homotopy category

homotopy category

0.1

Homotopy category, fundamental groups and fundamental groupoids

Let us consider first the

category

T

⁢

o

⁢

p

whose objects are

topological spaces

X

with a chosen

basepoint

x

∈

X

and whose

morphisms

are

continuous maps

X

→

Y

that associate the basepoint of

Y

to the
basepoint of

X

. The

fundamental group

of

X

specifies a

functor

T

⁢

o

⁢

p

→

𝐆

,
with

G

being the category of groups and

group homomorphisms

, which is called

the

fundamental group functor

.

0.2

Homotopy category

Next, when one has a suitably defined

relation

of

homotopy

between morphisms, or maps, in a category

U

, one can define the

homotopy category

h

⁢

U

as the category whose objects are the same as the objects of

U

, but with morphisms being defined by the

homotopy classes

of maps; this is in fact the homotopy category of

unbased spaces

.

0.3

Fundamental groups

We can further require that homotopies on

T

⁢

o

⁢

p

map each basepoint to a corresponding basepoint, thus leading to the definition of the

homotopy category

h

⁢

T

⁢

o

⁢

p

of based spaces

. Therefore, the fundamental group is a

homotopy invariant

functor on

T

⁢

o

⁢

p

, with the meaning that the latter functor factors through a functor

h

⁢

T

⁢

o

⁢

p

→

𝐆

. A

homotopy equivalence

in

U

is an

isomorphism

in

h

⁢

T

⁢

o

⁢

p

. Thus, based homotopy equivalence induces an isomorphism of fundamental groups.

0.4

Fundamental groupoid

In the general case when one does not choose a basepoint, a

fundamental groupoid

Π

1

⁢

(

X

)

of a topological space

X

needs to be defined as the category whose objects are
the base points of

X

and whose morphisms

x

→

y

are the

equivalence classes

of paths from

x

to

y

.

•

Explicitly, the objects of

Π

1

⁢

(

X

)

are the points of

X

Obj

⁢

(

Π

1

⁢

(

X

)

)

=

X

,

•

morphisms are homotopy classes of paths “rel

endpoints

” that is

Hom

Π

1

⁢

(

x

)

(

x

,

y

)

=

Paths

(

x

,

y

)

/

∼

,

where,

∼

denotes homotopy rel endpoints, and,

•

composition

of morphisms is defined

via

piecing together, or

concatenation

, of paths.

0.5

Fundamental groupoid functor

Therefore, the set of

endomorphisms

of an object

x

is precisely the fundamental group

π

⁢

(

X

,

x

)

. One can thus

construct

the

groupoid

of homotopy equivalence classes

; this construction can be then carried out by utilizing functors from the category

T

⁢

o

⁢

p

, or its

subcategory

h

⁢

U

,
to the

category of groupoids

and

groupoid homomorphisms

,

G

⁢

r

⁢

p

⁢

d

. One such functor
which associates to each topological space its fundamental (homotopy) groupoid is appropriately called the

fundamental groupoid functor

.

0.6

An example: the category of simplicial, or CW-complexes

As an important example, one may wish to consider the category of simplicial, or

C

⁢

W

-complexes and homotopy defined
for

C

⁢

W

-complexes. Perhaps, the simplest example is that of a one-dimensional

C

⁢

W

-complex, which is a graph.
As described above, one can define a functor from the

category of graphs

,

Grph

, to

G

⁢

r

⁢

p

⁢

d

and then define the fundamental

homotopy groupoids

of graphs,

hypergraphs

, or

pseudographs

. The case of

freely generated

graphs (one-dimensional

C

⁢

W

-complexes) is particularly simple and can be computed with a digital computer by a finite
algorithm using the finite groupoids associated with such

finitely generated

C

⁢

W

-complexes.

0.6.1

Remark

Related to this

concept

of homotopy category for unbased topological spaces, one can then prove the

approximation theorem for an arbitrary space

by considering a functor

Γ

:

𝐡𝐔

⟶

𝐡𝐔

,

and also the construction of an approximation of an arbitrary space

X

as the

colimit

Γ

⁢

X

of a

sequence

of cellular

inclusions

of

C

⁢

W

-complexes

X

1

,

…

,

X

n

, so
that one obtains

X

≡

c

⁢

o

⁢

l

⁢

i

⁢

m

⁢

[

X

i

]

.

Furthermore, the

homotopy groups

of the

C

⁢

W

-complex

Γ

⁢

X

are the colimits of the
homotopy groups of

X

n

, and

γ

n

+

1

:

π

q

⁢

(

X

n

+

1

)

⟼

π

q

⁢

(

X

)

is a group epimorphism.

References

1

May, J.P. 1999,

A Concise Course in Algebraic Topology.

, The University of Chicago Press: Chicago

2

R. Brown and G. Janelidze.(2004). Galois theory and a new

homotopy double groupoid

of a map of spaces.(2004).

Applied

Categorical

Structures

,

12

: 63-80. Pdf file in arxiv: math.AT/0208211

Title

homotopy category

Canonical name

HomotopyCategory

Date of creation

2013-03-22 18:17:07

Last modified on

2013-03-22 18:17:07

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

44

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55P99

Classification

msc 55R10

Classification

msc 55R05

Classification

msc 55R65

Classification

msc 55R37

Synonym

category of homotopy equivalence classes

Related topic

FundamentalGroupoidFunctor

Related topic

TopologicalSpace

Related topic

ApproximationTheoremForAnArbitrarySpace

Related topic

FundamentalGroupoid

Related topic

RiemannianMetric

Related topic

CohomologyGroupTheorem

Related topic

OmegaSpectrum

Related topic

CategoryOfGroupoids2

Defines

fundamental groupoid

Defines

fundamental group functor

Defines

homotopy category

Defines

fundamental groupoid of a topological space

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