Subject: comment on Van Kampen From: Gustavo Granja 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 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) TEL: (+1) 352-381-8497(home) FAX: (+1) 352-392-8357 URL: http://www.math.ufl.edu/~rudyak/