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, $\Pi$ 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 $2-category$ , whereas the category $T o p$ - as defined in the above quote- is not viewed as a $2-category$ . 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 $2-groupoid$ , the $cat1-group$ 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 $\^{e}$ tements $\'{E}$ tales et Groupe Fondamental (SGA1), chapter VI: Cat $\'{e}$ gories fibr $\'{e}$ 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