About: Fundamental groupoid

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In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids. — Alexander Grothendieck, Esquisse d'un Programme (Section 2, English translation)

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rdfs:label	Fundamentalgruppoid (de) Fundamental groupoid (en)
rdfs:comment	In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids. — Alexander Grothendieck, Esquisse d'un Programme (Section 2, English translation) (en) In der Mathematik, speziell der algebraischen Topologie, soll das Fundamentalgruppoid eines topologischen Raumes die Menge der Wegzusammenhangskomponenten und die Fundamentalgruppen (zu allen ) in einem einzigen algebraischen Objekt zusammenfassen. Das Fundamentalgruppoid ist ein Gruppoid, also eine Kategorie, in der jeder Morphismus ein Isomorphismus ist. Die Objekte sind die Punkte von , die Morphismen von nach sind die Homotopieklassen (relativ ) von stetigen Wegen mit . (de)
dcterms:subject	Algebraic topology Higher category theory
Wikipage page ID	55264190 (xsd:integer)
Wikipage revision ID	1090663023 (xsd:integer)
Link from a Wikipage to another Wikipage	Ronald Brown (mathematician) Algebraic topology Lie groupoid Esquisse d'un Programme Generalization Topological property Circle Edwin Spanier George W. Whitehead Conjecture Connected space Equivalence of categories Functor Fundamental group Path (topology) Additive group Local system Locally constant sheaf Alexander Grothendieck Algebraic topology Higher category theory Group homomorphism J. Peter May Homology (mathematics) Homotopy Homotopy hypothesis Smoothness Groupoid Integer Category theory Singleton (mathematics) Skeleton (category theory) Up to Space (mathematics) ∞-groupoid Universal Cover Seifert-Van Kampen theorem Homotopy type Weak homotopy equivalence
Link from a Wikipage to an external page	http://groupoids.org.uk/topgpds.html http://matematicas.unex.es/~navarro/res/esquisseeng.pdf http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf http://groupoids.org.uk/
sameAs	Fundamental groupoid Fundamental groupoid Fundamental groupoid Fundamental groupoid
dbp:wikiPageUsesTemplate	dbt:= dbt:ISBN dbt:Math dbt:Mvar dbt:Nlab dbt:Quote dbt:Category_theory dbt:Topology-stub
author	Alexander Grothendieck
id	fundamental+groupoid (en) fundamental+infinity-groupoid (en)
quote	[...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points, [,,,] (en)
source	Esquisse d'un Programme (en)
title	fundamental groupoid (en) fundamental infinity-groupoid (en)
has abstract	In der Mathematik, speziell der algebraischen Topologie, soll das Fundamentalgruppoid eines topologischen Raumes die Menge der Wegzusammenhangskomponenten und die Fundamentalgruppen (zu allen ) in einem einzigen algebraischen Objekt zusammenfassen. Das Fundamentalgruppoid ist ein Gruppoid, also eine Kategorie, in der jeder Morphismus ein Isomorphismus ist. Die Objekte sind die Punkte von , die Morphismen von nach sind die Homotopieklassen (relativ ) von stetigen Wegen mit . In diesem Gruppoid entspricht der Menge der Isomorphismusklassen von Objekten, während der Automorphismengruppe des Objekts entspricht. (de) In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more widely-known fundamental group; as such, it captures information about the homotopy type of a topological space. In terms of category theory, the fundamental groupoid is a certain functor from the category of topological spaces to the category of groupoids. [...] people still obstinately persist, when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under the symmetries of the situation, which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indispensable for understanding something, to work with fundamental groupoids with respect to a suitable packet of base points, [,,,] — Alexander Grothendieck, Esquisse d'un Programme (Section 2, English translation) (en)
prov:wasDerivedFrom	wikipedia-en:Fundamental_groupoid?oldid=1090663023&ns=0
page length (characters) of wiki page	9065 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Fundamental_groupoid
is Link from a Wikipage to another Wikipage of	Pasting lemma Algebraic topology Double groupoid Lie groupoid Pursuing Stacks Glossary of algebraic topology Glossary of category theory Stokes' theorem Fundamental group Path (topology) Locally constant sheaf Directed algebraic topology Simply connected space Grothendieck–Teichmüller group Group (mathematics) Higher category theory Homotopy theory Fibration Groupoid Category (mathematics) Semidirect product Seifert–Van Kampen theorem ∞-groupoid
is foaf:primaryTopic of	wikipedia-en:Fundamental_groupoid

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