

p

∘

D

=

p

, that is, such that the following diagram commutes.

Generated on Sat Feb 10 11:44:23 2018 by

LaTeXML
lifting of maps

lifting of maps

Let

p

:

E

→

B

and

f

:

X

→

B

be (

continuous

) maps. Then a

lifting

of

f

to

E

is a (continuous) map

f

~

:

X

→

E

such that

p

∘

f

~

=

f

. The terminology is justified by the following

commutative diagram

Generated on Sat Feb 10 11:44:28 2018 by

LaTeXML
lifting theorem

lifting theorem

Let

p

:

E

→

B

be a covering map and

f

:

X

→

B

be a (

continuous

)
map where

X

,

B

and

E

are

path connected

and

locally path connected

(

http://planetmath.org/LocallyConnected

).
Also let

x

∈

X

and

e

∈

E

be points such that

f

⁢

(

x

)

=

p

⁢

(

e

)

.
Then

f

lifts to a map

f

~

:

X

→

E

with

f

~

⁢

(

x

)

=

e

if and only if

π

1

⁢

(

f

)

maps

π

1

⁢

(

X

,

x

)

inside the image

π

1

⁢

(

p

)

⁢

(

π

1

⁢

(

E

,

e

)

)

, where

π

1

denotes the

fundamental
group functor

. Furthermore

f

~

is unique (provided it exists of course).

The following diagrams might be useful: To check

Generated on Sat Feb 10 11:44:35 2018 by

LaTeXML
properly discontinuous action

properly discontinuous action

Let

G

be a group and

E

a

topological space

on which

G

acts by

homeomorphisms

, that is there is a

homomorphism

ρ

:

G

→

Aut

⁡

(

E

)

, where
the latter denotes the group of self-homeomorphisms of

E

. The action is
said to be

properly discontinuous

if each point

e

∈

E

has a

neighborhood

U

with the

property

that all non trivial elements of

G

move

U

outside itself:

∀

g

∈

G

g

≠

id

⇒

g

⁢

U

∩

U

=

∅

.

For example, let

p

:

E

→

X

be a covering map, then the group of

deck
transformations

of

p

acts properly discontinuously on

E

. Indeed if

e

∈

E

and

D

∈

Aut

⁡

(

p

)

then one can take as

U

to be any neighborhood with
the property that

p

⁢

(

U

)

is evenly covered. The following shows that this
is the only example:

Theorem

.

Assume that

E

is a

connected

and

locally path connected

Hausdorff
space

. If the group

G

acts properly discontinuously on

E

then the

quotient map

p

:

E

→

E

/

G

is a covering map and

Aut

⁡

(

p

)

=

G

.

Title

properly discontinuous action

Canonical

name

ProperlyDiscontinuousAction

Date of creation

2013-03-22 13:28:11

Last modified on

2013-03-22 13:28:11

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

9

Author

Dr_Absentius (537)

Entry type

Definition

Classification

msc 55R05

Classification

msc 37B05

Related topic

DiscontinuousAction

Related topic

DeckTransformation

Defines

properly discontinuous

Defines

properly discontinuously

Generated on Sat Feb 10 11:44:40 2018 by

LaTeXML
regular covering

regular covering

Theorem 1

.

Let

p

:

E

→

X

be a covering map where

E

and

X

are

connected

and

locally path connected

and let

X

have a

basepoint

*

. The following are

equivalent

:

1.

The action of

Aut

⁡

(

p

)

, the

group of covering transformations

of

p

, is

transitive

on the fiber

p

-

1

⁢

(

*

)

,

2.

for some

e

∈

p

-

1

⁢

(

*

)

,

p

*

⁢

(

π

1

⁢

(

E

,

e

)

)

is a

normal subgroup

of

π

1

⁢

(

X

,

*

)

, where

p

*

denotes

π

1

⁢

(

p

)

,

3.

∀

e

,

e

′

∈

p

-

1

⁢

(

*

)

,

p

*

⁢

(

π

1

⁢

(

E

,

e

)

)

=

p

*

⁢

(

π

1

⁢

(

E

,

e

′

)

)

,

4.

there is a

discrete group

G

such that

p

is a principal

G

-bundle.

All the

elements

for the proof of this

theorem

are

contained

in the articles
about the

monodromy action

and the

deck transformations

.

Definition 2

.

A covering with the

properties

described in the previous theorem is called
a

regular

or

normal

covering. The term

Galois

covering is also used sometimes.

Title

regular covering

Canonical

name

RegularCovering

Date of creation

2013-03-22 13:27:38

Last modified on

2013-03-22 13:27:38

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

6

Author

Dr_Absentius (537)

Entry type

Definition

Classification

msc 55R05

Synonym

normal covering

Synonym

Galois covering

Generated on Sat Feb 10 11:44:42 2018 by

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4 surface bundles

4 surface bundles

Four Kleinbottle bundles

K

⊂

M

→

S

1

.

There are four because the extended

mapping class group

for the genus two, non

orientable surface

K

the

Klein bottle

, is

ℤ

2

⊕

ℤ

2

.

This group is generated by a Dehn-twist

τ

about the unique two-sided curve in

K

and by the

y-homeomorphism

, both representing two

isotopy

classes of order two.

These bundles are

•

K

×

S

1

, the trivial Cartesian product

•

K

×

τ

S

1

,

•

K

×

y

S

1

=

K

×

∼

I

O

∪

(

0

,

1

)

M

⁢

o

¨

×

S

1

,

•

K

×

y

⁢

τ

S

1

.

Where

K

×

∼

I

O

is the

orientable

twisted

I

-bundle over

K

, among the three

I

-bundles over

K

.The symbol

∪

(

0

,

1

)

is used to indicate that, the

meridian

in

∂

⁡

(

M

⁢

o

¨

×

S

1

)

is attached to the meridian of

∂

⁡

(

K

×

∼

I

O

)

, both 2-tori.

M

⁢

o

¨

is the Möbius band.

Now, since those

monodromies

are periodic then they are also

homeomorphic

respectively to the

Seifert fiber spaces

•

(

N

n

I

,

2

|

0

)

=

K

×

S

1

,

•

(

N

n

I

,

2

|

1

)

=

(

K

×

S

1

∖

int

W

)

∪

(

1

,

1

)

W

,

•

(

N

n

I

I

,

2

|

0

)

=

K

×

y

S

1

=

K

×

∼

I

O

∪

(

0

,

1

)

M

o

¨

×

S

1

and

•

(

N

n

I

I

,

2

|

1

)

=

(

K

×

y

S

1

∖

int

W

)

∪

(

1

,

1

)

W

Where

W

is a solid torus in the space and

∪

(

1

,

1

)

is the

Dehn surgery

: meridian of

∂

⁡

W

to the

longitude

of

∂

⁡

(

K

×

S

1

∖

int

⁢

W

)

.

The non trivial

homeomorphisms

were given by Per Orlik and Frank Raymond, in 1969.

Title

4 surface bundles

Canonical name

4SurfaceBundles

Date of creation

2013-03-22 16:01:40

Last modified on

2013-03-22 16:01:40

Owner

juanman (12619)

Last modified by

juanman (12619)

Numerical id

12

Author

juanman (12619)

Entry type

Feature

Classification

msc 55R10

Related topic

SurfaceBundleOverTheCircle

Generated on Sat Feb 10 11:44:45 2018 by

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associated bundle construction

associated bundle construction

Let

G

be a

topological group

,

π

:

P

→

X

a (right) principal

G

-bundle,

F

a

topological space

and

ρ

:

G

→

Aut

⁢

(

F

)

a
representation of

G

as

homeomorphisms

of

F

. Then the

fiber bundle

associated

to

P

by

ρ

, is a fiber bundle

π

ρ

:

P

×

ρ

F

→

X

with fiber

F

and group

G

that is defined as follows:

•

The

total space

is defined as

P

×

ρ

F

:=

P

×

F

/

G

where the (left) action of

G

on

P

×

F

is defined by

g

⋅

(

p

,

f

)

:=

(

p

⁢

g

-

1

,

ρ

⁢

(

g

)

⁢

(

f

)

)

,

∀

g

∈

G

,

p

∈

P

,

F

∈

F

.

•

The projection

π

ρ

is defined by

π

ρ

⁢

[

p

,

f

]

:=

π

⁢

(

p

)

,

where

[

p

,

f

]

denotes the

G

–orbit of

(

p

,

f

)

∈

P

×

F

.

Theorem 1

.

The above is well defined and defines a

G

–bundle over

X

with fiber

F

. Furthermore

P

×

ρ

F

has the same

transition functions

as

P

.

Sketch of proof.

To see that

π

ρ

is well defined just notice that for

p

∈

P

and

g

∈

G

,

π

⁢

(

p

⁢

g

)

=

π

⁢

(

p

)

. To see that the fiber is

F

notice that since
the principal action is simply

transitive

, given

p

∈

P

any orbit of the

G

–action on

P

×

F

contains a

unique

representative of the
form

(

p

,

f

)

for some

f

∈

F

. It is clear that an

open cover

that
trivializes

P

trivializes

P

×

ρ

F

as well.
To see that

P

×

ρ

F

has the same
transition functions as

P

notice that transition functions of

P

act on the
left and thus commute with the principal

G

–action on

P

.
∎

Notice that if

G

is a

Lie group

,

P

a smooth

principal bundle

and

F

is a

smooth manifold

and

ρ

maps inside the diffeomorphism group of

F

, the
above construction produces a smooth bundle. Also quite often

F

has extra

structure

and

ρ

maps into the homeomorphisms of

F

that preserve that
structure. In that case the above construction produces a “bundle of such
structures.” For example when

F

is a vector space and

ρ

⁢

(

G

)

⊂

GL

⁡

(

F

)

, i.e.

ρ

is a linear
representation of

G

we get
a

vector bundle

; if

ρ

⁢

(

G

)

⊂

SL

⁡

(

F

)

we get an

oriented

vector bundle, etc.

Title

associated bundle construction

Canonical name

AssociatedBundleConstruction

Date of creation

2013-03-22 13:26:46

Last modified on

2013-03-22 13:26:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 55R10

Defines

associated bundle

Generated on Sat Feb 10 11:44:47 2018 by

LaTeXML
bundle map

bundle map

Let

E

1

⁢

→

π

1

⁢

B

1

and

E

2

⁢

→

π

2

⁢

B

2

be

fiber bundles

for which there is a

continuous map

f

:

B

1

→

B

2

of

base spaces

. A

bundle map

(or

bundle morphism

) is a commutative square

\xymatrix

⁢

E

1

⁢

\ar

⁢

[

r

]

f

^

⁢

\ar

⁢

[

d

]

π

1

⁢

&

⁢

E

2

⁢

\ar

⁢

[

d

]

π

2

⁢

B

1

⁢

\ar

⁢

[

r

]

f

⁢

&

⁢

B

2

such that the

induced map

E

1

→

f

-

1

⁢

E

2

is a

homeomorphism

(here

f

-

1

⁢

E

2

denotes the pullback of

f

along the

bundle projection

π

2

).

Title

bundle map

Canonical name

BundleMap

Date of creation

2013-03-22 13:07:24

Last modified on

2013-03-22 13:07:24

Owner

RevBobo (4)

Last modified by

RevBobo (4)

Numerical id

5

Author

RevBobo (4)

Entry type

Definition

Classification

msc 55R10

Defines

bundle morphism

Generated on Sat Feb 10 11:44:49 2018 by

LaTeXML
fiber bundle

fiber bundle

Let

F

be a

topological space

and

G

be a

topological group

which acts on

F

on the left. A

fiber bundle

with fiber

F

and

structure group

G

consists of the following data:

•

a topological space

B

called the

base space

, a space

E

called the

total space

and a

continuous

surjective map

π

:

E

→

B

called the projection of the bundle,

•

an

open cover

{

U

i

}

of

B

along with a

collection

of continuous maps

{

ϕ

i

:

π

-

1

⁢

U

i

→

F

}

called

local trivializations

and

•

a collection of continuous maps

{

g

i

⁢

j

:

U

i

∩

U

j

→

G

}

called

transition functions

which

satisfy

the following

properties

1.

the map

π

-

1

⁢

U

i

→

U

i

×

F

given by

e

↦

(

π

⁢

(

e

)

,

ϕ

i

⁢

(

e

)

)

is a

homeomorphism

for each

i

,

2.

for all indices

i

,

j

and

e

∈

π

-

1

⁢

(

U

i

∩

U

j

)

,

g

j

⁢

i

⁢

(

π

⁢

(

e

)

)

⋅

ϕ

i

⁢

(

e

)

=

ϕ

j

⁢

(

e

)

and

3.

for all indices

i

,

j

,

k

and

b

∈

U

i

∩

U

j

∩

U

k

,

g

i

⁢

j

⁢

(

b

)

⁢

g

j

⁢

k

⁢

(

b

)

=

g

i

⁢

k

⁢

(

b

)

.

Readers familiar with Čech cohomology may recognize condition 3), it is often called the

cocycle condition

. Note, this imples that

g

i

⁢

i

⁢

(

b

)

is the

identity

in

G

for each

b

, and

g

i

⁢

j

⁢

(

b

)

=

g

j

⁢

i

⁢

(

b

)

-

1

.

If the total space

E

is homeomorphic to the

product

B

×

F

so that the

bundle projection

is essentially projection onto the first factor, then

π

:

E

→

B

is called a

trivial bundle

. Some examples of fiber bundles are

vector bundles

and covering spaces.

There is a notion of

morphism

of fiber bundles

E

,

E

′

over the same base

B

with the same structure group

G

. Such a morphism is a

G

-equivariant map

ξ

:

E

→

E

′

, making the following diagram commute

\xymatrix

⁢

E

⁢

\ar

⁢

[

r

⁢

r

]

ξ

⁢

\ar

⁢

[

d

⁢

r

]

π

⁢

&

⁢

&

⁢

E

′

⁢

\ar

⁢

[

d

⁢

l

]

π

′

⁢

&

⁢

B

⁢

&

.

Thus we have a category of fiber bundles over a fixed base with fixed structure group.

Title

fiber bundle

Canonical

name

FiberBundle

Date of creation

2013-03-22 13:07:06

Last modified on

2013-03-22 13:07:06

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

10

Author

bwebste (988)

Entry type

Definition

Classification

msc 55R10

Synonym

fibre bundle

Related topic

ReductionOfStructureGroup

Related topic

SectionOfAFiberBundle

Related topic

Fibration

Related topic

Fibration2

Related topic

HomotopyLiftingProperty

Related topic

SurfaceBundleOverTheCircle

Defines

trivial bundle

Defines

local trivializations

Defines

structure group

Defines

cocycle condition

Defines

local trivialization

Generated on Sat Feb 10 11:44:51 2018 by

LaTeXML
fibre map

fibre map

A

fibre map

is a map of

topological spaces

f

:

E

→

B

, for which there exists a space

F

, such that any point of

B

is contained in some

neighbourhood

U

satisfying the following: on

f

-

1

⁢

U

,

f

restricts to the

natural projection

F

×

U

→

U

, via some

homeomorphic

identification of

f

-

1

⁢

U

with

F

×

U

.

One class of examples are covering maps. Another example is the map

S

⁢

O

3

→

S

2

sending a

rotation

to the point it sends the “

North Pole

” to.

Title

fibre map

Canonical name

FibreMap

Date of creation

2013-03-22 14:32:27

Last modified on

2013-03-22 14:32:27

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

11

Author

whm22 (2009)

Entry type

Definition

Classification

msc 55R10

Synonym

fiber map

Related topic

FibreBundle

Related topic

LocallyTrivialBundle

Related topic

LongExactSequenceLocallyTrivialBundle

Related topic

Fibration2

Related topic

HomotopyLiftingProperty

Defines

fibre map

Generated on Sat Feb 10 11:44:53 2018 by

LaTeXML
locally trivial bundle

locally trivial bundle

A

locally trivial bundle

is a

continuous map

π

:

E

→

B

of

topological spaces

such that the following conditions
hold.
First, each point

x

∈

B

must have a

neighborhood

U

such that
the

inverse image

U

~

=

π

-

1

⁢

(

U

)

is

homeomorphic

to

U

×

π

-

1

⁢

(

x

)

.
Second, for some

homeomorphism

g

:

U

~

→

U

×

π

-

1

⁢

(

x

)

,
the

diagram

\xymatrix

U

~

\ar

[

r

]

(

0.3

)

g

\ar

[

d

]

π

&

U

×

π

-

1

(

x

)

\ar

[

d

]

id

×

{

x

}

U

\ar

[

r

]

id

&

U

must be commutative (

http://planetmath.org/CommutativeDiagram

).

Locally trivial bundles are useful because of their covering homotopy

property

and because each locally trivial bundle has an associated

long exact sequence

(

http://planetmath.org/LongExactSequenceLocallyTrivialBundle

) and Serre spectral sequence. Every

fibre bundle

is an example of a locally trivial bundle.

Title

locally trivial bundle

Canonical

name

LocallyTrivialBundle

Date of creation

2013-03-22 13:15:01

Last modified on

2013-03-22 13:15:01

Owner

mps (409)

Last modified by

mps (409)

Numerical id

10

Author

mps (409)

Entry type

Definition

Classification

msc 55R10

Related topic

Fibration

Related topic

Fibration2

Related topic

HomotopyLiftingProperty

Generated on Sat Feb 10 11:44:57 2018 by

LaTeXML
principal bundle

principal bundle

Let

E

be a

topological space

on which a

topological group

G

acts continuously and freely. The map

π

:

E

→

E

/

G

=

B

is called a

principal bundle

(or

principal

G

-bundle

) if the

projection
map

π

:

E

→

B

is a

locally trivial bundle

.

Any principal bundle with a

section

σ

:

B

→

E

is

trivial

, since the map

ϕ

:

B

×

G

→

E

given by

ϕ

⁢

(

b

,

g

)

=

g

⋅

σ

⁢

(

b

)

is an

isomorphism

. In particular, any

G

-bundle which is topologically trivial is also isomorphic to

B

×

G

as a

G

-space. Thus any

local trivialization

of

π

:

E

→

B

as a topological bundle is an equivariant

trivialization

.

Title

principal bundle

Canonical name

PrincipalBundle

Date of creation

2013-03-22 13:07:18

Last modified on

2013-03-22 13:07:18

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

8

Author

rmilson (146)

Entry type

Definition

Classification

msc 55R10

Defines

principal G-bundle

Generated on Sat Feb 10 11:44:58 2018 by

LaTeXML
pullback bundle

pullback bundle

If

π

:

E

→

B

is a bundle and

f

:

B

′

→

B

is an arbitrary

continuous
map

, then there exists a

pullback

, or induced, bundle

f

*

⁢

(

π

)

:

E

′

→

B

′

, where

E

′

=

{

(

e

,

b

)

∈

E

×

B

′

|

f

⁢

(

b

)

=

π

⁢

(

e

)

}

,

and

f

*

⁢

(

π

)

is the

restriction

of the

projection map

to

B

′

. There is a natural

bundle map

from

f

*

⁢

(

π

)

to

π

with the map

B

′

→

B

given
by

f

, and the map

φ

:

E

′

→

E

given by the restriction of projection.

If

π

is locally

trivial

, a principal

G

-bundle, or a

fiber bundle

, then

f

*

⁢

(

π

)

is as well.
The pullback

satisfies

the following

universal property

:

\xymatrix

&

\ar

[

d

d

l

]

X

\ar

[

d

d

r

]

\ar

@

-

-

>

[

d

]

&

&

\ar

[

d

l

]

f

*

⁢

π

E

′

\ar

[

d

r

]

φ

&

B

′

\ar

[

d

r

]

f

&

&

E

\ar

[

d

l

]

π

&

B

&

(i.e. given a diagram with the solid arrows, a map satisfying the dashed arrow exists).

Title

pullback bundle

Canonical name

PullbackBundle

Date of creation

2013-03-22 13:17:19

Last modified on

2013-03-22 13:17:19

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 55R10

Synonym

induced bundle

Generated on Sat Feb 10 11:45:00 2018 by

LaTeXML