Hypercovers in topology

2001/11/27 by Daniel Dugger, Daniel C. Isaksen, Dugger, Daniel +1 · 1 citation
Mathematics · #55U35 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AT #msc:55U35

paper · pdf · doi:10.48550/arxiv.math/0111287

20 pages

arxiv created 2001/11/27 · openalex publication_date 2001/11/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 A¹-homotopy theory of schemes over real and complex fields.

Citations

• Hypercovers and simplicial presheaves

Cited by

• Sheaves, Cosheaves and Applications

Related

• Etale realization on the A¹-homotopy theory of schemes
• THH and TC are (very) far from being homotopy functors
• The homotopy-invariance of constructible sheaves
• The topological realization of a simplicial presheaf
• The Up Topology for Mirrored Topological Posets