Subject:
comment on Van Kampen
From:
Gustavo Granja <ggranja@math.ist.utl.pt>
Date:
Thu, 24 Feb 2005 13:25:51 +0000 (WET)


It seems to me that the best way of thinking about the Van Kampen
theorem for arbitrary covers is the following:

1. Taking the fundamental groupoid is a left adjoint in a Quillen pair
and hence commutes with homotopy colimits.

2. The homotopy colimit of the Cech diagram of an open cover is the space inquestion. (For a very readable recent treatment generalizing Segal's result see D. Dugger and D. Isaksen, "Topological hypercovers and A1-realizations" Math. Zeit. 246 #4 (2004) 667--689)

3. There is a very simple formula (amounting to the usual descent category in the case when the diagram is the nerve of a cover) for the homotopy colimit of a diagram of groupoids in

Sharon Hollander "A homotopy theory for stacks", available on Hopf.

See also

Dror Farjoun, Emmanuel "Fundamental group of homotopy colimits." Adv. Math. 182 (2004), no. 1, 1--27. (Reviewer: Donald M. Davis)

for a slightly different perspective.

Gustavo Granja
--------------
Subject:
question
From:
Yuli Rudyak <rudyak@math.ufl.edu>
Date:
Mon, 7 Feb 2005 19:04:00 -0500 (EST)
To:
dmd1@lehigh.edu

Dear Don,

I have a question for the list.

Where can I find the van Kampen Theorem for the case when the intersection of
parts is not connected (like as we divide a circle in two segments and
intersection has two components).

Yuli

Dr. Yuli B. Rudyak
Department of Mathematics
University of Florida
358 Little Hall
PO Box 118105
Gainesville, FL 32611-8105
USA
TEL: (+1) 352-392-0281 ext. 319(office)
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URL: http://www.math.ufl.edu/~rudyak/