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- Discussion Type
- discussion topicanyonic braiding in momentum space -- references
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 22 minutes ago

a bare list of references, to be

`!include`

-ed into the References-section of relevant entries (such as at*braid group representation*and at*semi-metal*).Had originally compiled this list already last April (for this MO reply) but back then the nLab couldnt be edited

- Discussion Type
- discussion topictwisted arrow (∞,1)-category
- Category Latest Changes
- Started by Hurkyl
- Comments 6
- Last comment by Hurkyl
- Last Active 1 hour ago

Created an entry for this.

I’ve adopted the existing convention at nLab in the definition of $Tw(C)$ (which is also the definition I prefer).

Since the opposite convention is used a lot (e.g. by Lurie), I’ve decided it was worth giving it notation, the relation between the versions, and citing results in both forms. Since I didn’t have any better ideas, I’ve settled on $\overline{Tw}(C)$.

- Discussion Type
- discussion topicBartomeu Monserrat
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 hour ago

- Discussion Type
- discussion topicBo Peng
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 hour ago

- Discussion Type
- discussion topiccondensed module spectrum
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active 2 hours ago

- Discussion Type
- discussion topicYoneda lemma for (infinity,1)-categories
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 2 hours ago

For completeness I have added pointer to

- Emily Riehl, Dominic Verity, Section 6 of:
*Fibrations and Yoneda’s lemma in an $\infty$-cosmos*, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 (arXiv:1506.05500, doi:10.1016/j.jpaa.2016.07.003)

though there should really be some accompanying discussion of how this form of the statement is related to the usual one in terms of presheaves.

- Emily Riehl, Dominic Verity, Section 6 of:

- Discussion Type
- discussion topicliquid vector space
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active 2 hours ago

Created:

## Idea

An alternative to complete topological vector spaces in the framework of condensed mathematics.

Roughly, completeness is expressed as ability to integrate with respect to Radon measures.

This doesn’t quite work as stated, and to make this rigorous one has to bring L^p-spaces for $0\lt p\le 1$ (i.e., the non-convex case) into the picture.

## Definition

A condensed abelian group $V$ is

**$p$-liquid**($0\lt p\le 1$) if for every compact Hausdorff topological space $S$ and every morphism of condensed sets $f\colon S\to V$ there is a unique morphism of condensed abelian groups $M_{<p}(S)\to V$ that extends $f$ along the inclusion $S\to M_{<p}(S)$.Here for a compact Hausdorff topological space $S$ and for any $p$ such that $0<p\le 1$ we have

$M_{<p}(S)=\bigcup_{q<p}M_q(S),$where

$M_p(S)=\bigcup_{C>0}M(S)_{\ell^p\le C},$where

$M(S)_{\ell^p\le C}=\lim_i M(S_i)_{\ell^p\le C},$where $S_i$ are finite sets such that

$S = \lim_i S_i$and

$M(F)_{\ell^p\le C}$for a finite set $F$ denotes the subset of $\mathbf{R}^F$ consisting of sequence with l^p-norm at most $C$.

## References

- Discussion Type
- discussion topiccondensed E-infinity ring
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active 2 hours ago

- Discussion Type
- discussion topiccondensed spectrum
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 3 hours ago

- Discussion Type
- discussion topiccompact projective object
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 6
- Last comment by JonasFrey
- Last Active 4 hours ago

Created:

\tableofcontents

## Definition

An object $P$ of a category $C$ is a

**compact projective object**if its corepresentable functor $Hom(P,-)\colon C\to Set$ preserves all small sifted colimits.Equivalently, it is an object that is a compact object ($Hom(P,-)$ preserves all small filtered colimits) and a projective object ($Hom(P,-)$ preserves epimorphisms, which follows from its preservation of coequalizers).

## Examples

In the category of algebras over an algebraic theory, compact projective objects are retracts of free algebras.

Conversely, if a locally small category has enough compact projective objects (meaning that there is a set of compact projective objects that generates it under small colimits and reflects isomorphisms), then this category is equivalent to the category of algebras over an algebraic theory. Such a category is also known as a locally strongly finitely presentable category

## Related concepts

- Discussion Type
- discussion topicreal polynomial functions are pointwise continuous
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active 7 hours ago

polynomials are a concept from abstract algebra, and it is not true that all polynomials are continuous as (non-trivial) topological vector spaces over a field with a (non-trivial) metric space; polynomials over finite fields are one such counterexample: they are only continuous when equipped with the discrete or indiscrete topology and the finite field is equipped with the trivial metric.

This article is about polynomial functions over the real numbers and epsilontic pointwise continuity of such functions.

Anonymous

- Discussion Type
- discussion topiccondensed abelian group
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active 7 hours ago

Created:

## Definition

An abelian group object in the category of condensed sets.

## Properties

The category of condensed abelian groups enjoys excellent categorical properties for homological algebra:

It is an abelian category that admits all small limits and colimits;

In this category, filtered colimits and infinite products are exact. The latter property is rather rare.

It has enough compact projective objects: free condensed abelian groups on extremally disconnected compact Hausdorff topological spaces generate all condensed abelian groups under small colimits and their corepresentable functors reflect isomorphisms;

The previous property implies that condensed abelian groups have the same exactness properties as the category of abelian groups.

## Related concepts

## References

- Discussion Type
- discussion topiccondensed infinity-groupoid
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active 13 hours ago

- Discussion Type
- discussion topicreal rational functions are pointwise continuous
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 14 hours ago

- Discussion Type
- discussion topicLebesgue space
- Category Latest Changes
- Started by TobyBartels
- Comments 12
- Last comment by Dmitri Pavlov
- Last Active 16 hours ago

I added some more to Lebesgue space about the cases where $1 \lt p \lt \infty$ fails.

- Discussion Type
- discussion topiclattice (discrete subgroup)
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Guest
- Last Active 18 hours ago

found this article as well, going to merge lattice (in a vector space, etc.) into here

Anonymous

- Discussion Type
- discussion topiclattice (in a vector space, etc.) > history
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active 18 hours ago

removing duplicate redirect link lattice in a vector space from top of the article (already appears at the bottom of the article with the other redirects)

Anonymous

- Discussion Type
- discussion topicQuantum Information Meets Quantum Matter
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 19 hours ago

- Discussion Type
- discussion topicstring-net model
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 20 hours ago

- Discussion Type
- discussion topiccommutative von Neumann algebra
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 5
- Last comment by Urs
- Last Active 21 hours ago

- Discussion Type
- discussion topicEdward Marczewski
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active 21 hours ago

- Discussion Type
- discussion topiccrossed homomorphism
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by jim stasheff
- Last Active 22 hours ago

- Discussion Type
- discussion topiclinear extension of a partial order
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 22 hours ago

- Discussion Type
- discussion topictopological order
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 23 hours ago

added pointer to today’s

- Amit Jamadagni, Hendrik Weimer,
*An Operational Definition of Topological Order*(arXiv:2005.06501)

- Amit Jamadagni, Hendrik Weimer,

- Discussion Type
- discussion topicentanglement
- Category Latest Changes
- Started by DavidRoberts
- Comments 5
- Last comment by Urs
- Last Active 1 day ago

Had a bit of clean up of entanglement, check the previous revision to see the section I deleted.

- Discussion Type
- discussion topictopological entanglement entropy -- references
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 1 day ago

this is a bare list of references which used to be (and still is) at

*entanglement entropy*. But since the same references are now also needed at*long-range entanglement*, I am putting them in a separate page here, to be`!include`

-ed into both these entries

- Discussion Type
- discussion topicDold-Kan correspondence
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Urs
- Last Active 1 day ago

added reference to dendroidal version of Dold-Kan correspondence

- Discussion Type
- discussion topicninebrane group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 1 day ago

put table in new “related concepts” section in analogy with fivebrane group article

Anonymous

- Discussion Type
- discussion topicbasis theorem
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 1 day ago

- Discussion Type
- discussion topiccondensed mathematics
- Category Latest Changes
- Started by David_Corfield
- Comments 19
- Last comment by DavidRoberts
- Last Active 1 day ago

- Discussion Type
- discussion topictrace-class map
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 1 day ago

- Discussion Type
- discussion topictopological phase of matter
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active 1 day ago

added these pointers on classification of topological phases of matter via tensor network states:

C. Wille, O. Buerschaper, Jens Eisert,

*Fermionic topological quantum states as tensor networks*, Phys. Rev. B 95, 245127 (2017) (arXiv:1609.02574)Andreas Bauer, Jens Eisert, Carolin Wille,

*Towards a mathematical formalism for classifying phases of matter*(arXiv:1903.05413)

- Discussion Type
- discussion topicDuan-Lu Zhou
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicBei Zeng
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicTarun Grover
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicGeorges Reeb
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicReeb sphere theorem
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 1 day ago

started something at

*Reeb sphere theorem*

- Discussion Type
- discussion topicReeb graph
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicGurjeet Singh
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicquotient bimodule
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 1 day ago

- Discussion Type
- discussion topictriple category
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by maxsnew
- Last Active 2 days ago

Stub for triple category.

- Discussion Type
- discussion topicdimension
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 2 days ago

created stub for dimension

- Discussion Type
- discussion topicspherical category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicEnrico Betti
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category: people`

-entry, for the moment just to attribute the naming at*Betti number*

- Discussion Type
- discussion topicRenato Betti
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category: people`

-entry for satisfying a link requested at*locally internal category*(hope I have identified the correct author)

- Discussion Type
- discussion topicclosed bicategory
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topiclong-range entanglement
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

starting something, to go along with

*topological order*and*short-range entanglement*. But for the moment this remains a stub, nothing to see here yet.

- Discussion Type
- discussion topicentanglement entropy
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 2 days ago

added pointer to today’s

- Matthew Headrick,
*Lectures on entanglement entropy in field theory and holography*(arXiv:1907.08126)

- Matthew Headrick,

- Discussion Type
- discussion topiclocally internal category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicSpan
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by varkor
- Last Active 2 days ago

I thought it was ridiculous that Span redirected to (infinity,n)-category of correspondences, so I made a stubby page for it instead.

- Discussion Type
- discussion topicPhilip Ball
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicexercise
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 2 days ago

touching this ancient and abandoned entry in reaction to the discussion here:

I think an entry with this title deserves to exist (even if its current content is unsatisfactory). To make this point, I am cross-linking it now with all of the following entries which all exist (even though all of them leave a lot of room for improvement):

In this vein, I have removed the

`category: meta`

-tag from this entry.

- Discussion Type
- discussion topicCenter for Quantum and Topological Systems
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topictransport
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicfield
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Urs
- Last Active 2 days ago

I added to field a mention of some other constructive variants of the definition, with a couple more references.

- Discussion Type
- discussion topiccodomain fibration
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 2 days ago

have added to

*codomain fibration*a brief paragraph on the $(\infty,1)$-version here and that it’s a coCartesian fibration.

- Discussion Type
- discussion topicpersistent cohomotopy
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicfilter of a ring
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 2 days ago

- Discussion Type
- discussion topicascending chain condition
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 2 days ago

- Discussion Type
- discussion topicArtinian bimodule
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 2 days ago