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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundle bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity 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 internal-categories k-theory lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology multicategories nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics planar 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 type type-theory universal variational-calculus

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- Discussion Type
- discussion topicaction monad
- Category Latest Changes
- Started by PaoloPerrone
- Comments 7
- Last comment by Urs
- Last Active 44 minutes ago

- Discussion Type
- discussion topicMaterial set theory
- Category Latest Changes
- Started by TobyBartels
- Comments 89
- Last comment by atmacen
- Last Active 53 minutes ago

I have reorganised set theory and spun off material set theory.

- Discussion Type
- discussion topicalgebra over a monad
- Category Latest Changes
- Started by Richard Williamson
- Comments 7
- Last comment by PaoloPerrone
- Last Active 7 hours ago

- Discussion Type
- discussion topiclattice gauge theory
- Category Latest Changes
- Started by Tim_Porter
- Comments 7
- Last comment by Urs
- Last Active 8 hours ago

- Discussion Type
- discussion topicflavour anomaly
- Category Latest Changes
- Started by Urs
- Comments 103
- Last comment by Urs
- Last Active 10 hours ago

- Discussion Type
- discussion topicflavour (particle physics)
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active 10 hours ago

- Discussion Type
- discussion topicinternal hom
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Mike Shulman
- Last Active 10 hours ago

at internal hom the following discussion was sitting. I hereby move it from there to here

Here's some discussion on notation:

$Cat(x \times y,z) \cong Cat(x,CAT(y,z)).$*Ronnie*: I have found it convenient in a number of categories to use the convention that if say the set of morphisms is $hom(x,y)$ then the internal hom when it exists is $HOM(x,y)$. In particular we have the exponential law for categoriesThen one can get versions such as $CAT_a(y,z)$ if $y,z$ are objects over $a$.

Of course to use this the name of the category needs more than one letter. Also it obviates the use of those fonts which do not have upper and lower case, so I have tended to use mathsf, which does not work here!

How do people like this? Of course, panaceas do not exist.

*Toby*: I see, that fits with using $\CAT$ as the $2$-category of categories but $\Cat$ as the category of categories. (But I'm not sure if that's a good thing, since I never liked that convention much.) I only used ’Hom’ for the external hom here since Urs had already used ’hom’ for the internal hom.Most of the time, I would actually use the same symbol for both, just as I use the same symbol for both a group and its underlying set. Every closed category is a concrete category (represented by $I$), and the underlying set of the internal hom is the external hom. So I would distinguish them only when looking at the theorems that relate them, much as I would bother parenthesising an expression like $a b c$ only when stating the associative law.

*Ronnie*: In the case of crossed complexes it would be possible to use $Crs_*(B,C)$ for the internal hom and then $Crs_0(B,C)$ is the actual set of morphisms, with $Crs_1(B,C)$ being the (left 1-) homotopies.But if $G$ is a groupoid does $x \in G$ mean $x$ is an arrow or an object? The group example is special because a group has only one object.

If $G$ is a group I like to distinguish between the group $Aut(G)$ of automorphisms, and the crossed module $AUT(G)$, some people call it the

*actor*, which is given by the inner automorphism map $G \to Aut(G)$, and this seems convenient. Similarly if $G$ is a groupoid we have a group $Aut(G)$ of automorphisms but also a group groupoid, and so crossed module, $AUT(G)$, which can be described as the maximal subgroup object of the monoid object $GPD(G,G)$ in the cartesian closed closed category of groupoids.*Toby*: ’But if $G$ is a groupoid does $x \in G$ mean $x$ is an arrow or an object?’: I would take it to mean that $x$ is an object, but I also use $\mathbf{B}G$ for the pointed connected groupoid associated to a group $G$; I know that groupoid theorists descended from Brandt wouldn't like that. I would use $x \in \Arr(G)$, where $\Arr(G)$ is the arrow category (also a groupoid now) of $G$, if you want $x$ to be an arrow. (Actually I don't like to use $\in$ at all to introduce a variable, preferring the type theorist's colon. Then $x: G$ introduces $x$ as an object of the known groupoid $G$, $f: x \to y$ introduces $f$ as a morphism between the known objects $x$ and $y$, and $f: x \to y: G$ introduces all three variables. This generalises consistently to higher morphisms, and of course it invites a new notation for a hom-set: $x \to y$.)

continued in next comment…

- Discussion Type
- discussion topicWolfgang Altmannshofer
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 10 hours ago

brief

`category:people`

-entry for hyperlinking references at*flavour problem*and*flavour anomaly*

- Discussion Type
- discussion topicGino Isidori
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 12 hours ago

brief

`category:people`

-entry for hyperlinking references at*flavour (particle physics)*and at*flavour anomaly*

- Discussion Type
- discussion topiclength scale
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 12 hours ago

- Discussion Type
- discussion topicQCD cosmology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 13 hours ago

- Discussion Type
- discussion topicAndrzej Buras
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 13 hours ago

brief

`category:people`

-entry for hyperlinking references at*kaon*and at*flavour anomaly*

- Discussion Type
- discussion topickaon
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 14 hours ago

starting a stub, for the moment just to record som references on potential flavour anomalies in kaon-decays:

- Andrzej J. Buras,
*The Revival of Kaon Flavour Physics*(arXiv:1609.05711)

- Andrzej J. Buras,

- Discussion Type
- discussion topicM-theory
- Category Latest Changes
- Started by David_Corfield
- Comments 14
- Last comment by Urs
- Last Active 16 hours ago

A stub for M-theory. What’s supposed to be so mysterious about it? Is it that people don’t even know what form it would take?

- Discussion Type
- discussion topicScience of Logic
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 16 hours ago

added English translation of this bit

PN§260 Der Raum ist in sich selbst der Widerspruch des gleichgültigen Auseinanderseins und der unterschiedlosen Kontinuität, die reine Negativität seiner selbst und das Übergehen zunächst in die Zeit. Ebenso ist die Zeit, da deren in Eins zusammengehaltene entgegengesetzte Momente sich unmittelbar aufheben, das unmittelbare Zusammenfallen in die Indifferenz, in das ununterschiedene Außereinander oder den Raum.

Space is in itself the contradiction of the indifferent being-apart and of the difference-less continuity, the pure negativity of itself and the transformation, first of all, to time. In the same manner time – since its opposite moments, held together in unity, immeditely sublate themselves – is the undifferentiated being-apart or: space.

And polished a little around and following this bit.

- Discussion Type
- discussion topicM2-M5 brane bound state
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 18 hours ago

- Discussion Type
- discussion topicquantum anomaly
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 21 hours ago

added to quantum anomaly

an uncommented link to Liouville cocycle

a paragraph with the basic idea of fermioninc anomalies

the missing reference to Witten’s old article on spin structures and fermioninc anomalies.

The entry is still way, way, stubby. But now a little bit less than a minute ago ;-

- Discussion Type
- discussion topicsmall category
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Mike Shulman
- Last Active 22 hours ago

Todd points out elsewhere that there is a problem with the following sentence in the section

*Smallness in the context of universes*:$C$ is

**essentially $U$-small**if there is a bijection from its set of morphisms to an element of $U$ (the same for the set of objects follows); this condition is non-evil.(introduced in revision 11).

It looks to me that first of all this is not the right condition – the right condition must mention equivalence of categories to a U-small category.

- Discussion Type
- discussion topicstructural set theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 25
- Last comment by nLab edit announcer
- Last Active 1 day ago

In response to discussions at set theory, created structural set theory with a tentative formal definition of when a set theory is "structural."

- Discussion Type
- discussion topicnerve
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by nLab edit announcer
- Last Active 1 day ago

- Discussion Type
- discussion topichorizontal chord diagram
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicstate on a star-algebra
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 1 day ago

I have expanded the Idea section at

*state on a star-algebra*and added a bunch of references.The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.

- Discussion Type
- discussion topicPieter Hofstra
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active 1 day ago

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

Added the recent

- Pieter Hofstra, Philip Scott,
*Aspects of categorical recursion theory*, (arXiv:2001.05778)

- Pieter Hofstra, Philip Scott,

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

added the following to the References-section of the entries

*Euclidean geometry*,*synthetic geometry*and*Coq*:A textbook account of the axiomatization of Euclidean geometry is

- Wolfram Schwabhäuser, W.Szmielew, Alfred Tarski,
*Mathematische Methoden in der Geometrie*, Springer 1983

Full formalization of this book in Coq (as synthetic geometry but following Tarski’s work) is discussed at

- Wolfram Schwabhäuser, W.Szmielew, Alfred Tarski,

- Discussion Type
- discussion topicgeometric type theory
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by AlexisHazell
- Last Active 1 day ago

I came to think that the term

*geometric type theory*for the type theory internal toi sheaf toposes should exists. Thanks to Bas Spitter for pointing out that Steve Vickers had already had the same idea (now linked to at the above entry).Also created

*geometric homotopy type theory*in this vein, with some evident comments.

- Discussion Type
- discussion topichyperanalytic function
- Category Latest Changes
- Started by aleksr
- Comments 9
- Last comment by Urs
- Last Active 1 day ago

This is a base topic of my contribution. It introduces a new function that gives series whose coefficients are powers of fine structure constant. Furthermore each member represents natural physical interaction. It can be treated as natural physics that introduces natural particles.

May be I made a lot of mistakes. I will correct them.

- Discussion Type
- discussion topicgeometry of physics -- categories and toposes
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active 1 day ago

I’ll be preparing here notes for my lectures

*Categories and Toposes (schreiber)*, later this month.

- Discussion Type
- discussion topic2-Lawvere theory
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Joe Moeller
- Last Active 2 days ago

created stub for 2-Lawvere theory

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

added a further quote from

interview with Mike Duff by Graham Fermelo,

*The universe speaks in numbers – Interview 14*(web):(7:04) The problem we face is that we have a patchwork understanding of M-theory, like a quilt. We understand this corner and that corner, but what’s lacking is the overarching big picture. So directly or indirectly, my research hopes to explain what M-theory really is. We don’t know what it is.

In a certain sense, and this is not a popular statement, I think it’s premature to be asking: “What are the empirical consequences”, because it’s not yet in a mature enough state, where we can sensibly make falsifiable prediction.