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, $\mathrm{\Pi}$ 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^{} $Top$- 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$\mathrm{\xea}$tements $\mathrm{\xc9}$tales et Groupe Fondamental (SGA1), chapter VI: Cat$\mathrm{\xe9}$gories fibr$\mathrm{\xe9}$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 |