Hypercovers in topology

Daniel Dugger, Daniel C. Isaksen

55U35, 14F20, 14F42

Department of Mathematics
Purdue University
West Lafayette, IN 47907

Department of Mathematics 
University of Notre Dame
Notre Dame, IN 46556

ddugger@math.purdue.edu
isaksen.1@nd.edu

We show that if U is a hypercover of a topological
space X then the natural map from hocolim U to X is a weak
equivalence.  This fact is used to construct topological realization
functors for the A^1-homotopy theory of schemes over real and
complex fields.