iterating the fundamental groupoid does not work, so.....

iterating the fundamental groupoid does not work, so.....

[Ronnie Brown]

I would like to make a further point about the topological fundamental
groupoid of X.  The information on this shows that iterating the
fundamental groupoid does NOT lead to higher dimensional information on X.

This was one of the facts leading to the Brown-Higgins construction of
the homotopy double groupoid
\rho(X,X_1,X_0) , where X_0 \subset X_1 \subset X, which in dimension 2
consists of homotopy classes rel vertices of maps of I^2 into X which
take the edges of I^2 into X_1 and the vertices into X_0. This does
inherit the obvious compositions of squares in two directions to become
a strict double groupoid, and with which one can prove a 2-d van Kampen
theorem.

This was done in 1974, and published in 1978, in the teeth of
opposition, which possibly explains why the new nonabelian calculations
which resulted have not been generally recognised or taken up.

Ronnie Brown

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Re: iterating the fundamental groupoid does not work, so.....

[David Roberts]

On 29 April 2010 05:48, Ronnie Brown <ronnie.profbrown@btinternet.com>wrote:

> I would like to make a further point about the topological fundamental
> groupoid of X.  The information on this shows that iterating the
> fundamental groupoid does NOT lead to higher dimensional information on X.
>
>
If this means what I think it means: applying Pi_1 to the arrows and objects
of the topologised fundamental groupoid, then I agree. In a suitable
bicategory of topological groupoids (where internal weak equivalences a la
Bunge-Pare or Everaert-Kieboom***-*van der Linden are formally inverted) the
topologised fundamental groupoid is equivalent to a groupoid sans topology -
in fact it is equivalent to itself where the topology is replaced by the
discrete topology. The topologised fundamental groupoid in this way encodes
only the 1-type of the space.


David Roberts


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