aarjaneiro/planetmath huggingface dataset:
suspension isomorphism

suspension isomorphism

Proposition 1

.

Let

X

be a

topological space

. There is a

natural isomorphism

s

:

H

n

+

1

⁢

(

S

⁢

X

)

→

H

n

⁢

(

X

)

,

where

S

⁢

X

stands for the

unreduced suspension

of

X

.

If

X

has a

basepoint

, there is a natural isomorphism

s

:

H

~

n

+

1

⁢

(

Σ

⁢

X

)

→

H

~

n

⁢

(

X

)

,

where

Σ

⁢

X

is
the reduced suspension.

A

similar

proposition

holds with

homology

replaced by

cohomology

.

In fact, these propositions follow from the

Eilenberg-Steenrod axioms

without the dimension axiom, so they hold for any generalized (co)homology theory in place of integral (co)homology.

Title

suspension isomorphism

Canonical

name

SuspensionIsomorphism

Date of creation

2013-03-22 13:25:58

Last modified on

2013-03-22 13:25:58

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

4

Author

antonio (1116)

Entry type

Theorem

Classification

msc 55N99

Related topic

Suspension

Generated on Sat Feb 10 11:42:32 2018 by

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cofibration

cofibration

If

X

is a

topological space

and

A

is a

subspace

of

X

, the

inclusion map

i

:

A

→

X

is a

cofibration

if it has the

homotopy extension property

with respect to any space

Y

.

Title

cofibration

Canonical name

Cofibration

Date of creation

2013-03-22 12:13:32

Last modified on

2013-03-22 12:13:32

Owner

RevBobo (4)

Last modified by

RevBobo (4)

Numerical id

6

Author

RevBobo (4)

Entry type

Definition

Classification

msc 55P05

Related topic

Fibration2

Generated on Sat Feb 10 11:42:34 2018 by

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homotopy extension property

homotopy extension property

Let

X

be a

topological space

and

A

a

subspace

of

X

. Suppose there is a

continuous map

f

:

X

→

Y

and a

homotopy of maps

F

:

A

×

I

→

Y

. The inclusion map

i

:

A

→

X

is said to have the

homotopy extension property

if there exists a continuous map

F

′

such that the following diagram commutes:

Generated on Sat Feb 10 11:42:39 2018 by

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homotopy equivalence

homotopy equivalence

Definition

Suppose that

X

and

Y

are

topological spaces

and

f

:

X

→

Y

is a

continuous map

.
If there exists a
continuous map

g

:

Y

→

X

such that

f

∘

g

≃

i

⁢

d

Y

(i.e.

f

∘

g

is

http://planetmath.org/node/1584

homotopic

to the

identity
mapping

on

Y

),
and

g

∘

f

≃

i

⁢

d

X

, then

f

is a

homotopy equivalence

.
This homotopy equivalence is sometimes called

strong homotopy equivalence

to distinguish it from

weak homotopy equivalence

.

If there exist a homotopy equivalence between the topological
spaces

X

and

Y

, we say that

X

and

Y

are

homotopy equivalent

, or that

X

and

Y

are of the same

homotopy type

.
We then write

X

≃

Y

.

0.0.1

Properties

1.

Any

homeomorphism

f

:

X

→

Y

is obviously a homotopy equivalence with

g

=

f

-

1

.

2.

For topological spaces, homotopy equivalence is an

equivalence relation

.

3.

A topological space

X

is (by definition)

contractible

,
if

X

is homotopy equivalent to a point, i.e.,

X

≃

{

x

0

}

.

References

1

A. Hatcher,

Algebraic Topology

, Cambridge University Press, 2002. Also available

http://www.math.cornell.edu/ hatcher/AT/ATpage.html

online.

Title

homotopy equivalence

Canonical name

HomotopyEquivalence

Date of creation

2013-03-22 12:13:22

Last modified on

2013-03-22 12:13:22

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

14

Author

matte (1858)

Entry type

Definition

Classification

msc 55P10

Related topic

HomotopyOfMaps

Related topic

WeakHomotopyEquivalence

Related topic

Contractible

Related topic

HomotopyInvariance

Related topic

ChainHomotopyEquivalence

Related topic

PathConnectnessAsAHomotopyInvariant

Related topic

TheoremOnCWComplexApproximationOfQuantumStateSpacesInQAT

Defines

homotopy equivalent

Defines

homotopically equivalent

Defines

homotopy type

Defines

strong homotopy equivalence

Generated on Sat Feb 10 11:42:41 2018 by

LaTeXML
nullhomotopic map

nullhomotopic map

Let

f

:

X

→

Y

be a

continuous map

. We say that

f

is

nullhomotopic

if there is a

homotopy

from

f

to a constant map

c

, that is,

f

≃

c

.

Title

nullhomotopic map

Canonical name

NullhomotopicMap

Date of creation

2013-03-22 14:00:26

Last modified on

2013-03-22 14:00:26

Owner

drini (3)

Last modified by

drini (3)

Numerical id

6

Author

drini (3)

Entry type

Definition

Classification

msc 55P10

Generated on Sat Feb 10 11:42:43 2018 by

LaTeXML
weak homotopy equivalence

weak homotopy equivalence

A

continuous map

f

:

X

→

Y

between

path-connected

based
topological spaces

is said to be a

weak homotopy equivalence

if for each

k

≥

1

it induces an

isomorphism

f

*

:

π

k

⁢

(

X

)

→

π

k

⁢

(

Y

)

between the

k

th

homotopy groups

.

X

and

Y

are then said to be

weakly
homotopy equivalent.

Remark 1

.

It is

not

enough for

π

k

⁢

(

X

)

to be isomorphic to

π

k

⁢

(

Y

)

for all

k

.

The definition requires these isomorphisms
to be induced by a space-level map

f

.

Remark 2

.

More generally, two spaces

X

and

Y

are defined to be weakly homotopy equivalent if there is a

sequence

of spaces and maps

X

→

X

1

←

X

2

→

X

3

←

⋯

→

X

n

←

Y

in which each map is a weak homotopy equivalence.

Title

weak homotopy equivalence

Canonical name

WeakHomotopyEquivalence

Date of creation

2013-03-22 13:25:45

Last modified on

2013-03-22 13:25:45

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

9

Author

antonio (1116)

Entry type

Definition

Classification

msc 55P10

Synonym

weak equivalence

Related topic

HomotopyEquivalence

Related topic

WeakHomotopyAdditionLemma

Related topic

ApproximationTheoremForAnArbitrarySpace

Related topic

OmegaSpectrum

Related topic

WhiteheadTheorem

Defines

weakly homotopy equivalent

Defines

weakly equivalent

Generated on Sat Feb 10 11:42:45 2018 by

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Whitehead theorem

Whitehead theorem

Theorem 1

(J.H.C. Whitehead)

If

f

:

X

→

Y

is a

weak homotopy equivalence

and

X

and

Y

are

path-connected

and of the

homotopy type

of

CW complexes

, then

f

is a strong homotopy equivalence.

Remark 1

It is essential to the theorem that

isomorphisms

between

π

k

⁢

(

X

)

and

π

k

⁢

(

Y

)

for all

k

are induced by a map

f

:

X

→

Y

;

if an isomorphism exists which is not induced by a map, it need not be the case that the spaces are homotopy equivalent.

For example, let

X

=

ℝ

⁢

P

m

×

S

n

and

Y

=

ℝ

⁢

P

n

×

S

m

.

Then the two spaces have isomorphic

homotopy groups

because they both have a

universal covering space

homeomorphic

to

S

m

×

S

n

,

and it is a double

covering

in both cases. However, for

m

<

n

,

X

and

Y

are not homotopy equivalent, as can be seen, for example, by using

homology

:

H

m

⁢

(

X

;

ℤ

/

2

⁢

ℤ

)

≅

ℤ

/

2

⁢

ℤ

,

but

H

m

⁢

(

Y

;

ℤ

/

2

⁢

ℤ

)

≅

ℤ

/

2

⁢

ℤ

⊕

ℤ

/

2

⁢

ℤ

.

(Here,

ℝ

⁢

P

n

is

n

-dimensional real projective space, and

S

n

is the

n

-sphere.)

Title

Whitehead theorem

Canonical name

WhiteheadTheorem

Date of creation

2013-03-22 13:25:48

Last modified on

2013-03-22 13:25:48

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

10

Author

antonio (1116)

Entry type

Theorem

Classification

msc 55P10

Classification

msc 55P15

Classification

msc 55Q05

Related topic

ConjectureApproximationTheoremHoldsForWhitneyCrMNSpaces

Related topic

WeakHomotopyEquivalence

Related topic

ApproximationTheoremAppliedToWhitneyCrMNSpaces

Generated on Sat Feb 10 11:42:48 2018 by

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(path) connectness as a homotopy invariant

(path) connectness as a homotopy invariant

Theorem.

Let

X

and

Y

be arbitrary

topological spaces

with

Y

(path)

connected

. If there are maps

f

:

X

→

Y

and

g

:

Y

→

X

such that

g

∘

f

:

X

→

X

is

homotopic

to the identity map, then

X

is (path) connected.

Proof:

Let

f

:

X

→

Y

and

g

:

Y

→

X

be maps satisfying theorem’s assumption. Furthermore let

X

=

⋃

X

i

be a decomposition of

X

into (path)

connected components

. Since

Y

is (path) connected, then

g

⁢

(

Y

)

⊆

X

i

for some

i

. Thus

(

g

∘

f

)

⁢

(

X

)

⊆

X

i

. Now let

H

:

I

×

X

→

X

be the

homotopy

from

g

∘

f

to the identity map. Let

α

x

:

I

→

X

be a path defined by the formula:

α

x

⁢

(

t

)

=

H

⁢

(

t

,

x

)

. Since for all

x

∈

X

we have

α

x

⁢

(

0

)

∈

X

i

and

I

is

path connected

, then

α

x

⁢

(

I

)

⊆

X

i

. Therefore

H

⁢

(

I

×

X

)

⊆

X

i

, but

H

⁢

(

{

1

}

×

X

)

=

X

which implies that

X

i

=

X

, so

X

is (path) connected.

□

Straightforward application of this theorem is following:

Corollary.

Let

X

and

Y

be

homotopy equivalent

spaces. Then

X

is (path) connected if and only if

Y

is (path) connected.

Title

(path) connectness as a

homotopy invariant

Canonical name

pathConnectnessAsAHomotopyInvariant

Date of creation

2013-03-22 18:02:15

Last modified on

2013-03-22 18:02:15

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

8

Author

joking (16130)

Entry type

Theorem

Classification

msc 55P10

Related topic

Homotopy

Related topic

homotopyequivalence

Related topic

path

Related topic

connectedspace

Generated on Sat Feb 10 11:42:50 2018 by

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simply connected

simply connected

A

topological space

is said to be

simply connected

if it is

path connected

and the

fundamental group

of the space is trivial (i.e. the one element group). What this means, basically, is that every path on the space can be shrunk to a point. This is

equivalent

to saying that every path is

contractible

. A simply connected space can be visualized as a space with no “holes”.

Some basic examples of a simply connected space are the

unit disc

in

ℝ

2

,

S

2

or the

Riemann sphere

. Non-examples of a simply connected space are the circle, the

annulus

, and a punctured plane (a plane with a point removed). In each of the non-examples, any

closed curve

around the “hole” is a path that can not be shrunk to a point.

Title

simply connected

Canonical name

SimplyConnected

Date of creation

2013-03-22 11:59:33

Last modified on

2013-03-22 11:59:33

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 55P15

Related topic

SemilocallySimplyConnected

Generated on Sat Feb 10 11:42:51 2018 by

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Eilenberg-MacLane space

Eilenberg-MacLane space

Let

π

be a

discrete group

. A

based topological space

X

is called an

Eilenberg-MacLane space

of type

K

⁢

(

π

,

n

)

, where

n

≥

1

,

if all the

homotopy groups

π

k

⁢

(

X

)

are

trivial

except for

π

n

⁢

(

X

)

,

which is

isomorphic

to

π

.

Clearly, for such a space to exist when

n

≥

2

,

π

must be

abelian

.

Given any group

π

,

with

π

abelian if

n

≥

2

,

there exists an Eilenberg-MacLane space of type

K

⁢

(

π

,

n

)

.

Moreover, this space can be constructed as a

CW complex

. It turns out that any two Eilenberg-MacLane spaces of type

K

⁢

(

π

,

n

)

are

weakly homotopy equivalent

. The

Whitehead theorem

then implies that there is a unique

K

⁢

(

π

,

n

)

space up to

homotopy equivalence

in the category of topological spaces of the

homotopy type

of a CW complex. We will henceforth restrict ourselves to this

category

. With a slight abuse of notation, we refer to any such space as

K

⁢

(

π

,

n

)

.

An important property of

K

⁢

(

π

,

n

)

is that, for

π

abelian,
there is a

natural isomorphism

H

n

⁢

(

X

;

π

)

≅

[

X

,

K

⁢

(

π

,

n

)

]

of contravariant set-valued

functors

, where

[

X

,

K

⁢

(

π

,

n

)

]

is the set of

homotopy classes

of based maps from

X

to

K

⁢

(

π

,

n

)

.

Thus one says that the

K

⁢

(

π

,

n

)

are

representing spaces

for

cohomology

with coefficients in

π

.

Remark 1

.

Even when the group

π

is

nonabelian

, it can be seen that the set

[

X

,

K

⁢

(

π

,

1

)

]

is naturally isomorphic to

Hom

(

π

1

⁢

(

X

)

,

π

)

/

π

;

that is,
to

conjugacy classes

of

homomorphisms

from

π

1

⁢

(

X

)

to

π

.

In fact, this
is a way to define

H

1

⁢

(

X

;

π

)

when

π

is nonabelian.

Remark 2

.

Though the above description does not include the case

n

=

0

,

it is natural to define a

K

⁢

(

π

,

0

)

to be any space homotopy equivalent to

π

.

The above statement about cohomology then becomes true for the

reduced

zeroth cohomology functor.

Title

Eilenberg-MacLane space

Canonical name

EilenbergMacLaneSpace

Date of creation

2013-03-22 13:25:42

Last modified on

2013-03-22 13:25:42

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

6

Author

antonio (1116)

Entry type

Definition

Classification

msc 55P20

Synonym

Eilenberg-Mac Lane space

Related topic

NaturalTransformation

Related topic

LoopSpace

Related topic

HomotopyGroups

Related topic

RepresentableFunctor

Related topic

FundamentalGroupoid2

Related topic

CohomologyGroupTheorem

Related topic

ProofOfCohomologyGroupTheorem

Related topic

OmegaSpectrum

Generated on Sat Feb 10 11:42:54 2018 by

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H-space

H-space

A

topological space

X

is said to be an

H-space

(or

Hopf-space

)
if there exists
a

continuous

binary operation

φ

:

X

×

X

→

X

and a point

p

∈

X

such that the functions

X

→

X

defined by

x

↦

φ

⁢

(

p

,

x

)

and

x

↦

φ

⁢

(

x

,

p

)

are both

homotopic

to the identity map via

homotopies

that leave

p

fixed.
The element

p

is sometimes referred to as an ‘

identity

’,
although it need not be an

identity element

in the usual sense.
Note that the definition implies that

φ

⁢

(

p

,

p

)

=

p

.

Topological groups

are examples of H-spaces.

If

X

is an H-space with ‘identity’

p

,
then the

fundamental group

π

1

⁢

(

X

,

p

)

is

abelian

.
(However, it is possible for the fundamental group to be

non-abelian

for other choices of basepoint, if

X

is not

path-connected

.)

Title

H-space

Canonical name

Hspace

Date of creation

2013-03-22 16:18:18

Last modified on

2013-03-22 16:18:18

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

4

Author

yark (2760)

Entry type

Definition

Classification

msc 55P45

Synonym

Hopf-space

Synonym

H space

Synonym

Hopf space

Generated on Sat Feb 10 11:42:55 2018 by

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equivalent statements to statement that sphere is not contractible

equivalent statements to statement that sphere is not contractible

Let

V

be a

normed space

. Recall the definition of the sphere and the ball in

V

:

𝕊

=

{

v

∈

V

;

∥

v

∥

=

1

}

;

𝔹

=

{

v

∈

V

;

∥

v

∥

≤

1

}

.

Proposition

.

The following are

equivalent

:

(

1

)

𝕊

is not

contractible

;

(

2

)

for each continous map

F

:

𝔹

→

𝔹

there exists

x

∈

𝔹

such that

F

⁢

(

x

)

=

x

;

(

3

)

there is no

retraction

from

𝔹

onto

𝕊

.

Proof.

The proof of this proposition probably can be found in some books about

topology

. I present here the proof from my lecture due to Prof.

G

⁢

o

´

⁢

rniewicz

.

(

1

)

⇒

(

2

)

Assume there exists a continous map

F

:

𝔹

→

𝔹

such that for each

x

∈

𝔹

we have

F

⁢

(

x

)

≠

x

. Define a map

H

:

𝕊

×

[

0

,

1

]

→

𝕊

as follows:

H

(

x

,

t

)

=

{

x

-

2

⁢

t

⁢

F

⁢

(

x

)

∥

x

-

2

⁢

t

⁢

F

⁢

(

x

)

∥

,

if

⁢

0

≤

t

≤

1

2

(

2

-

2

⁢

t

)

⁢

x

-

F

⁢

(

(

2

-

2

⁢

t

)

⁢

x

)

∥

(

2

-

2

⁢

t

)

⁢

x

-

F

⁢

(

(

2

-

2

⁢

t

)

⁢

x

)

∥

if

⁢

1

2

≤

t

≤

1

Thanks to the condition

F

⁢

(

x

)

≠

x

this map is well defined and it is easy to check that this is a

homotopy

from the

identity map

to

constant map

. But

𝕊

is not contractible.

Contradiction

.

(

2

)

⇒

(

3

)

Assume there exists a retraction

r

:

𝔹

→

𝕊

. Define a map

F

:

𝔹

→

𝔹

by the

formula

F

⁢

(

x

)

=

-

r

⁢

(

x

)

. This map has no

fixed point

. Contradiction.

(

3

)

⇒

(

1

)

Assume that

𝕊

is contractible and take any homotopy

H

:

𝕊

×

[

0

,

1

]

→

𝕊

from constant map to identity map, i.e. for all

x

∈

𝕊

we have

H

⁢

(

x

,

0

)

=

x

0

(for some

x

0

∈

𝕊

) and

H

⁢

(

x

,

1

)

=

x

. Define a map

r

:

𝔹

→

𝕊

as follows:

r

(

x

)

=

{

x

0

,

if

⁢

∥

x

∥

≤

1

2

H

⁢

(

x

∥

x

∥

,

2

⁢

∥

x

∥

-

1

)

if

⁢

∥

x

∥

≥

1

2

It is easy to see that this formula defines a retraction from

𝔹

onto

𝕊

. Contradiction.

□

Note that this proposition does not state that any of the conditions

(

1

)

,

(

2

)

,

(

3

)

hold. It only states that they are equivalent. It is well known that all of them are true if

V

is finite dimensional and all are false if

V

is infinite dimensional.

Title

equivalent statements to statement that sphere is not contractible

Canonical name

EquivalentStatementsToStatementThatSphereIsNotContractible

Date of creation

2013-03-22 18:07:53

Last modified on

2013-03-22 18:07:53

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

10

Author

joking (16130)

Entry type

Theorem

Classification

msc 55P99

Generated on Sat Feb 10 11:42:58 2018 by

LaTeXML
every map into sphere which is not onto is nullhomotopic

every map into sphere which is not onto is nullhomotopic

Proposition

. Let

X

be a

topological space

and

f

:

X

→

𝕊

n

a continous map from

X

to

n

-dimensional sphere which is not onto. Then

f

is

nullhomotopic

.

Proof

. Assume that there is

y

0

∈

𝕊

n

such that

y

0

∉

im

⁢

(

f

)

. It is well known that there is a

homeomorphism

ϕ

:

𝕊

n

∖

{

y

0

}

→

ℝ

n

. Then we have an

induced map

ϕ

∘

f

:

X

→

ℝ

n

.

Since

ℝ

n

is

contractible

, then there is

c

∈

ℝ

n

such that

ϕ

∘

f

is

homotopic

to the

constant map

in

c

(denoted with the same symbol

c

). Let

ψ

:

ℝ

n

→

𝕊

n

be a map such that

ψ

⁢

(

x

)

=

ϕ

-

1

⁢

(

x

)

(note that

ψ

is not the

inverse

of

ϕ

because

ψ

is not onto) and take any

homotopy

H

:

I

×

X

→

ℝ

n

from

ϕ

∘

f

to

c

. Then we have a homotopy

F

:

I

×

X

→

𝕊

n

defined by the

formula

F

=

ψ

∘

H

. It is clear that

F

⁢

(

0

,

x

)

=

ψ

⁢

(

H

⁢

(

0

,

x

)

)

=

ψ

⁢

(

ϕ

⁢

(

f

⁢

(

x

)

)

)

=

f

⁢

(

x

)

;

F

⁢

(

1

,

x

)

=

ψ

⁢

(

H

⁢

(

1

,

x

)

)

=

ψ

⁢

(

c

)

∈

𝕊

n

.

Thus

F

is a homotopy from

f

to a constant map.

□

Corollary

. If

A

⊆

𝕊

n

is a

deformation retract

of

𝕊

n

, then

A

=

𝕊

n

.

Proof

. If

A

⊆

X

then by deformation retraction (associated to

A

) we understand a map

R

:

I

×

X

→

X

such that

R

⁢

(

0

,

x

)

=

x

for all

x

∈

X

,

R

⁢

(

1

,

a

)

=

a

for all

a

∈

A

and

R

⁢

(

1

,

x

)

∈

A

for all

x

∈

X

. Thus a deformation retract is a subset

A

⊆

X

such that there is a deformation retraction

R

:

I

×

X

→

X

associated to

A

.

Assume that

A

is a deformation retract of

𝕊

n

and

A

≠

𝕊

n

. Let

R

:

I

×

𝕊

n

→

𝕊

n

be a deformation retraction. Then

r

:

𝕊

n

→

𝕊

n

such that

r

⁢

(

x

)

=

R

⁢

(

1

,

x

)

is homotopic to the

identity map

(by definition of a deformation retract), but on the other hand it is homotopic to a constant map (it follows from the proposition, since

r

is not onto, because

A

is a

proper subset

of

𝕊

n

). Thus the identity map is homotopic to a constant map, so

𝕊

n

is contractible.

Contradiction

.

□

Title

every map into sphere which is not onto is nullhomotopic

Canonical name

EveryMapIntoSphereWhichIsNotOntoIsNullhomotopic

Date of creation

2013-03-22 18:31:41

Last modified on

2013-03-22 18:31:41

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

8

Author

joking (16130)

Entry type

Theorem

Classification

msc 55P99

Generated on Sat Feb 10 11:43:00 2018 by

LaTeXML
functorial morphism

functorial morphism

Functorial morphism

is another name for

natural transformation

which was, and still is,
employed especially in the

context

of

category theory

and applications developed by Charles Ehresmann, the ‘Nicolas Bourbaki’ group and other French schools of mathematics; this is also a natural, English

translation

of the same concept from French, that is a ‘

morphism between functors

’, viz. (ref.

[

4

]

).

References

1

A. Hatcher,

Algebraic Topology

, Cambridge University Press, 2002.

2

S. Mac Lane,

Categories

for the Working Mathematician

(2nd edition), Springer-Verlag, 1997.

3

C. Ehresmann, Trends Toward Unity in Mathematics.,

Cahiers de Topologie et Geometrie Differentielle

8

: 1-7, 1966.

4

C. Ehresmann,

Catégories et

Structures

. Dunod: Paris , 1965.

5

C. Ehresmann, Catégories doubles des quintettes: applications covariantes
,

C.R.A.S. Paris

,

256

: 1891-1894, 1963.

6

C. Ehresmann,

Oeuvres complètes et commentées:
Amiens, 1980-84

, 1984 (

edited and commented by Andrée Ehresmann

).

7

S. Eilenberg and S. Mac Lane.,

Natural Isomorphisms

in Group Theory.,

American Mathematical Society

43

: 757-831, 1942.

8

S. Eilenberg and S. Mac Lane, The General Theory of Natural Equivalences,

Transactions of the American Mathematical Society

58

: 231-294, 1945.

9

P. Gabriel, Des catégories abéliennes,

Bull. Soc.Math. France

90

: 323-448, 1962.

10

A. Grothendieck, and J. Dieudoné,

Eléments de geometrie algébrique.

,

Publ. Inst. des Hautes Etudes de Science

,

4

, 1960.

11

N. Popescu,

Abelian Categories

with Applications to Rings and Modules.

, New York and London: Academic Press.,
1973, 2nd edn. 1975,

(English translation by I.C. Baianu)

.

Title

functorial morphism

Canonical name

FunctorialMorphism

Date of creation

2013-03-22 18:14:53

Last modified on

2013-03-22 18:14:53

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

20

Author

bci1 (20947)

Entry type

Feature

Classification

msc 55P99

Classification

msc 55R10

Classification

msc 55R65

Classification

msc 55R37

Classification

msc 18A05

Classification

msc 18A25

Classification

msc 18-00

Synonym

morphisms between functors

Synonym

morphism

in the 2–category of functorsnatural

transformations

Related topic

NaturalTransformation

Related topic

FundamentalGroupoidFunctor

Related topic

Monad

Related topic

EilenbergMacLaneSpace

Related topic

Sheaf2

Related topic

FundamentalGroupoid

Related topic

HomotopyDoubleGroupoidOfAHausdorffSpace

Defines

transformations

Generated on Sat Feb 10 11:43:02 2018 by

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