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- Discussion Type
- discussion topiccategory
- Category Latest Changes
- Started by Mike Shulman
- Comments 13
- Last comment by Urs
- Last Active Jun 18th 2021

- Discussion Type
- discussion topictotally ordered abelian group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 18th 2021

- Discussion Type
- discussion topictotally ordered ring
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 18th 2021

- Discussion Type
- discussion topiceffective epimorphism
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicshear map
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicSegal condition
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Jun 18th 2021

finally a stub for

*Segal condition*. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).

- Discussion Type
- discussion topicpseudolattice ordered ring
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicpseudolattice ordered abelian group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicYoneda lemma
- Category Latest Changes
- Started by Urs
- Comments 102
- Last comment by Urs
- Last Active Jun 18th 2021

Todd,

you added to Yoneda lemma the sentence

In brief, the principle is that the identity morphism $id_x: x \to x$ is the universal generalized element of $x$. This simple principle is surprisingly pervasive throughout category theory.

Maybe it would be good to expand on that. One might think that the universal property of a genralized element is that every other one factors through it uniquely. That this is true for the generalized element $id_x$ is a tautological statement that does not need or imply the Yoneda lemma, it seems.

- Discussion Type
- discussion topicLiouville theory
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jun 18th 2021

started something stubby at

*Liouville theory*, for the moment just so as to record some references and provide for a minimum of cross-links (e.g. with Chern-Simons gravity).(also created a stub for quantum Teichmüller theory in the course of this, but nothing there yet except a pointer to reviews)

- Discussion Type
- discussion topicColin Guillarmou
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 18th 2021

for hyperlinking references at conformal bootstrap

- Discussion Type
- discussion topicVincent Vargas
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 18th 2021

for hyperlinking references at Liouville theory

- Discussion Type
- discussion topicRémi Rhodes
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 18th 2021

for hyperlinking references at Liouville theory

- Discussion Type
- discussion topicAntti Kupiainen
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 18th 2021

for hyperlinking references at Liouville theory

- Discussion Type
- discussion topicFrançois David
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 18th 2021

for hyperlinking references at Liouville theory

- Discussion Type
- discussion topicEuler angle
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by nLab edit announcer
- Last Active Jun 17th 2021

- Discussion Type
- discussion topicconcordance
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 17th 2021

made more explicit (here) the back-link to

*topological vector bundle*for classical concordance for topological vector bundles

- Discussion Type
- discussion topicQuillen equivalence
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Tim_Porter
- Last Active Jun 17th 2021

I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.

- Discussion Type
- discussion topicparacompact locale
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 16th 2021

Created with the following content:

### Definition

A locale is

**paracompact**if it is regular and every open cover has a locally finite refinement.### Properties

Paracompact locales are very closely related to fully normal locales. In fact, for regular locales these two properties are equivalent.

Any metrizable locale is paracompact.

Any Lindelöf locale is paracompact.

A locale is paracompact if and only if it admits a complete uniformity.

The full subcategory of paracompact locales is a reflective subcategory of the category of completely regular locales as well as the category of all [[locales].

In particular, the inclusion functor from paracompact locales to locales preserves small limits, so in particular, products of paracompact locales are paracompact.

This last property clearly distinguishes paracompact locales from paracompact spaces, since products of paracompact spaces need not be paracompact.

### Related concets

- Discussion Type
- discussion topicpartition of unity
- Category Latest Changes
- Started by DavidRoberts
- Comments 10
- Last comment by Dmitri Pavlov
- Last Active Jun 16th 2021

Cleaned up partition of unity and fine sheaf a bit, so I could link to them from this MO answer to the question ’Why are there so many smooth functions?’.

- Discussion Type
- discussion topicparacompact Hausdorff spaces equivalently admit subordinate partitions of unity
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Dmitri Pavlov
- Last Active Jun 16th 2021

I have spelled out the proof at

*paracompact Hausdorff spaces equivalently admit subordinate partitions of unity*.This uses Urysohn’s lemma and the shrinking lemma, whose proofs are not yet spelled out on the $n$Lab.

- Discussion Type
- discussion topicfinal functor
- Category Latest Changes
- Started by Tom Hirschowitz
- Comments 9
- Last comment by Urs
- Last Active Jun 16th 2021

- Discussion Type
- discussion topicprincipal bundle
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Jun 16th 2021

I have added to

*principal bundle*a remark on their

*definition*As quotients;statements about (classes of) (counter-)examples of quotients

Thanks for pointers to the literature from this MO thread!

- Discussion Type
- discussion topiccohesive topos
- Category Latest Changes
- Started by Urs
- Comments 90
- Last comment by Urs
- Last Active Jun 16th 2021

created cohesive topos.

wrote an Idea-section that is meant to explain why the concept is very natural, trying to provide some of the chat that one cannot find in the terse (but beautiful in its own way) article by Lawvere

spelled out the definition in some detail, here, too, trying to fill in things that Lawvere is glossing over, making it all very explicit;

started an Examples-section:

copied over the discussion that $Sh(CartSp)$ is a connected topos. checking the remaining axioms for cohesive topos are easy, but i have not typed that yet

included a little discussion of how diffeological spaces fit in, following our conversation in another thread

started an analogous section for $\infty Sh(CartSp)$, but just a stub so far

but added in a section that goes rgrough the various items in Lawvere’s definition and discusses their meaning in a cohesive oo-topos

- Discussion Type
- discussion topicdiffeological groupoid
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 16th 2021

added pointer to the second original article by Souriau on diffeological groups:

- Jean-Marie Souriau,
*Groupes différentiels et physique mathématique*, In: Denardo G., Ghirardi G., Weber T. (eds.) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 201. Springer 1984 (doi:10.1007/BFb0016198)

- Jean-Marie Souriau,

- Discussion Type
- discussion topicVladimir Drinfel'd
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jun 16th 2021

Added the writing

- Vladimir Drinfeld,
*Prismatization*(arXiv:2005.04746)

from James Borger’s comment

But I am sure that there’s a rich, rich overlap between the higher-categorical world and the biring/plethory world. Just to mention one data point, in Drinfeld’s recent paper “Prismatization”, he explains how ring stacks give rise to cohomology theories. He’s particularly interested in prismatic cohomology, but crystalline, de Rham, Dolbeault are super fun baby cases. A ring stack is, sort of by definition, just Spec of a derived biring which is concentrated in two degrees. So “slightly categorified birings” = de-Rham-like cohomology theories.

- Vladimir Drinfeld,

- Discussion Type
- discussion topicreal number
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Todd_Trimble
- Last Active Jun 15th 2021

I looked at

*real number*and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:A

*real number*is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a*number field*, denoted $\mathbb{R}$. The underlying set is the*completion*of the ordered field $\mathbb{Q}$ of rational numbers: the result of adjoining to $\mathbb{Q}$ suprema for every bounded subset with respect to the natural ordering of rational numbers.The set of real numbers also carries naturally the structure of a topological space and as such $\mathbb{R}$ is called the

*real line*also known as*the continuum*. Equipped with both the topology and the field structure, $\mathbb{R}$ is a topological field and as such is the uniform completion of $\mathbb{Q}$ equipped with the absolute value metric.Together with its cartesian products – the Cartesian spaces $\mathbb{R}^n$ for natural numbers $n \in \mathbb{N}$ – the real line $\mathbb{R}$ is a standard formalization of the idea of

*continuous space*. The more general concept of (smooth)*manifold*is modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (see*spacetime*), or at least in classical physics. See at*geometry of physics*for more on this.

- Discussion Type
- discussion topicfixpoint
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Todd_Trimble
- Last Active Jun 15th 2021

- Discussion Type
- discussion topicparametric right adjoint
- Category Latest Changes
- Started by Mike Shulman
- Comments 13
- Last comment by Todd_Trimble
- Last Active Jun 15th 2021

Added alternative terminology “local right adjoint” and “strongly cartesian monad” from Berger-Mellies-Weber. They claim the former “has become the more accepted terminology” than “parametric right adjoint”; does anyone know other references to support this? (I think it’s certainly more logical, in that it fits with the general principle of “local” meaning “on slice categories” — not to be confused with the different general principle of “local” meaning “in hom-objects”.)

- Discussion Type
- discussion topicsynthetic mathematics
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by nLab edit announcer
- Last Active Jun 15th 2021

felt the desire to have an entry on the general idea (if any) of

*synthetic mathematics*, cross-linking with the relevant examples-entries.This has much room for being further expanded, of course.

- Discussion Type
- discussion topicGrothendieck ring
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jun 15th 2021

- Discussion Type
- discussion topicpermutation representation
- Category Latest Changes
- Started by zskoda
- Comments 106
- Last comment by David_Corfield
- Last Active Jun 15th 2021

Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a

*finite*group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the**permutation representation**in the standard representation theory books like in- Claudio Procesi,
*Lie groups, an approach through invariants and representations*, Universitext, Springer 2006, gBooks

I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.

Edit: new (related) entries for Claudio Procesi and Arun Ram.

- Claudio Procesi,

- Discussion Type
- discussion topicQuillen adjunction
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Jun 15th 2021

there used to be, all along, a section titled “Derived adjunction”, which however fell short of saying anything about the derived adjunction as such.

Have added the statement now, with pointer to a new stand-alone entry

*derived adjunction*.

- Discussion Type
- discussion topicfinitely complete category
- Category Latest Changes
- Started by Oscar_Cunningham
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jun 14th 2021

- Discussion Type
- discussion topiccartesian model category
- Category Latest Changes
- Started by Zhen Lin
- Comments 7
- Last comment by Mike Shulman
- Last Active Jun 14th 2021

I added an explicit definition of cartesian model category to cartesian closed model category to highlight the convention that the terminal object is assumed cofibrant.

- Discussion Type
- discussion topicstuff type
- Category Latest Changes
- Started by John Baez
- Comments 1
- Last comment by John Baez
- Last Active Jun 14th 2021

- Discussion Type
- discussion topicbeta-gamma system
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active Jun 14th 2021

added a definition to

*beta-gamma system*

- Discussion Type
- discussion topicplethory
- Category Latest Changes
- Started by Joe Moeller
- Comments 5
- Last comment by David_Corfield
- Last Active Jun 14th 2021

- Discussion Type
- discussion topicStone-Čech compactification
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Jun 13th 2021

have splitt off an entry

*Stone-Cech compactification*from the entry*compactum*, (where it was a bit hidden) such as to make the content here a bit easier to find and to link to.

- Discussion Type
- discussion topicJohn Cartmell
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by David_Corfield
- Last Active Jun 13th 2021

- Discussion Type
- discussion topicultrafilter theorem
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Guest
- Last Active Jun 13th 2021

- Discussion Type
- discussion topicspecies
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by David_Corfield
- Last Active Jun 13th 2021

- Discussion Type
- discussion topic(homotopical) species
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by David_Corfield
- Last Active Jun 13th 2021

at species it says that this is a presheaf of sets on $core(FinSet)$. At structure type this then makes me expect the words “is a presheaf of groupoids” on $core(FinSet)$. Is there a deeper reason why it does not say that?

It seems clear that the Gepner-Kock homotopical species are precisely the $(\infty,1)$-presheaves on $core(FinSet)$, i.e. the $\infty$-groupoid valued ones.

I’d think it would be good to emphasize this presheaf-point of view in our entries.

- Discussion Type
- discussion topicgraded monoid
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jun 12th 2021

- Discussion Type
- discussion topicmodel structure on simplicial presheaves
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Jun 12th 2021

fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.

- Discussion Type
- discussion topic(n+1,1)-category of n-truncated objects
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 12th 2021

- Discussion Type
- discussion topicsimplicial localization
- Category Latest Changes
- Started by Tim_Porter
- Comments 7
- Last comment by Urs
- Last Active Jun 12th 2021

Someone deleted the contents of the entry simplicial localization on th 4 April, then another reinstated it on the 5th. Curious!

- Discussion Type
- discussion topicmodel structure on sSet-categories
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Urs
- Last Active Jun 12th 2021

started adding some genuine substance to model structure on sSet-categories (which used to be just a template).

- Discussion Type
- discussion topiclocally presentable (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 12th 2021

added pointer to

- Lior Yanovski,
*The Monadic Tower for $\infty$-Categories*(arXiv:2104.01816)

This would reduce the question of Quillen right adjoints representing right $\infty$-adjoints between presentable $\infty$-categories to coreflections and monadic functors…

- Lior Yanovski,

- Discussion Type
- discussion topicJohn D. Berman
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 12th 2021

brief

`category:people`

-entry for hyperlinking references at*equivariant stable homotopy theory*and*enriched (infinity,1)-category theory*

- Discussion Type
- discussion topicequivariant stable cohomotopy
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 12th 2021

added references (also to

*Burnside ring*):Erkki Laitinen,

*On the Burnside ring and stable cohomotopy of a finite group*, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (jstor:24491306, Laitinen79.pdf:file)Wolfgang Lück,

*The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups*(arXiv:math/0504051)

- Discussion Type
- discussion topicenriched (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 12th 2021

I thought we needed an entry enriched (infinity,1)-category, so I created one. Added an Idea-section that mentions the evident general abstract definition (which hasn’t been worked out) and mentions the evident model (which has).

I have used links to this now at table - models for (infinity,1)-operads in an attempt to clarify the “general pattern” of the table (now the first part of the table itself).

I notice/rememberd that we have

*two*equally orphaned and equally stubby entries, titled*weak enrichment*and titled*homotopical enrichment*. Something should be done about that unfortunate state of affairs, but for the moment I just added more links between these.There was also this ancient discussion, which we don’t need to keep there:

[begin old forwarded discussion]

Urs: can anyone point me to – or write an entry containing – a discussion of systematical “homotopical enrichment” where we enrich over a homotopical category systematically weakening everything up to coherent homotopy. If/when we have this we should also link it to (infinity,n)-category, as that is built by iteratively doing homotopical enrichement starting with Top.

Mike: If anyone ever does anything like that, I would love to see it. As far as I know there is no general theory. You can define Segal categories in any homotopical category with finite products. You can define complete Segal spaces in any model category, at least, and less may suffice. And you can define $A_\infty$-categories in any monoidal homotopical category. But the problem is finding some way to get a handle on them, like lifting a model structure to them. Of course, people have iterated the existing definitions to get notions of $n$-category and of $(\infty,n)$-category (Simpson-Tamsamani, Trimble, Barwick, Lurie, etc), but I’ve never seen a general theory. Peter May and I have been planning for a while to think about iterating enriched $A_\infty$-categories.

[end old forwarded discussion]

- Discussion Type
- discussion topiccartesian multicategory
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Sam Staton
- Last Active Jun 11th 2021

Created cartesian multicategory.

- Discussion Type
- discussion topiccondensed mathematics
- Category Latest Changes
- Started by David_Corfield
- Comments 14
- Last comment by DavidRoberts
- Last Active Jun 11th 2021

- Discussion Type
- discussion topicsimplicially enriched category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 11th 2021

added pointer to:

- Philip Hirschhorn, Section 9.1.1 of:
*Model Categories and Their Localizations*, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

- Philip Hirschhorn, Section 9.1.1 of:

- Discussion Type
- discussion topicenriched model category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 11th 2021

added publication data for:

- Bertrand Guillou, Peter May,
*Enriched model categories and presheaf categories*, New York J. Math. 26 (2020) 37–9 (arXiv:1110.3567, , NYJM:2020/26-3)

- Bertrand Guillou, Peter May,

- Discussion Type
- discussion topicJohn Schwarz
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 11th 2021

added pointer to:

- Interview by D. Zierler, AIP Oral History Interviews, July 2020

- Discussion Type
- discussion topicsimplicial Quillen adjunction
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 10th 2021

felt like splitting off simplicial Quillen adjunction, so that one can better link to it

- Discussion Type
- discussion topicintegration over supermanifolds
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 10th 2021

added pointer to Belopolsky97b etc., regarding “picture number” induced by a choice of integral top-forms.

- Discussion Type
- discussion topicintegral form
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 10th 2021

added disambiguation with the notion of “integral form” in the sense of integration over supermanifolds