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fundamental groupoid
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(Definition)
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Definition 1 Given a topological space  the fundamental groupoid  of  is defined as follows:
It is easily checked that the above defined category is indeed a groupoid with the inverse of (a morphism represented by) a path being (the homotopy class of) the “reverse” path. Notice that for  , the group of automorphisms of  is the fundamental group of  with basepoint  ,
It is straightforward to check that the above indeed defines a functor. Therefore  can (and should) be regarded as a functor from the category of topological spaces to the category of groupoids. This functor is not really homotopy invariant but it is “homotopy invariant up to homotopy” in the sense that the following holds.
A reader who understands the meaning of the statement should be able to give an explicit construction and supply the proof without much trouble.
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"fundamental groupoid" is owned by Dr_Absentius.
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(view preamble)
Cross-references: proof, natural transformation, induces, homotopy invariant, commutative diagram, functor, induced, continuous function, basepoint, fundamental group, automorphisms, group, inverse, groupoid, category, concatenation, composition, endpoints, paths, classes, homotopy, morphisms, points, objects, topological space
There are 8 references to this entry.
This is version 4 of fundamental groupoid, born on 2003-01-29, modified 2004-01-24.
Object id is 3941, canonical name is FundamentalGroupoid.
Accessed 2551 times total.
Classification:
| AMS MSC: | 55P99 (Algebraic topology :: Homotopy theory :: Miscellaneous) |
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Pending Errata and Addenda
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![$ [\alpha]\colon\thinspace x\to y$](http://images.planetmath.org:8080/cache/objects/3941/l2h/img22.png "$ [\alpha]\colon\thinspace x\to y$"){kind=small}
![$ \Pi_1(f)([\alpha])\colon\thinspace f(x)\to f(y)$](http://images.planetmath.org:8080/cache/objects/3941/l2h/img23.png "$ \Pi_1(f)([\alpha])\colon\thinspace f(x)\to f(y)$"){kind=small}
![$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{{I}\ar[d]_{\alpha}\ar@{-->}[dr]^{\Pi_1(f)(\alpha)}\ {X}\ar[r]_f&{Y} } } \end{xy}$](http://images.planetmath.org:8080/cache/objects/3941/l2h/img24.png "$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{{I}\ar[d]_{\alpha}\ar@{-->}[dr]^{\Pi_1(f)(\alpha)}\ {X}\ar[r]_f&{Y} } } \end{xy}$")