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• 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

• 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 $\overline{Tw}(C)$.

• brief category:people-entry for hyperlinking references

• brief category:people-entry for hyperlinking references

• Page created, but author did not leave any comments.

Anonymous

• For completeness I have added pointer to

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.

• 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_{ that extends $f$ along the inclusion $S\to M_{.

Here for a compact Hausdorff topological space $S$ and for any $p$ such that $0 we have

$M_{

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

• Page created, but author did not leave any comments.

Anonymous

• the (infinity,1)-category of condensed infinity-groupoids has all finite (infinity,1)-limits, so it should have spectrum objects.

Anonymous

• 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

• 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.

Anonymous

• hopefully the given definition makes sense and is equivalent to the definition found in Scholze’s “Lectures on analytic geometry”, somebody more knowledgeable at $(\infty,1)$-category theory could double check.

Anonymous

• these are real-valued rational functions and continuity here is pointwise continuity

Anonymous

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

• 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

• finally found this textbook, am giving it it’s own category:reference-entry hereby, for ease of cross-linking

have started to make a hyperlinked index for the chapters.

• starting something (prodded by the comment here), but just a bare minimum for the moment

• Created a stub. Will be expanded eventually.

• brief category:people-entry for hyperlinking references

• starting an entry, for the moment mainly in order to record the fact that “crossed homomorphisms” are equivalently homomorphic sections of the corresponding semidirect product group projection. This is obvious, but is there a reference that makes it explicit?

• bare minimum, for the moment just so that one may link to it

• Amit Jamadagni, Hendrik Weimer, An Operational Definition of Topological Order (arXiv:2005.06501)
• Had a bit of clean up of entanglement, check the previous revision to see the section I deleted.

• 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

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

Anonymous

Anonymous

• Only a stub at the moment, but I thought we needed to start a page on this. Looks like it’s going to become important.

• Page created, but author did not leave any comments.

Anonymous

• brief category:people-entry for hyperlinking references

• brief category: people-entry for hyperlinking references

• brief category: people-entry for hyperlinking references

• brief category: people-entry for hyperlinking references

• a bare minimum

• brief category:people-entry for hyperlinking references

• Page created, but author did not leave any comments.

Anonymous

• Made explicit the notion of quantum dimension, and made that term redirect here (for the time being).

• brief category: people-entry, for the moment just to attribute the naming at Betti number

• brief category: people-entry for satisfying a link requested at locally internal category

(hope I have identified the correct author)

• Created stub.

• Matthew Headrick, Lectures on entanglement entropy in field theory and holography (arXiv:1907.08126)
• Spelled out the (equivalent) definition of locally small indexed category, and noted the equivalence of 2-categories explicitly.

Eigil Rischel

• brief category:people-entry for hyperlinking references

• 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.

• I am taking the liberty of creating a category: reference-entry in order to have a way to hyperlink references to our new research center here in NYUAD, which is slowly but surely entering into tangible existence.

• Page created, but author did not leave any comments.

Anonymous

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

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

• starting someting – not done yet but need to save

• Page created, but author did not leave any comments.

Anonymous

• Page created, but author did not leave any comments.

Anonymous

• Page created, but author did not leave any comments.

Anonymous