Categorias de feixes, álgebras de incidência e equivalências derivadas

Abstract

A finite partially ordered set (poset) X carries a natural structure of a topological space, so we can consider the category of sheaves over X with values in an abelian category A which can be identified with the category of covariant functors from the Hasse diagram of X into A. In particular, when A is the category of finite dimensional vector spaces over a field k, the category of sheaves over X with values in A is equivalent to the category of finite dimensional right modules over the incidence algebra of X over k. In this work we present a detailed study of the category of sheaves over posets with values in an abelian category A based on the article [20] of Sefi Ladkani and, more specifically, as main objective, we show a construction in which the author used ideas of algebraic topology and algebraic geometry to obtain derivedequivalences between the incidence algebra of a poset X and incidence algebras of posets induced by closed subsets of X.