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-categories 2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory 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 limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal modal-logic model model-category-theory monads monoidal monoidal-category-theory morphism motives motivic-cohomology natural 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 string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory 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 topicinternal (co-)limit
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 23rd 2021

added pointer to:

- Francis Borceux, Chapter 8 in Vol 1
*Basic Category Theory*of:*Handbook of Categorical Algebra*, Cambridge University Press (1994) (doi:10.1017/CBO9780511525858)

- Francis Borceux, Chapter 8 in Vol 1

- Discussion Type
- discussion topicSketches of an Elephant
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active Mar 23rd 2021

- Discussion Type
- discussion topicdcpo
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by martinescardo
- Last Active Mar 22nd 2021

- Discussion Type
- discussion topicparacompact topological space
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Mar 22nd 2021

Added to

*paracompact topological space*in the*Properties*-section the statement that every bounded hypercover over a paracompact space is refined by the Čech nerve of a plain open cover… by appealing to a lemma in HTT. (Thanks to Danny Stevenson for pointing this out.)

- Discussion Type
- discussion topicnumerable open cover
- Category Latest Changes
- Started by arsmath
- Comments 17
- Last comment by Urs
- Last Active Mar 22nd 2021

- Discussion Type
- discussion topicnerve theorem
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Mar 22nd 2021

created nerve theorem

linked to it from homotopy groups in an (infinity,1)-topos, where it had implicitly been mentioned before, but not made explicit.

Apart from stating the theorem, I wrote a section that explains what’s going on from the nPOV. As far as I can see, at least.

- Discussion Type
- discussion topiccategory of simplices
- Category Latest Changes
- Started by Mike Shulman
- Comments 10
- Last comment by John Dougherty
- Last Active Mar 22nd 2021

I added a bit to category of simplices, including the fact that the category of nondegenerate simplices is final and thus colimits can be computed using only that, and that the nerve of the category of simplices itself is colimit-preserving.

- Discussion Type
- discussion topicToposes, Triples, and Theories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 22nd 2021

- Discussion Type
- discussion topicgroup object
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 22nd 2021

Added to group object the Yoneda-embedding-style definition and added supergroup to the list of examples.

- Discussion Type
- discussion topicPierre Ocvirk
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 22nd 2021

brief

`category:people`

-entry for hyperlinking references at*cosmic structure formation*and*cosmic reionization*

- Discussion Type
- discussion topiccosmic reionization
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 22nd 2021

a stub in

*cosmology*, for the moment just so as to satisfy links at*cosmic structure formation*

- Discussion Type
- discussion topicstructure formation
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Mar 22nd 2021

- Discussion Type
- discussion topicLocally Presentable and Accessible Categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Urs
- Last Active Mar 22nd 2021

I have created Locally Presentable and Accessible Categories, with redirect LPAC, since we seem to refer to it a lot. I also added it as a reference to compact object, where it was surprisingly absent.

- Discussion Type
- discussion topicSheaves in Geometry and Logic
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 22nd 2021

added some more keywords to the “index” (came here to add

*action object*, but added a few more while I was at it)

- Discussion Type
- discussion topicmodule object
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 22nd 2021

expanded the Idea-section and added a reference:

- Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of:
*Internal object actions*, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)

- Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of:

- Discussion Type
- discussion topiccategorical algebra -- contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 22nd 2021

- Discussion Type
- discussion topicaction object > history
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 22nd 2021

starting something, to be another item of

*internalization*along with*monoid object*and*group object*.Not done yet, but need to save.

(Our entry “action” does cover internal notions of actions, but it seems hard to bring that old entry into a good shape, and in any case splitting off an entry on the internal notion might just be the first step of bringing it into shape)

The evident question for compiling this entry is: What are the canonical original references?

Eckmann-Hilton in their articles originally introducing the notion of internal group objects (here) have all the ingredients in hand to say “action object”, but it seems they don’t (?).

I found the notion stated clearly and explicitly, albeit somewhat in passing, in:

- Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of:
*Internal object actions*, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)

but if anyone has other/better references, let’s add them.

- Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of:

- Discussion Type
- discussion topicHQFT
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Urs
- Last Active Mar 22nd 2021

I have edited at

*HQFT*, touched the general formatting and structuring a good bit, trying to clean it up and beautify it a bit, and added a brief cross-pointer to the cobordisms hypothesis for cobordisms with maps into a base manifold.

- Discussion Type
- discussion topicholographic light front QCD
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Mar 22nd 2021

am splitting this off from

*holographic QCD*, since the latter is getting too crowded. Prompted by today’s- Mohammad Ahmady,
*Holographic light-front QCD in B meson phenomenology*(arXiv:2001.00266)

- Mohammad Ahmady,

- Discussion Type
- discussion topichadrons as KK-modes of 5d Yang-Mills theory -- references
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Mar 22nd 2021

- Discussion Type
- discussion topiccartesian model category
- Category Latest Changes
- Started by Zhen Lin
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Mar 21st 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 topicpolarization identity
- Category Latest Changes
- Started by TobyBartels
- Comments 7
- Last comment by TobyBartels
- Last Active Mar 21st 2021

- Discussion Type
- discussion topicquotient stack
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by nLab edit announcer
- Last Active Mar 21st 2021

I noticed that there was no entry

*quotient stack*, so I quickly started one, just to be able to point to it from elswhere.

- Discussion Type
- discussion topicEckmann-Hilton argument
- Category Latest Changes
- Started by Urs
- Comments 70
- Last comment by Urs
- Last Active Mar 21st 2021

added to Eckmann-Hilton argument the formal proposition formulated in any 2-category.

BTW, doesn’t anyone have a gif with the nice picture proof?

- Discussion Type
- discussion topicinductive dimension
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 5
- Last comment by Dmitri Pavlov
- Last Active Mar 21st 2021

- Discussion Type
- discussion topicsketch
- Category Latest Changes
- Started by David_Corfield
- Comments 8
- Last comment by Urs
- Last Active Mar 21st 2021

I pasted in something Mike wrote on sketches and accessible models to sketch. But now it needs tidying up, and I’m wondering if it might have been better placed at accessible category. Alternatively we start a new page on sketch-theoretic model theory. Ideas?

- Discussion Type
- discussion topicnotions of dimension -- contents
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Mar 21st 2021

- Discussion Type
- discussion topicmonoid in a monoidal category
- Category Latest Changes
- Started by anqurvanillapy
- Comments 7
- Last comment by Urs
- Last Active Mar 21st 2021

- Discussion Type
- discussion topicbi-pointed object
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Mar 21st 2021

- Discussion Type
- discussion topictwo-valued object
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by nLab edit announcer
- Last Active Mar 21st 2021

two-valued objects are topos-theoretic analogues of the set $\mathbf{2}$ with two elements in $Set$, and the categorical semantics of the type of booleans. Different from subobject classifiers. Still to do: categorical semantics of binary coproduct types and binary product types using dependent sums and products and two-valued objects.

Anonymous

- Discussion Type
- discussion topicequivariant Tietze extension theorem
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 21st 2021

- Discussion Type
- discussion topicabsolute retract
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 21st 2021

splitting off this definition from

*neighborhood retract*, for ease of linking, and in order to record the characterization AR = ANR+contractible

- Discussion Type
- discussion topiclocally finite cell complex
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 21st 2021

- Discussion Type
- discussion topictwo-valued type
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Mar 21st 2021

- Discussion Type
- discussion topicdimension of a separable metric space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 21st 2021

just for completeness, to go alongside

*dimension of a cell complex*and*dimension of a manifold*

- Discussion Type
- discussion topicdimension of a manifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 21st 2021

Am finally giving this its own entry, alongside

*dimension of a cell complex*(this here used to just redirect to

*dimension*)

- Discussion Type
- discussion topiccovering dimension
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 21st 2021

have created covering dimension

- Discussion Type
- discussion topicseparable metric space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 21st 2021

- Discussion Type
- discussion topicfolklore
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active Mar 21st 2021

felt like archiving a quote by Paul Taylor somewhere, it is now at

*folklore*.Besides being funny, it is actually a useful comment for the newbie, and so I linked to it from

*category theory*.

- Discussion Type
- discussion topicjoin of simplicial sets
- Category Latest Changes
- Started by Urs
- Comments 89
- Last comment by Dmitri Pavlov
- Last Active Mar 20th 2021

I rephrased and restructured the Definition-section at join of simplicial sets a bit, in an attempt to make the exposition clearer. Please check if I succeeded.

- Discussion Type
- discussion topicexponential object
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Mike Shulman
- Last Active Mar 20th 2021

I had started an entry “exponentiation” but then thought better of it and instead expanded the existing exponential object: added an examples-section specifically for $Set$ and made some remarks on exponentiation of numbers.

- Discussion Type
- discussion topicOlof Hanner
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Tim_Porter
- Last Active Mar 20th 2021

brief

`category:people`

-entry for hyperlinking references at*absolute neighbourhood retract*

- Discussion Type
- discussion topicdifferential cohomology diagram
- Category Latest Changes
- Started by David_Corfield
- Comments 7
- Last comment by Urs
- Last Active Mar 20th 2021

Added

For discussion of the hexagon in an equivariant setting, see

- Andreas Kübel, Andreas Thom,
*Equivariant Differential Cohomology*, (arXiv:1510.06392)

- Andreas Kübel, Andreas Thom,

- Discussion Type
- discussion topicMichael Boardman
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Urs
- Last Active Mar 20th 2021

- Discussion Type
- discussion topicMeasure Theory
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

- Discussion Type
- discussion topicRiesz space
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

- Discussion Type
- discussion topicYosida duality
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

- Discussion Type
- discussion topicPaul Halmos
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

- Discussion Type
- discussion topicVladimir Bogachev
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

- Discussion Type
- discussion topicmeasure theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

added publication data to:

- Simon Henry,
*Measure theory over boolean toposes*, Mathematical Proceedings of the Cambridge Philosophical Society Volume 163 Issue 1, 2016 (arXiv:1411.1605, doi:10.1017/S0305004116000700)

- Simon Henry,

- Discussion Type
- discussion topicCauchy complete category
- Category Latest Changes
- Started by David_Corfield
- Comments 20
- Last comment by Mike Shulman
- Last Active Mar 19th 2021

- Asked a question there.

- Discussion Type
- discussion topicclassical model structure on simplicial sets
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

I have split off an entry

*classical model structure on simplicial sets*from “model structure on simplicial set”. This entry should eventually contain detailed, self-contained and polished discussion of the definition, verification and key properties of the standard Kan-Quillen model structure.So far I have inserted fair bit of background material regarding (minimal) fibrations and geometric realizations, essentially the material in chapter 1 of Goerss-Jardine. A bunch of little proofs are spelled out, but not yet the more laborious ones. Discussion of the verification of the axioms is not yet in the entry, but the key parts of the Quillen equivalence to $Top_{Quillen}$ are (modulo relying on previous lemmas that don’t have proofs spelled out yet).

The somewhat random list of properties of $sSet_{Quillen}$ that used to be sitting at “model structure on simplicial sets” I have copied over to a section “Basic properties”, just for completenes, but this now needs re-organization to give decent logical flow.

For the moment I have to leave it at that, need to take care of something else now for a little bit.

- Discussion Type
- discussion topicpolyhedron
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

- Discussion Type
- discussion topicpolytope
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 19th 2021

Moved some content from polyhedron.

- Discussion Type
- discussion topicglobal quotient orbifold
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Mar 19th 2021

- Discussion Type
- discussion topicgood orbifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 19th 2021

- Discussion Type
- discussion topicphilosophy
- Category Latest Changes
- Started by zskoda
- Comments 9
- Last comment by zskoda
- Last Active Mar 19th 2021

Several recent updates to literature at philosophy, the latest being

- Mikhail Gromov,
*Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2.*, pdf;*Structures, Learning and Ergosystems: Chapters 1-4, 6*(2011) pdf

which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…

- Mikhail Gromov,

- Discussion Type
- discussion topictopos
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by David_Corfield
- Last Active Mar 19th 2021

Added a table of contents to topos, a section on "special classes" and one on "higher toposes".

- Discussion Type
- discussion topicepimorphism
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by zskoda
- Last Active Mar 19th 2021

felt like adding a handful of basic properties to epimorphism

- Discussion Type
- discussion topiccartesian bicategory
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Mike Shulman
- Last Active Mar 18th 2021

Added the references

Aurelio Carboni, Bob Walters,

*Cartesian Bicategories, I*, articleAurelio Carboni, Max Kelly, Bob Walters, Richard Wood,

*Cartesian Bicategories II*, (arXiv:0708.1921)