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单词 FundamentalGroupoid
释义

fundamental groupoid


Definition 1.

Given a topological spaceMathworldPlanetmath X the fundamental groupoidMathworldPlanetmathPlanetmathPlanetmathΠ1(X) of X is defined asfollows:

  • The objects of Π1(X) are the points of X

    Obj(Π1(X))=X,
  • morphismsMathworldPlanetmath are homotopy classes of paths “rel endpoints” that is

    HomΠ1(X)(x,y)=Paths(x,y)/,

    where, denotes homotopyMathworldPlanetmathPlanetmath rel endpoints, and,

  • compositionMathworldPlanetmath of morphisms is defined via concatenation of paths.

It is easily checked that the above defined category is indeed a groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathwith the inverseMathworldPlanetmathPlanetmathPlanetmath of (a morphism represented by) a path being (the homotopyclass of) the “reverse” path.Notice that for xX, the group of automorphismsPlanetmathPlanetmathPlanetmathPlanetmath of x is thefundamental groupMathworldPlanetmathPlanetmath of X with basepoint x,

HomΠ1(X)(x,x)=π1(X,x).
Definition 2.

Let f:XY be a continuous function between two topological spaces.Then there is an induced functorMathworldPlanetmath

Π1(f):Π1(X)Π1(Y)

defined as follows

  • on objects Π1(f) is just f,

  • on morphisms Π1(f) is given by “composing with f”, that isif α:I X is a path representing the morphism[α]:xy then a representative ofΠ1(f)([α]):f(x)f(y)is determined by the following commutative diagramMathworldPlanetmath

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更新时间:2025/5/3 12:29:37