Pyknotic objects, I. Basic notions

TL;DR

This paper introduces pyknotic objects, a new framework of sheaves on compacta that incorporate topological data, facilitating algebraic and homotopical analysis with applications to local fields.

Contribution

It develops the foundational theory of pyknotic objects, providing basic definitions and examples to bridge topology with algebra and homotopy theory.

Findings

Pyknotic objects unify sheaf theory with topological data.

They enable new approaches to derived categories of local fields.

Basic examples illustrate the utility of pyknotic structures.

Abstract

Pyknotic objects are (hyper)sheaves on the site of compacta. These provide a convenient way to do algebra and homotopy theory with additional topological information present. This appears, for example, when trying to contemplate the derived category of a local field. In this article, we present the basic theory of pyknotic objects, with a view to describing a simple set of everyday examples.