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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty 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 constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration 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 lie-theory limit limits linear linear-algebra locale localization logic mathematics measure 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 type type-theory universal variational-calculus

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- Discussion Type
- discussion topicAdams operation
- Category Latest Changes
- Started by John Baez
- Comments 6
- Last comment by Urs
- Last Active Jan 7th 2021

I added a clearer “The idea” section for Adams operation, and changed the word “functorial” to “natural” in a number of places, because while various sources do say the Adams operations are functorial, they must really be natural transformations from the functor $K: Top \to AbGp$ to itself.

- Discussion Type
- discussion topicheap
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jan 7th 2021

- Discussion Type
- discussion topicquantum information
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by zskoda
- Last Active Jan 6th 2021

Aleks Kissinger has contacted me about his aims to start a collection of nLab entries on quantum information from the point of view of the Bob Coecke school.

Being very much delighted about this offer, I created a template entry quantum information for his convenience.

- Discussion Type
- discussion topicquantum computation
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by zskoda
- Last Active Jan 6th 2021

stub for

*quantum computation*

- Discussion Type
- discussion topicgood open cover
- Category Latest Changes
- Started by Todd_Trimble
- Comments 9
- Last comment by Dmitri Pavlov
- Last Active Jan 6th 2021

In response to an email sent to Urs by Andrew Marshall, a slight amendment to good open cover was made in the proof that paracompact manifolds admit good open covers.

- Discussion Type
- discussion topicsynthetic homotopy theory
- Category Latest Changes
- Started by David_Corfield
- Comments 5
- Last comment by David_Corfield
- Last Active Jan 6th 2021

Well I made a start, basing the entry on Urs’s FOM comment.

- Discussion Type
- discussion topiclax monoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by Mike Shulman
- Last Active Jan 6th 2021

- Discussion Type
- discussion topicfactorization system in a 2-category
- Category Latest Changes
- Started by Mike Shulman
- Comments 16
- Last comment by Mike Shulman
- Last Active Jan 6th 2021

Created factorization system in a 2-category.

- Discussion Type
- discussion topicinternal logic of an (oo,1)-topos
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Mike Shulman
- Last Active Jan 6th 2021

- Discussion Type
- discussion topicsynthetic mathematics
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by David_Corfield
- Last Active Jan 6th 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 topicLectures on Étale Cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 6th 2021

added jstor:j.ctt1bpmbk1

- Discussion Type
- discussion topicinternal logic
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Richard Williamson
- Last Active Jan 5th 2021

created at internal logic an Examples-subsection and spelled out at Internal logic in Set how by turning the abstract-nonsense crank on the topos Set, one does reproduce the standard logic.

- Discussion Type
- discussion topicPainleve transcendent
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jan 5th 2021

Some important references added, including recent geometric study

- Leonid O. Chekhov, Marta Mazzocco, Vladimir N. Rubtsov,
*Painlevé monodromy manifolds, decorated character varieties, and cluster algebras*, IMRN**24**(2017) 7639–7691 doi

and a section of papers studying the mathematics of Painlevé equations and of Fuchsian $j_\Gamma$-functions from model theoretic perspective (strongly minimal structures), including a new paper in Annals 2020.

- Leonid O. Chekhov, Marta Mazzocco, Vladimir N. Rubtsov,

- Discussion Type
- discussion topicCisinski model structure
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Hurkyl
- Last Active Jan 5th 2021

created Cisinski model structure

- Discussion Type
- discussion topicreduced cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 5th 2021

- Discussion Type
- discussion topicsuper-Cartan geometry
- Category Latest Changes
- Started by David_Corfield
- Comments 27
- Last comment by Urs
- Last Active Jan 5th 2021

I see super-Cartan geometry is taking shape. Will Clifford algebras make an appearance in the The super-Klein geometry: super-Minkowski spacetime section?

Is there a higher super-Cartan way of thinking about what is at 3-category of fermionic conformal nets, about the String 2-group and superstrings, as here about the spin group and fermions.

- Discussion Type
- discussion topiccategoricity
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by zskoda
- Last Active Jan 4th 2021

New entry categoricity and a slight extension of the stub abstract elementary class.

- Discussion Type
- discussion topicaxioms of choice
- Category Latest Changes
- Started by Mike Shulman
- Comments 45
- Last comment by DavidRoberts
- Last Active Jan 4th 2021

I wrote a bit about the axiom of multiple choice and about SOSHWIS (a terrible name, but I didn't have time to think of anything better), with comments at axiom of choice and anafunctor and COSHEP.

- Discussion Type
- discussion topic(classical) axiom of multiple choice
- Category Latest Changes
- Started by Brian Pinsky
- Comments 2
- Last comment by Urs
- Last Active Jan 4th 2021

- Discussion Type
- discussion topicPicard-Vessiot theory
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Jan 4th 2021

- Discussion Type
- discussion topicHowTo
- Category Latest Changes
- Started by Urs
- Comments 44
- Last comment by Richard Williamson
- Last Active Jan 4th 2021

Have added to HowTo a description for how to label equations

In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.

- Discussion Type
- discussion topick-ary factorization system
- Category Latest Changes
- Started by Fosco
- Comments 6
- Last comment by Richard Williamson
- Last Active Jan 4th 2021

I corrected a couple og microscopic typos at k-ary factorization system, and then I noticed that something is unclear in the definition: first of all the family of factorization system is asked to be strong (= uniqueness of solution to any lifting problem) or weak (existence, no uniqueness)? And when the definition says

$M_1 \subseteq \dots \subseteq M_{\kappa-1}$

**whenever this is meaningful**(equivalently, $E_{k-1} \subseteq\dots\subseteq E_{1}$)what does it precisely mean? Are we asking that right classes be nested?

Thirdly, it is my humble opinion that saying

A discrete category has a (necessarily unique) $(-1)$-ary factorisation system.

is formally incorrect: discrete categories are groupoids where the only arrows are identities, so this is a particular kind of 0-ary factorization system.

Instead, negative thinking suggests that (-1)-ary factorization systems live in non-unital categories, and detect precisely the case where the class of isomorphisms is empty (recall that in a WFS $(L,R)$ the intersection $L\cap R$ consists of all isomorphisms; if in a 0-ary factorization system we had $L=R=L\cap R=Iso(\mathbf C)$, morally in a (-1)-ary system the intersection has to be empty, giving a category without identities -i.e. a particular kind of “plot”, in the jargon of this paper which I finally convinced my friend Salvatore to put on the arXiv-, and more precisely an associative, “strongly nonunital” plot).

This leads to another question: how can be the notion of (W)FS be extended to Mitchell’s semicategories (with empty or partially defined identity function)?

- Discussion Type
- discussion topicM-theory
- Category Latest Changes
- Started by David_Corfield
- Comments 18
- Last comment by Urs
- Last Active Jan 4th 2021

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 topiccomposition algebra
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jan 4th 2021

added pointer to today’s article by Ross Street:

A general abstract formulation of Rost 96 in terms of string diagrams in additive braided monoidal categories is in

- Ross Street,
*Vector product and composition algebras in braided monoidal additive categories*(arXiv:1812.04143)

- Ross Street,

- Discussion Type
- discussion topicaxiom of multiple choice
- Category Latest Changes
- Started by Brian Pinsky
- Comments 2
- Last comment by Todd_Trimble
- Last Active Jan 3rd 2021

- Discussion Type
- discussion topicsix operations
- Category Latest Changes
- Started by Urs
- Comments 20
- Last comment by zskoda
- Last Active Jan 3rd 2021

I am starting something at six operations.

(Do we already have an nLab page on this? I seemed to remember something, but can’t find it.)

- Discussion Type
- discussion topicultrafilter
- Category Latest Changes
- Started by Todd_Trimble
- Comments 7
- Last comment by DavidRoberts
- Last Active Jan 3rd 2021

I added a bit to the section on the ultrafilter monad in ultrafilter. This could stand to be fleshed out still more. The immediate reason for my editing here was to put down the notion of “compact Hausdorff object” (which is used in a remark at BoolAlg).

- Discussion Type
- discussion topicnumber
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jan 3rd 2021

- Discussion Type
- discussion topicPythagoras
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Jan 3rd 2021

- Discussion Type
- discussion topicBecker-Gottlieb transfer
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jan 3rd 2021

added links to journal pages/pdf-s of:

James Becker, Daniel Gottlieb,

*The transfer map and fiber bundles*, Topology , 14 (1975) (pdf, doi:10.1016/0040-9383(75)90029-4)James Becker, Daniel Gottlieb,

*Vector fields and transfers*Manuscr. Math. , 72 (1991) pp. 111–130 (pdf, doi:10.1007/BF02568269)

and added pointer to:

- Dai Tamaki, Akira Kono, Section 4.5 in:
*Generalized Cohomology*, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)