A Simplicial approach to the Sheaf Theoretic construction of Intersection Cohomology

Sebastian Andrés Cea Garcia (1, 2)

1 UPJV - Université de Picardie Jules Verne
2 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

Intersection (co)homology is a way to enhance classical (co)homology, allowing us to use a famous result called Poincaré duality on a large class of spaces known as stratified pseudomanifolds. There is a theoretically powerful way to arrive at intersection (co)homology by a classifying sheaves that satisfy what are called the Deligne axioms. There is a successful way to construct a simplicial intersection (co)homology, exposed in the works of D. Chataur, D. Tanré and M. Saralegi-Araguren, but a simplicial manifestation of the Deligne axioms has remained under shadows until now. This paper draws on constructions made by these authors, showing a simplicial manifestation of the Deligne axioms. This consists on presenting categories of "simplicial sheaves", localizing them appropriately and then stating "simplicial Deligne axioms". All this for different simplicial structures one can encounter. We finalize by presenting sheaves that satisfy the axioms on simplicial complexes.

Domaines: Mathématiques [math]; Topologie algébrique [math.AT]

Sebastian Andrés Cea Garcia. A Simplicial approach to the Sheaf Theoretic construction of Intersection Cohomology. 2021. ⟨hal-03379849⟩