Difference between revisions of "Fundamental groupoid"
From Encyclopedia of Mathematics
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| − | A groupoid (a category in which all morphisms are isomorphisms) defined from a topological space | + | {{TEX|done}} |
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| + | 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]] $\pi_1(X,x_0)$. | ||
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====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 {{ZBL|0612.20032}}</TD></TR> | ||
| + | </table> | ||
Latest revision as of 11:05, 17 March 2023
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 Zbl 0612.20032 |
How to Cite This Entry:
Fundamental groupoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_groupoid&oldid=12895
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