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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 (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 .
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:
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 (i.e., the non-convex case) into the picture.
A condensed abelian group is -liquid () if for every compact Hausdorff topological space and every morphism of condensed sets there is a unique morphism of condensed abelian groups that extends along the inclusion .
Here for a compact Hausdorff topological space and for any such that we have
where
where
where are finite sets such that
and
for a finite set denotes the subset of consisting of sequence with l^p-norm at most .
Created:
\tableofcontents
An object of a category is a compact projective object if its corepresentable functor preserves all small sifted colimits.
Equivalently, it is an object that is a compact object ( preserves all small filtered colimits) and a projective object ( preserves epimorphisms, which follows from its preservation of coequalizers).
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
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
Created:
An abelian group object in the category of condensed sets.
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.
I added some more to Lebesgue space about the cases where fails.
found this article as well, going to merge lattice (in a vector space, etc.) into here
Anonymous
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
added pointer to today’s
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
added reference to dendroidal version of Dold-Kan correspondence
put table in new “related concepts” section in analogy with fivebrane group article
Anonymous
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)
started something at Reeb sphere theorem
Stub for triple category.
created stub for dimension
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)
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.
added pointer to today’s
I thought it was ridiculous that Span redirected to (infinity,n)-category of correspondences, so I made a stubby page for it instead.
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 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 -version here and that it’s a coCartesian fibration.