fundamental groupoid
Definition 1.
Given a topological space the fundamental groupoid
of is defined as
follows:
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•
The objects of are the points of
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•
morphisms
are homotopy classes of paths “rel endpoints” that is
where, denotes homotopy
rel endpoints, and,
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•
composition
of morphisms is defined via concatenation of paths.
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 ,
Definition 2.
Let be a continuous function between two topological spaces.
Then there is an induced functor
defined as follows
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•
on objects is just ,
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•
on morphisms is given by “composing with ”, that is if is a path representing the morphism then a representative of is determined by the following commutative diagram