Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

Want to take part in these discussions? Sign in if you have an account, or apply for one below

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicIgor V. Kanatchikov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 22nd 2023

- Discussion Type
- discussion topicn-plectic geometry
- Category Latest Changes
- Started by perezl.alonso
- Comments 3
- Last comment by perezl.alonso
- Last Active Aug 22nd 2023

- Discussion Type
- discussion topicPeter-Weyl theorem
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by zskoda
- Last Active Aug 22nd 2023

an essentially empty stub, for the moment just to satisfy a link long requested at

*harmonic analysis*

- Discussion Type
- discussion topicrepresentative function
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Aug 22nd 2023

If $G$ is an arbitrary monoid with multiplication $m:G\times G\to G$ then $m$ induces a map $m^*:Fun(G,k)\to Fun(G\times G,k)$, $m^*(f):f\mapsto f\circ m$. We say that $f\in Fun(G,k)$ is representative if $m^*(f)$ is in the image of the canonical map $Fun(G,k)\otimes Fun(G,k)\hookrightarrow Fun(G\times G,k)$. Equivalently, $f$ is representative if the span of all functions $g\cdot f : h\mapsto f(h\cdot g)$ is finite dimensional. It follows then that $m^*(f)$ is in fact in (the image of) $R(G)\otimes R(G)$ where $R(G)$ is the space of all representative functions on $G$.

Peter-Weyl theorem says that the continuous representative functions form a dense subspace of the space of all continuous functions on a compact Lie group $G$.

- Discussion Type
- discussion topicone
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Aug 22nd 2023

added a pointer to Euclid and his

*monad teminology*.

- Discussion Type
- discussion topicarrow (in computer science)
- Category Latest Changes
- Started by mattecapu
- Comments 3
- Last comment by Urs
- Last Active Aug 22nd 2023

- Discussion Type
- discussion topicJohn Hughes
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 22nd 2023

- Discussion Type
- discussion topictight and loose morphisms
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by varkor
- Last Active Aug 22nd 2023

- Discussion Type
- discussion topiclocal reflexive coequalizer
- Category Latest Changes
- Started by mattecapu
- Comments 1
- Last comment by mattecapu
- Last Active Aug 22nd 2023

- Discussion Type
- discussion topichyperalgebra
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Aug 22nd 2023

This entry largely overlaps with the new entry distribution on an affine algebraic group, but for the moment, due different tradition and minor differences in scope and in definitions , the entries are (at least temporarily) separate.

Some minor additions and changes at the page.

- Discussion Type
- discussion topicgroup scheme
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Aug 22nd 2023

I have edited group scheme and algebraic group slightly. To the latter I added Example-pointers to multiplicative group and additive group

- Discussion Type
- discussion topicaffine group scheme
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Urs
- Last Active Aug 22nd 2023

- Discussion Type
- discussion topicdistribution
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by zskoda
- Last Active Aug 22nd 2023

I found the section-outline of the entry

*distribution*was a bit of a mess. So I have now edited it (just the secion structure, nothing else yet):a) There are now two subsections for “Operations on distributions”,

b) in “Related concepts” I re-titled “Variants” into “Currents” (for that’s what the text is about) and gave “Hyperfunctions and Coulombeau distributions” its own subsection title.

c) split up the References into “General” and “On Coulombeau functions”.

(I hope that this message is regarded as boring and non-controversial.)

- Discussion Type
- discussion topicdifferential forms in synthetic differential geometry
- Category Latest Changes
- Started by maxsnew
- Comments 5
- Last comment by Urs
- Last Active Aug 22nd 2023

- Discussion Type
- discussion topic2-groupoid of Lie 2-algebra valued forms
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Aug 21st 2023

- Discussion Type
- discussion topiclist of functorial field theories
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Urs
- Last Active Aug 21st 2023

Created:

This article is meant to give an exhaustive list of explicitly constructed nontopological functorial field theories in dimension 2 and higher. All currently known explicit constructions are nonextended, and with the exception of the Kandel construction, have dimension 2.

## Free field theories

## Posthuma

- Hessel Posthuma,
*The Heisenberg group and conformal field theory*, arXiv.

## Kandel

- Santosh Kandel,
*Functorial Quantum Field Theory in the Riemannian setting*, arXiv.

## Tener

- James E. Tener,
*Construction of the unitary free fermion Segal CFT*, arXiv.

## Field theories with interaction

### Pickrell

- Doug Pickrell,
*P(phi)_2 quantum field theories and Segal’s axioms*, arXiv.

### Liouville field theory

- Colin Guillarmou, Antti Kupiainen, Rémi Rhodes, Vincent Vargas,
*Segal’s axioms and bootstrap for Liouville Theory*, arXiv.

- Hessel Posthuma,

- Discussion Type
- discussion topicDaniel Altschuler
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Aug 21st 2023

- Discussion Type
- discussion topicLaurent Freidel
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Aug 21st 2023

- Discussion Type
- discussion topicmultiplicative unitary
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Aug 21st 2023

I am starting a page about the pentagon relation for multiplicative unitaries and related mathematics. The page for pentagon relation should be a separate page, as one does not really need the real forms and unitarity condition for the pentagon to work; this pentagon relations is sometimes called pentagon equation. $n$Lan uses pentagon equation as a redirect to pentagon identity from the axioms of (coherent) monoidal category, which is usually called pentagon identity indeed, and the terms relations and equation are more used in the context of dilogarithms, quantum groups, operator algebras and alike subjects, all related. The pentagon coherence is in fact related to all of these in a large subset of cases which can be directly expressed categorically, but the literature is quite different in flavour and eventually I will build 3 different pages with redirects and other superstructure, and references to the related terms like Drinfeld associator.

- Discussion Type
- discussion topicKac algebra
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Aug 21st 2023

- Discussion Type
- discussion topicGeorge Kac
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Aug 21st 2023

- Discussion Type
- discussion topicfinite set
- Category Latest Changes
- Started by David_Corfield
- Comments 12
- Last comment by Urs
- Last Active Aug 21st 2023

Added

For a treatment in homotopy type theory see

- Dan Frumin, Herman Geuvers, Léon Gondelman, Niels van der Weide,
*Finite Sets in Homotopy Type Theory*, (pdf)

- Dan Frumin, Herman Geuvers, Léon Gondelman, Niels van der Weide,

- Discussion Type
- discussion topicFrobenius algebra
- Category Latest Changes
- Started by John Baez
- Comments 40
- Last comment by Urs
- Last Active Aug 21st 2023

- I added some more information under Frobenius algebra. I would like to add the axioms in picture form, but I haven't figure out how to upload pictures yet. I'm sure I could figure it out if I wanted...

- Discussion Type
- discussion topicTambara-Yamagami category
- Category Latest Changes
- Started by perezl.alonso
- Comments 4
- Last comment by perezl.alonso
- Last Active Aug 21st 2023

- Discussion Type
- discussion topicrelative monad
- Category Latest Changes
- Started by mattecapu
- Comments 15
- Last comment by Urs
- Last Active Aug 20th 2023

- Discussion Type
- discussion topicextended functorial field theory
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Dmitri Pavlov
- Last Active Aug 20th 2023

added pointer to:

- Lukas Müller,
*Extended Functorial Field Theories and Anomalies in Quantum Field Theories*(arXiv:2003.08217)

- Lukas Müller,

- Discussion Type
- discussion topicinvertible field theory
- Category Latest Changes
- Started by David_Corfield
- Comments 5
- Last comment by Dmitri Pavlov
- Last Active Aug 20th 2023

- Discussion Type
- discussion topicrelative adjoint functor
- Category Latest Changes
- Started by Peter Heinig
- Comments 2
- Last comment by Urs
- Last Active Aug 20th 2023

Made some some small improvements (ordering of sections, note on how the definition defaults to the usual definition of adjoints, fixing broken link in the references, etc) in relative adjoint functor.

- Discussion Type
- discussion topicsecond-countable spaces are Lindelöf
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 4
- Last comment by nLab edit announcer
- Last Active Aug 20th 2023

- Discussion Type
- discussion topicGabriele Lobbia
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 20th 2023

- Discussion Type
- discussion topicJohn Wheeler
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Aug 20th 2023

added today’s

- Kip Thorne,
*John Archibald Wheeler: A Biographical Memoir*(arXiv:1901.06623)

- Kip Thorne,

- Discussion Type
- discussion topicRuediger Schack
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Aug 20th 2023

- Discussion Type
- discussion topicCarlton Caves
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 20th 2023

- Discussion Type
- discussion topicMaximilian Schlosshauer
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 20th 2023

- Discussion Type
- discussion topicBayesian interpretation of quantum mechanics
- Category Latest Changes
- Started by TobyBartels
- Comments 51
- Last comment by Urs
- Last Active Aug 20th 2023

The Bayesian interpretation of quantum mechanics is correct. So there!

- Discussion Type
- discussion topicparacompact Hausdorff spaces equivalently admit subordinate partitions of unity
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by nLab edit announcer
- Last Active Aug 20th 2023

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 topicSven van Nigtevecht
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 19th 2023

- Discussion Type
- discussion topicmonoidal double category
- Category Latest Changes
- Started by mattecapu
- Comments 7
- Last comment by mattecapu
- Last Active Aug 19th 2023

- Discussion Type
- discussion topiccut rule
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by J-B Vienney
- Last Active Aug 19th 2023

some bare minimum at

*cut rule*

- Discussion Type
- discussion topicquasi-Frobenius algebra
- Category Latest Changes
- Started by perezl.alonso
- Comments 1
- Last comment by perezl.alonso
- Last Active Aug 18th 2023

created page just to hyperlink a result in weak Hopf algebra, should fill in

- Discussion Type
- discussion topicweak bialgebra
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by perezl.alonso
- Last Active Aug 18th 2023

To support mentioning weak wreath product in a parallel discussion with Urs, I created a stub for weak bialgebra with redirect weak Hopf algebra.

- Discussion Type
- discussion topicnonabelian group cohomology
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by jesuslop
- Last Active Aug 18th 2023

`<div> <p>created <a href="https://ncatlab.org/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a></p> <p>the secret title of this entry is "Schreier theory done right". (where "right" is right from the <a href="https://ncatlab.org/nlab/show/nPOV">nPOV</a>)</p> <p>this is the first part of the answer to</p> <blockquote> What is going on at <a href="https://ncatlab.org/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>? </blockquote> <p>The second part of the answer is the statement:</p> <blockquote> The same. </blockquote> <p>;-)</p> <p>I'll expand on that eventually.</p> </div>`

- Discussion Type
- discussion topicF-category
- Category Latest Changes
- Started by varkor
- Comments 2
- Last comment by varkor
- Last Active Aug 18th 2023

- Discussion Type
- discussion topichorizontal composition
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Aug 18th 2023

the graphics at the old entry

*horizontal composition*comes out wrong on my system. What’s going on? This is included as SVG.

- Discussion Type
- discussion topicCat
- Category Latest Changes
- Started by Peter Heinig
- Comments 16
- Last comment by Urs
- Last Active Aug 18th 2023

you can define $\Cat$ to be the 2-category of all $U'$-small categories, where $U'$ is some Grothendieck universe containing $U$. That way, you have $\Set \in \Cat$ without contradiction.

Do you agree with changing this to

” you can define $\Cat$ to be the 2-category of all $U'$-small categories, where $U'$ is some Grothendieck universe containing $U$. That way, for every small category $J$, you have the category $\Set^J$ an object of $\Cat$ without contradiction. This way, e.g. the diagram in Cat used in this definition of comma categories is defined. “

?

Reason: motivation is to have the pullback-definition of a comma category in (For others, it’s about the diagram here) defined, or rather, having Cat provide a way to make it precise. Currently, the diagrammatic definition can either be read formally, as a device to encode the usual definition of comma categories, or a reader can try to consult Cat in order to make it precise. Then they will first find only the usual definition of Cat having small objects only, which does not take care of the large category

$Set^I$

used in the pullback-definition. Then perhaps they will read all the way up to Grothendieck universes, but find that option not quite sufficient either since it only mentions Set, but not $Set^{Interval}$ . It seems to me that large small-presheaf-categories such as $Set^{Interval}$ can be accomodated, too, though.

(Incidentally, tried to find a “canonical” thread for the article “Cat”, by using the search, but to no avail. Therefore started this one.)

- Discussion Type
- discussion topicwhiskering
- Category Latest Changes
- Started by Peter Heinig
- Comments 10
- Last comment by Urs
- Last Active Aug 18th 2023

Added a pointer to adjoint functors and triangle identities to the entry whiskering. Feel that an encyclopedia entry on that operation should mention these two other entries.

- Discussion Type
- discussion topicRoger Godement
- Category Latest Changes
- Started by Todd_Trimble
- Comments 5
- Last comment by Urs
- Last Active Aug 18th 2023

Wrote up a quick article on Roger Godement.

- Discussion Type
- discussion topic2-category
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Urs
- Last Active Aug 18th 2023

promted by demand from my Basic-Course-On-Category-Theory-Students I expanded the entry 2-category:

mentioned more relations to other concepts in the Idea-section;

added an Examples-section with a bunch of (classes of) examples;

added a list of references. Please add more if you can think of more!

- Discussion Type
- discussion topicJosef Janyška
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 18th 2023

- Discussion Type
- discussion topicnatural bundle
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Aug 18th 2023

Wrote some minimum at

*natural bundle*.

- Discussion Type
- discussion topicChuu-Lian Terng
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 18th 2023

- Discussion Type
- discussion topicjet group
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Aug 18th 2023

created a bare minimum at

*jet group*

- Discussion Type
- discussion topicJean-François Pommaret
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 18th 2023

- Discussion Type
- discussion topicjet bundle
- Category Latest Changes
- Started by Urs
- Comments 61
- Last comment by Urs
- Last Active Aug 18th 2023

stub for jet bundle

- Discussion Type
- discussion topictorsor
- Category Latest Changes
- Started by nLab edit announcer
- Comments 42
- Last comment by zskoda
- Last Active Aug 18th 2023

- Discussion Type
- discussion topicmonad (disambiguation)
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by Urs
- Last Active Aug 18th 2023

I have added some accompanying text to the list of links at

*monad (disambiguation)*.One question: in the entry

*Gottfried Leibniz*it is claimed that the term “monad” for a functor on a category with monoid structure also follows Leibniz’s notion of monads. Is this really so? What’s a reference for this claim?I am asking because I don’t see how the notion of monoid in the endomorphisms of a category would be related to what Leibniz was talking about. What’s the idea, if there is one?

- Discussion Type
- discussion topiceven number
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Aug 18th 2023

I have added the adjoint modality of $Even \dashv Odd$ on $(\mathbb{Z}, \leq)$.

This example is from

*adjoint modality*(here). But it was actually a little wrong there. I have fixed it and expanded there and then copied over to here.

- Discussion Type
- discussion topicprime number
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Aug 18th 2023

I just see that in this entry it said

Classically, 1 was also counted as a prime number, …

If this is really true, it would be good to see a historic reference. But I’d rather the entry wouldn’t push this, since it seems misguided and, judging from web discussion one sees, is a tar pit for laymen to fall into.

The sentence continued with

…$[$ the number 1 is $]$ too prime to be prime.

and that does seem like a nice point to make. So I have edited the entry to now read as follows, but please everyone feel invited to have a go at it:

A

*prime number*is a natural number which cannot be written as a product of two smaller numbers, hence a natural number greater than 1, which is divisible only by 1 and by itself.This means that every natural number $n \in \mathbb{N}$ is, up to re-ordering of factors,

$n \;=\; 2^{n_1} 3^{n_2} 5^{n_3} 7^{n_4} 11^{ n_5 } \cdots$*uniquely*expressed as a product of a tuple of prime numbers:This is called the

*prime factorization*of $n$.Notice that while the number $1 \in \mathbb{N}$ is, clearly, only divisible by one and by itself, hence might look like it deserves to be counted as a prime number, too, this would break the uniqueness of this prime factorization. In view of the general phenomenon in classifications in mathematics of objects being too simple to be simple one might say that 1 is “too prime to be prime”.

- Discussion Type
- discussion topicD=2 CFT as functorial field theory -- references
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Aug 17th 2023

a bare list of references, to be

`!include`

-ed into lists of references of relevant entries (such as*2d CFT*,*2d SCFT*,*conformal cobordism category*,*modular functor*and maybe elsewhere)

- Discussion Type
- discussion topicquantum lambda-calculus
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Aug 17th 2023