请输入您要查询的字词:

 

单词 FundamentalGroupoidFunctor
释义

fundamental groupoid functor

\\xyoption

curve

The following quote indicates how fundamental groupoidsMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/FundamentalGroupoid)can be alternatively defined via the Yoneda-Grothendieck construction specified bythe 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 Grothendieckin 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 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.
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/3 15:23:09