fundamental groupoid functor



The following quote indicates how fundamental groupoidsMathworldPlanetmathPlanetmathPlanetmath ( 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 functorMathworldPlanetmath from the category of topological spaces to the category of groupoidsPlanetmathPlanetmath. This functor is not really homotopy invariant but it is “homotopy invariant up to homotopyMathworldPlanetmathPlanetmath” 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 G2, as defined previously, is in fact a 2-category, whereas the categoryMathworldPlanetmath 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 extensionPlanetmathPlanetmathPlanetmath of the Galois theory involving groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath viewed as single object categories with invertible morphismsMathworldPlanetmath, 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 groupoidPlanetmathPlanetmath of a surjectivePlanetmathPlanetmath fibrationMathworldPlanetmath 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 groupoidPlanetmathPlanetmath is that it provides an algebraic model of a foliated bundle ([1]). A natural extension of the double groupoid is a double categoryPlanetmathPlanetmath 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].


  • 1 R. Brown and G. Janelidze.(2004). Galois theory and a new homotopy double groupoid of a map of spaces.(2004). Applied Categorical StructuresMathworldPlanetmath,12: 63-80. Pdf file in arxiv: math.AT/0208211 .
  • 2 Alexander Grothendieck. 1971, Revêtements Étales et Groupe Fondamental (SGA1), chapter VI: Catégories fibrées 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