nLab condensed spectrum

Contents

Context

Topology

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

Introduction

Basic concepts

open subset, closed subset, neighbourhood
topological space, locale
base for the topology, neighbourhood base
finer/coarser topology
closure, interior, boundary
separation, sobriety
continuous function, homeomorphism
uniformly continuous function
embedding
open map, closed map
sequence, net, sub-net, filter
convergence
category Top
- convenient category of topological spaces

Universal constructions

initial topology, final topology
subspace, quotient space,
fiber space, space attachment
product space, disjoint union space
mapping cylinder, mapping cocylinder
mapping cone, mapping cocone
mapping telescope
colimits of normal spaces

Extra stuff, structure, properties

nice topological space
metric space, metric topology, metrisable space
Kolmogorov space, Hausdorff space, regular space, normal space
sober space
compact space, proper map

sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
compactly generated space
second-countable space, first-countable space
contractible space, locally contractible space
connected space, locally connected space
simply-connected space, locally simply-connected space
cell complex, CW-complex
pointed space
topological vector space, Banach space, Hilbert space
topological group
topological vector bundle, topological K-theory
topological manifold

Examples

empty space, point space
discrete space, codiscrete space
Sierpinski space
order topology, specialization topology, Scott topology
Euclidean space
- real line, plane
cylinder, cone
sphere, ball
circle, torus, annulus, Moebius strip
polytope, polyhedron
projective space (real, complex)
classifying space
configuration space
path, loop
mapping spaces: compact-open topology, topology of uniform convergence
- loop space, path space
Zariski topology
Cantor space, Mandelbrot space
Peano curve
line with two origins, long line, Sorgenfrey line
K-topology, Dowker space
Warsaw circle, Hawaiian earring space

Basic statements

Hausdorff spaces are sober
schemes are sober
continuous images of compact spaces are compact
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces are normal
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
closed injections are embeddings
proper maps to locally compact spaces are closed
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and sigma-compact spaces are paracompact
locally compact and second-countable spaces are sigma-compact
second-countable regular spaces are paracompact
CW-complexes are paracompact Hausdorff spaces

Theorems

Urysohn's lemma
Tietze extension theorem
Tychonoff theorem
tube lemma
Michael's theorem
Brouwer's fixed point theorem
topological invariance of dimension
Jordan curve theorem

Analysis Theorems

Heine-Borel theorem
intermediate value theorem
extreme value theorem

topological homotopy theory

left homotopy, right homotopy
homotopy equivalence, deformation retract
fundamental group, covering space
fundamental theorem of covering spaces
homotopy group
weak homotopy equivalence
Whitehead's theorem
Freudenthal suspension theorem
nerve theorem
homotopy extension property, Hurewicz cofibration
cofiber sequence
Strøm model category
classical model structure on topological spaces

$(\infty,1)$ -Category theory

(∞,1)-category theory

Background

category theory
higher category theory
(n,r)-category

Basic concepts

(∞,1)-category
hom-objects
equivalences in/of $(\infty,1)$ -categories
sub-(∞,1)-category
- reflective sub-(∞,1)-category
- reflective localization
opposite (∞,1)-category
over (∞,1)-category
- join of quasi-categories
(∞,1)-functor
fibrations

Universal constructions

limit
- terminal object
adjoint functors

Local presentation

locally presentable

Theorems

(∞,1)-Yoneda lemma
(∞,1)-Grothendieck construction
adjoint (∞,1)-functor theorem
(∞,1)-monadicity theorem

Extra stuff, structure, properties

stable (∞,1)-category
(∞,1)-topos

Models

category with weak equivalences
model category
derivator
quasi-category
- model structure for quasi-categories
- model structure for Cartesian fibrations
relation to simplicial categories
Kan complex
- model structure for Kan complexes

Stable homotopy theory

stable homotopy theory

homological algebra, higher algebra

Introduction

Ingredients

homotopy theory

loop space object
suspension object
looping and delooping
stable (∞,1)-category
stable (∞,1)-category of spectra
- spectrum
- stable homotopy category
smash product of spectra

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Higher category theory

higher category theory

category theory
homotopy theory

Basic concepts

k-morphism, coherence
looping and delooping
looping and suspension

Basic theorems

homotopy hypothesis-theorem
delooping hypothesis-theorem
periodic table
stabilization hypothesis-theorem
exactness hypothesis
holographic principle

Applications

applications of (higher) category theory
higher category theory and physics

Models

(n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
  - n-fold complete Segal space
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
  - Kan complex
    - algebraic Kan complex
    - simplicial T-complex
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
  - poset = (0,1)-category
  - 2-poset = (1,2)-category
- n-groupoid = (n,0)-category
  - 2-groupoid, 3-groupoid
categorification/decategorification
geometric definition of higher category
algebraic definition of higher category
stable homotopy theory

Morphisms

k-morphism
- 2-morphism
transfor
- natural transformation
- modification

Functors

functor
2-functor
- pseudofunctor
- lax functor
(∞,1)-functor

Universal constructions

2-limit
(∞,1)-adjunction
(∞,1)-Kan extension
- (∞,1)-limit
(∞,1)-Grothendieck construction

Extra properties and structure

cosmic cube
- k-tuply monoidal n-category
- strict ∞-category, strict ∞-groupoid
stable (∞,1)-category
(∞,1)-topos

1-categorical presentations

homotopical category
model category theory
enriched category theory

Higher algebra

higher algebra

universal algebra

Algebraic theories

algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory
monad / (∞,1)-monad
operad / (∞,1)-operad

Algebras and modules

algebra over a monad

∞-algebra over an (∞,1)-monad
algebra over an algebraic theory

∞-algebra over an (∞,1)-algebraic theory
algebra over an operad

∞-algebra over an (∞,1)-operad
action, ∞-action
representation, ∞-representation
module, ∞-module
associated bundle, associated ∞-bundle

Higher algebras

monoidal (∞,1)-category
symmetric monoidal (∞,1)-category of spectra
A-∞ algebra
- A-∞ ring, A-∞ space
C-∞ algebra
E-∞ ring, E-∞ algebra
- ∞-module, (∞,1)-module bundle
- multiplicative cohomology theory
L-∞ algebra
- deformation theory

Model category presentations

model structure on simplicial T-algebras / homotopy T-algebra
model structure on operads

model structure on algebras over an operad

Geometry on formal duals of algebras

Isbell duality
derived geometry

Theorems

Deligne conjecture
delooping hypothesis
monoidal Dold-Kan correspondence

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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, …

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 $H\mathbb{Z}$ -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 $\mathcal{U}$ .

References

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

Peter Scholze, Lectures on condensed mathematics (2019) $[$ pdf $]$

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

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nLab condensed spectrum

Context

Topology

(∞,1)(\infty,1)-Category theory

Stable homotopy theory

Ingredients

Contents

Higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Higher algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Higher linear algebra

Contents

Idea

Definition

See also

References

$(\infty,1)$ -Category theory