2013/04/17 by Daniel S. Freed, Michael J. Hopkins · 1 citation
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Mathematics #Pure mathematics #Functor #Homotopy #Equivariant map #Simplicial set #Sheaf #Invariant (physics) #Algebra over a field #Cofibration #Hodge theory #Chern class #n-connected #Homotopy category #Homotopy lifting property #Cohomology
paper · pdf · doi:10.1090/s0273-0979-2013-01415-0
openalex publication_date 2013/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that Chern–Weil forms are the only natural differential forms associated to a connection on a principal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof. Other arguments come from classical invariant theory. We identify the Weil algebra as the de Rham complex of a specific simplicial sheaf, and similarly give a new interpretation of the Weil model in equivariant de Rham theory. There is an appendix proving a general theorem about set-theoretic transformations of polynomial functors.