Namespaces
Variants
Actions

Difference between revisions of "Fundamental groupoid"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(LaTeX)
 
Line 1: Line 1:
A groupoid (a category in which all morphisms are isomorphisms) defined from a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422201.png" />; the objects are the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422202.png" />, and the morphisms from an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422203.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422204.png" /> are the homotopy classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422205.png" /> of paths starting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422206.png" /> and ending at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422207.png" />; composition is the product of classes of paths. The group of automorphisms of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422208.png" /> is the same as the [[Fundamental group|fundamental group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f042/f042220/f0422209.png" />.
+
{{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