nLab condensed spectrum

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

(,1)(\infty,1)-Category theory

Stable homotopy theory

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Higher algebra

Higher linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents

Idea

By a condensed spectrum one should generally mean a spectrum in the context of condensed mathematics, hence a condensed object internal to spectra.

A very special case of this general notion has almost been considered in Scholze (2019), Footnote 12: The condensed simplicial abelian groups alluded to there there may be understood, under the stable Dold-Kan correspondence, to represent (the condensed version of) the particular case of connective H H\mathbb{Z} -module spectra.

Definition

A condensed spectrum is a hypercomplete (infinity,1)-sheaf of spectra on the pro-étale (infinity,1)-site of the point, small relative to a universe 𝒰\mathcal{U}.

See also

References

A discussion of condensed spectra in the literature seems not to be available yet. For general background on condensed mathematics see:

Last revised on July 5, 2022 at 11:21:00. See the history of this page for a list of all contributions to it.