# Difference between revisions of "Fundamental groupoid"

From Encyclopedia of Mathematics

(Importing text file) |
(LaTeX) |
||

Line 1: | Line 1: | ||

− | A groupoid (a category in which all morphisms are isomorphisms) defined from a topological space | + | {{TEX|done}} |

+ | |||

+ | A [[groupoid]] (a category in which all morphisms are isomorphisms) defined from a topological space $X$; the objects are the points of $X$, and the morphisms from an object $x_0$ to $x_1$ are the homotopy classes $\mathrm{rel} \{0,1\}$ of paths starting at $x_0$ and ending at $x_1$; composition is the product of classes of paths. The group of automorphisms of an object $x_0$ is the same as the [[fundamental group]] $\pi_1(X,x_0)$. | ||

Line 7: | Line 9: | ||

====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Brown, "From groups to groupoids: a brief survey" ''Bull. London Math. Soc.'' , '''19''' (1987) pp. 113–134</TD></TR></table> | + | <table> |

+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Brown, "From groups to groupoids: a brief survey" ''Bull. London Math. Soc.'' , '''19''' (1987) pp. 113–134</TD></TR> | ||

+ | </table> |

## Latest revision as of 19:46, 28 December 2014

A groupoid (a category in which all morphisms are isomorphisms) defined from a topological space $X$; the objects are the points of $X$, and the morphisms from an object $x_0$ to $x_1$ are the homotopy classes $\mathrm{rel} \{0,1\}$ of paths starting at $x_0$ and ending at $x_1$; composition is the product of classes of paths. The group of automorphisms of an object $x_0$ is the same as the fundamental group $\pi_1(X,x_0)$.

#### Comments

A useful survey of the applications of fundamental groupoids can be found in [a1].

#### References

[a1] | R. Brown, "From groups to groupoids: a brief survey" Bull. London Math. Soc. , 19 (1987) pp. 113–134 |

**How to Cite This Entry:**

Fundamental groupoid.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Fundamental_groupoid&oldid=12895

This article was adapted from an original article by A.V. Khokhlov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article