fundamental groupoid functor

fundamental groupoid functor

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The following quote indicates how

fundamental groupoids

(

http://planetmath.org/FundamentalGroupoid

)
can be alternatively defined

via

the Yoneda-Grothendieck construction specified by
the

fundamental groupoid functor

as in reference

[

1

]

.

0.1

Fundamental groupoid functor:

“Therefore the

fundamental groupoid

,

Π

can (and should) be regarded as a

functor

from the category of topological spaces to the

category of groupoids

. This functor is not really

homotopy invariant

but it is
“homotopy invariant up to

homotopy

” in the sense that the following holds:

Theorem 0.1

.

“A homotopy between two

continuous maps

induces a

natural transformation

between the corresponding functors.”
(

provided without proof

).

0.2

Remarks

On the other hand, the category of groupoids

G

2

, as defined previously, is in fact a

2

-

c

⁢

a

⁢

t

⁢

e

⁢

g

⁢

o

⁢

r

⁢

y

, whereas the

category

T

⁢

o

⁢

p

- as defined in the above quote- is not viewed as a

2

-

c

⁢

a

⁢

t

⁢

e

⁢

g

⁢

o

⁢

r

⁢

y

. An alternative approach involves the

representation

of the category

T

⁢

o

⁢

p

via

the Yoneda-Grothendieck construction as recently reported by Brown and Janelidze. This leads then to an

extension

of the Galois theory involving

groupoids

viewed as single object categories with

invertible

morphisms

, and also to a more useful definition of the

fundamental groupoid functor

, as reported by Brown and Janelidze (2004); they have used the generalised Galois Theory to

construct

a

homotopy double groupoid

of a

surjective

fibration

of Kan

simplicial sets

, and proceeded to utilize the latter to construct a new homotopy double groupoid of a map of spaces, which includes constructions by several other authors of a

2

-

g

⁢

r

⁢

o

⁢

u

⁢

p

⁢

o

⁢

i

⁢

d

, the

c

⁢

a

⁢

t

⁢

1

-

g

⁢

r

⁢

o

⁢

u

⁢

p

or crossed modules. Another advantage of such a

categorical

construction utilizing a

double groupoid

is that it provides an algebraic model of a foliated bundle (

[

1

]

).
A natural extension of the double groupoid is a

double category

that is not restricted to the condition of all invertible morphisms of the double groupoid; (for further details see ref.

[

1

]

).
Note also that an alternative definition of the fundamental functor(s) was introduced by Alexander Grothendieck
in ref.

[

2

]

.

References

1

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 .

2

Alexander Grothendieck. 1971, Rev

ê

tements

É

tales et Groupe Fondamental (SGA1),
chapter VI: Cat

é

gories fibr

é

es et descente,

Lecture Notes in Math.

,

224

, Springer–Verlag: Berlin.

Title

fundamental groupoid functor

Canonical name

FundamentalGroupoidFunctor

Date of creation

2013-03-22 18:12:03

Last modified on

2013-03-22 18:12:03

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

48

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55R10

Classification

msc 55R65

Classification

msc 22A22

Classification

msc 55P99

Classification

msc 20L05

Classification

msc 18A30

Classification

msc 55R37

Synonym

fundamental groupoid

Related topic

FundamentalGroupoid

Related topic

2Category

Related topic

TopologicalSpace

Related topic

HigherDimensionalAlgebraHDA

Related topic

FundamentalGroupoid2

Related topic

HomotopyDoubleGroupoidOfAHausdorffSpace

Related topic

QuantumFundamentalGroupoids

Related topic

HomotopyCategory

Related topic

GrothendieckCategory

Related topic

2CategoryOfDoubleGroupoids

Related topic

DoubleCategory3

Defines

fundamental groupoid functor

Defines

double groupoid

Defines

double category

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