# 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 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 $Top$- as defined in the above quote- is not viewed as a $2-category$. An alternative approach involves the representation of the category $Top$ 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