nLab
pyknotic set

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Algebra

Contents

Idea and definition

Pyknotic sets (from the ancient Greek πυκνός, meaning thick, dense, or compact) aim to provide a convenient setting in the framework of homotopical algebra for working with algebraic objects that have some sort of topology on them.

Definition

A pyknotic set is a sheaf of sets on the site of compact Hausdorff topological spaces whose underlying set has rank smaller than the smallest strongly inaccessible cardinal. The Grothendieck topology on the latter category is generated by finite jointly effective epimorphic families of maps.

Pyknotic sets form a coherent topos, whereas the closely related category of condensed sets does not form a topos (BarHai 19, Sec. 0.3).

The global sections functor is given by evaluation at the one-point compactum. This has a left adjoint sending a set, XX, to the discrete pyknotic set attached to XX, and a right adjoint to the indiscrete pyknotic set attached to XX. The left adjoint, the discrete functor, does not preserve limits and so does not possess a further left adjoint. The topos of pyknotic sets is thus not cohesive (BarHai 19, 2.2.14).

Compactly generated topological spaces embed fully faithfully into pyknotic sets, in a manner that preserves limits.

One key point, however, is that the relationship between compactly generated topological spaces and pyknotic sets is dual to the relationship between compactly generated topological spaces and general topological spaces: in topological spaces, compactly generated topological spaces are stable under colimits but not limits; in pyknotic sets,compactly generated topological spaces are stable under limits but not colimits (BarHai 19, 0.2.3).

This construction generalizes to pyknotic objects for any (∞,1)-category.

Examples

For example locally compact abelian groups, normed rings, and complete locally convex topological vector spaces are pyknotic objects.

References

Last revised on June 11, 2019 at 15:37:22. See the history of this page for a list of all contributions to it.