A Brief Note for Sheaf Structures on Posets

TL;DR

This note introduces cellular sheaves on posets with Alexandrov topology, explaining their connection to sheaves in algebraic geometry using elementary category theory and topology.

Contribution

It provides an elementary explanation of how cellular sheaves on posets relate to sheaves in algebraic geometry, accessible with basic algebra and topology.

Findings

Cellular sheaves are sheaves on topological spaces derived from posets.

Elementary approach simplifies understanding of sheaf connections in algebraic geometry.

Clarifies the role of Alexandrov topology in sheaf theory.

Abstract

This note is a part of the lecture notes of a graduate student algebraic geometry seminar held at the department of mathematics in National Taiwan Normal University, 2020 Falls. It aims to introduce an example of sheaves defined on posets equipped with the Alexandrov topology, called the cellular sheaves. A cellular sheaf is a functor from the category of a poset to the category of specific algebraic structures (e.g. the category of groups). Strictly speaking, even equipping the poset with the Alexandrov topology, it is just the definition of a pre-sheaf on the Alexandrov topological space. By checking details, cellular sheaves are actually sheaves on topological spaces. This is a well-known fact in sheaf theory via the Kan extension, while it requires readers who are familiar with the category theory. In this note, we follow an elementary approach to describe the connection between cellular sheaves and sheaves concerned in algebraic geometry, where only basic commutative algebra and point-set topology are required as the background knowledge.