Vipul: Created page with '==Definition== The '''fundamental groupoid''' of a topological space $X$ is defined as follows: * As a set, it is the set of all homotopy classes of [[defining i...'

2010-12-24T02:05:52Z

Created page with '==Definition== The '''fundamental groupoid''' of a topological space <math>X</math> is defined as follows: * As a set, it is the set of all homotopy classes of [[defining i...'

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==Definition==

The '''fundamental groupoid''' of a [[topological space]] <math>X</math> is defined as follows:

* As a set, it is the set of all homotopy classes of [[defining ingredient::path]]s between points in <math>X</math> (i.e., functions from the [[defining ingredient::closed unit interval]] to <math>X</math>), where two paths are homotopic if there is a [[homotopy]] between them that ''preserves endpoints'' at every stage of the homotopy.
* The partial multiplication is defined by concatenation of paths where the right endpoint of the left path coincides with the left endpoint of the right path. Specifically, if <math>f_1,f_2: [0,1] \to X</math> are paths, such that <math>f_1(1) = f_2(0)</math>, then <math>f_1 * f_2</math> is defined as (up to homotopy):

<math>\! (f_1 * f_2)(t) = \lbrace \begin{array}{rl} f_1(2t), & 0 \le t \le 1/2\\ f_2(2t - 1), & 1/2 < t \le 1 \\\end{array}</math>

===Why it is a groupoid===

{| class="sortable" border="1"
! Condition !! How it is shown !! Page detailing relevant homotopy
|-
| well defined || if <math>f_1</math> and <math>g_1</math> are homotopic, and <math>f_2</math> and <math>g_2</math> are homotopic, then the existence of <math>f_1 * f_2</math> implies the existence of <math>g_1 * g_2</math>, and they are homotopic || [[homotopy between composites of homotopic paths]]
|-
| existence of identity element || for a path <math>f</math>, the left identity element is a constant loop that stays fixed at <math>f(0)</math>, the right identity is a constant loop that stays fixed at <math>f(1)</math> ||
|-
| existence of inverses || for a path <math>f</math>, the inverse path is the path <math>f^{-1}(t) := f(1 - t)</math>, that traverses <math>f</math> backward. || similar to the loops case: [[homotopy between constant loop and composite of loop with inverse]]
|-
| associativity || consider the products <math>f_1 * (f_2 * f_3)</math> and <math>(f_1 * f_2) * f_3</math>. If either of these products is defined, so is the other, and they are homotopic as paths || similar to the loops case: [[homotopy between composites associated differently]]
|}

Fundamental groupoid - Revision history

Vipul: Created page with '==Definition== The '''fundamental groupoid''' of a topological space X is defined as follows: * As a set, it is the set of all homotopy classes of [[defining i...'

Vipul: Created page with '==Definition== The '''fundamental groupoid''' of a topological space $X$ is defined as follows: * As a set, it is the set of all homotopy classes of [[defining i...'