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fundamental groupoid
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(Definition)
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Definition 1 Given a topological space $X$ the fundamental groupoid $\Pi_1(X)$ of $X$ 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 $x \in X$ , the group of automorphisms of $x$ is the fundamental group of $X$ with basepoint $x$ , $$\mathrm{Hom}_{\Pi_1(X)}(x,x)=\pi_1(X,x)\,.$$
Definition 2 Let $f\co X\to Y$ be a continuous function between two topological spaces. Then there is an induced functor $$\Pi_1(f)\co \Pi_1(X)\to\Pi_1(Y)$$ defined as follows
- on objects $\Pi_1(f)$ is just $f$ ,
- on morphisms $\Pi_1(f)$ is given by ``composing with $f$ '', that is if $\alpha\co I\to$ $ X$ is a path representing the morphism $[\alpha]\co x\to y$ then a representative of $\Pi_1(f)([\alpha])\co f(x)\to f(y)$ is determined by the following commutative diagram
It is straightforward to check that the above indeed defines a functor. Therefore $\Pi_1$ 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 CWoo. [ full author list (2) | owner history (1) ]
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Cross-references: proof, natural transformation, induces, homotopy invariant, category of groupoids, 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 6 references to this entry.
This is version 6 of fundamental groupoid, born on 2003-01-29, modified 2008-09-07.
Object id is 3941, canonical name is FundamentalGroupoid.
Accessed 3730 times total.
Classification:
| AMS MSC: | 55P99 (Algebraic topology :: Homotopy theory :: Miscellaneous) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix{{I}\ar[d]_{\alpha}\ar@{-->}[dr]^{\Pi_1(f)(\alpha)}\ {X}\ar[r]_f&{Y} }$](http://images.planetmath.org:8080/cache/objects/3941/js/img1.png "$\displaystyle \xymatrix{{I}\ar[d]_{\alpha}\ar@{-->}[dr]^{\Pi_1(f)(\alpha)}\ {X}\ar[r]_f&{Y} }$")