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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion 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 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 limits linear linear-algebra locale localization logic manifolds 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 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

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- Discussion Type
- discussion topicentanglement island proposal for black hole paradox -- references
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 20th 2022

this is a bare list of references, meant to be

`!include`

-ed into the relevant References-sections at*black hole information paradox*and*Bekenstein-Hawkind entropy*

- Discussion Type
- discussion topicunivalence axiom
- Category Latest Changes
- Started by spitters
- Comments 39
- Last comment by nLab edit announcer
- Last Active Nov 20th 2022

- Discussion Type
- discussion topicquantum error correction
- Category Latest Changes
- Started by Urs
- Comments 38
- Last comment by Urs
- Last Active Nov 20th 2022

- Discussion Type
- discussion topicdifferential category
- Category Latest Changes
- Started by Tim_Porter
- Comments 22
- Last comment by J-B Vienney
- Last Active Nov 20th 2022

- Discussion Type
- discussion topicRobert Griffiths > history
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 20th 2022

I had accidentally duplicated this page, but merging it now into the new one:

*Robert B. Griffiths*

- Discussion Type
- discussion topicH. Dieter Zeh
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 20th 2022

- Discussion Type
- discussion topicErich Joos
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 20th 2022

- Discussion Type
- discussion topicquantum decoherence
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Nov 20th 2022

added pointer to today’s

- Chris Nagele, Oliver Janssen, Matthew Kleban,
*Decoherence: A Numerical Study*(arXiv:2010.04803)

- Chris Nagele, Oliver Janssen, Matthew Kleban,

- Discussion Type
- discussion topicRobert B. Griffiths
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 20th 2022

- Discussion Type
- discussion topicW. Forrest Stinespring
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 20th 2022

- Discussion Type
- discussion topicfamily of sets
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active Nov 20th 2022

- Discussion Type
- discussion topicfamily of subsets
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 20th 2022

added table of contents, headers, redirects, and a link to family of sets and union

Anonymous

- Discussion Type
- discussion topicfoundation of mathematics
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

at

*foundation of mathematics*I have tried to start an Idea-section.Also, I am hereby moving a bunch of old discussion boxes from there to here:

[ begin forwarded discussion ]

+– {: .query}

*Urs asks*: Concerning the last parenthetical remark: I suppose in this manner one could imagine $(n+1)$-categories as a foundation for $n$-categories? What happens when we let $n \to \infty$?*Toby answers*: That goes in the last, as yet unwritten, section. =–+– {: .query}

*Urs asks*: Can you say what the problem is?*Toby answers*: I'd say that it proved to be overkill; ETCS is simpler and no less conceptual. In ETCC (or whatever you call it), you can neatly define a group (for example) as a category with certain properties rather than as a set with certain structure. But then you still have to define a topological space (for example) as a set with certain structure (where a set is defined to be a discrete category, of course). I think that Lawvere himself still wants an ETCC, but everybody else seems to have decided to stick with ETCS.*Roger Witte*asks: Surely in ETCC, you define complete Heyting algebras as particular kinds of category and then work with Frames and Locales (ie follow Paul Taylor’s leaf and apply Stone Duality). You should be able to get to Top by examining relationships between Loc and Set. I thought Top might be the the comma category of forgetful functor from loc to set op and the contravariant powerset functor. Thus a Topological space would consist of a triple S, L, f where S is a set, L is a locale and f is a function from the objects of the locale to the powerset of S. A continuous function from S, L, f to S’, L’, f’ is a pair g, h where g is a function from the powerset of S’ to the powerset of S and g is a frame homomorphism from L’ to L and*(I don’t know how to draw the commutation square)*. However I think this has too many spaces since lattice structures other than the inclusion lattice can be used to define open sets.*Toby*: It's straightforward to define a topological space as a set equipped with a subframe of its power set. So you can define it as a set $S$, a frame $F$, and a frame monomorphism $f\colon F \to P(S)$, or equivalently as a set $S$, a locale $L$, and an epimorphism $f\colon L \to Disc(S)$ of locales, where $Disc(S)$ is the discrete space on $S$ as a locale. (Your ’However, […]’ sentence is because you didn't specify epimorphism/monomorphism.) This is a good perspective, but I don't think that it's any cleaner in ETCC than in ETCS.*Roger Witte*says Thanks, Toby. I agree with your last sentence but my point is that this approach is equally clean and easy in both systems. The clean thing about ETCC is the uniformity of meta theory and model theory as category theory. The clean thing about ETCS is that we have just been studying sets for 150 years, so we have a good intuition for them.I was responding to your point ’ETCC is less clean because you have to define some things (eg topological spaces) as sets with a structure’. But you can define and study the structure without referring to the sets and then ’bolt on’ the sets (almost like an afterthought).

Mike Shulman: In particular cases, yes. I thought the point Toby was trying to make is that only some kinds of structure lend themselves to this naturally. Groups obviously do. Perhaps topological spaces were a poorly chosen example of something that doesn’t, since as you point out they can naturally be defined via frames. But consider, for instance, a metric space. Or a graph. Or a uniform space. Or a semigroup. All of these structures can be easily defined in terms of sets, but I don’t see a natural way to define them in terms of categories without going through discrete categories = sets.

*Toby*: Roger, I don't understand how you intend to bolt on sets at the end. If I define a topological space as a set $S$, a frame $F$, and a frame monomorphism from $F$ to the power frame of $S$, how do I remove the set from this to get something that I can bolt the set onto afterwards? With semigroups, I can see how, from a certain perspective, it's just as well to study the Lawvere theory of semigroups as a cartesian category, but I don't see what to do with topological spaces.*Roger Witte*says If we want to found mathematics in ETCC we want to work on nice categories rather than nice objects. Nice objects in not nice categories are hard work (and probably ’evil’ to somke extent). Thus the answer to Toby is that to do topology in ETCC you do as much as possible in Locale theory (ie pointless topology) and then when you finally need to do stuff with points, you create Top as a comma like construction (ie you never take away the points but you avoid introducing them as long as possible). Is it not true that the only reason you want to introduce points is so that you can test them for equality/inequality (as opposed to topological separation)?Mike, I spent about two weeks trying to figure out how to get around Toby’s objection ’topology’ and now you chuck four more examples at me. My gut feeling is that the category of directed graphs is found by taking the skeleton of CAT, that metric locales are regular locales with some extra condition to ensure a finite basis, that Toby can mak

[ to be continued in next comment ]

- Discussion Type
- discussion topiccategory theory and foundations
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

moved material about category theory and foundations from foundations of mathematics to its own page

Anonymous

- Discussion Type
- discussion topicdivided power algebra
- Category Latest Changes
- Started by Théo de Oliveira S.
- Comments 24
- Last comment by J-B Vienney
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicsymmetric monoidal category
- Category Latest Changes
- Started by Urs
- Comments 38
- Last comment by J-B Vienney
- Last Active Nov 19th 2022

added to symmetric monoidal category a new Properties-section As models for connective spectra with remarks on the theorems by Thomason and Mandell.

- Discussion Type
- discussion topiccategory with class structure
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicRécoltes et semailles
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicsize issues
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicuniversal class
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicKarl Kraus
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicuniverse
- Category Latest Changes
- Started by Urs
- Comments 67
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

I noticed that there was a neglected stub entry universe that failed to link to the fairly detailed (though left in an unpolished state full of forgotten discussions) Grothendieck universe.

I renamed the former to universe > history and made “universe” redirect to “Grothendieck universe”

- Discussion Type
- discussion topicMorse-Kelley set theory
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

Added:

## References

A definitive source (by one of the authors of the theory) is

- Anthony P. Morse,
*A theory of sets*, Pure and Applied Mathematics XVIII, Academic Press (1965), xxxi+130 pp. Second Edition, Pure and Applied Mathematics 108, Academic Press (1986), xxxii+179 pp. ISBN: 0-12-507952-4

- Anthony P. Morse,

- Discussion Type
- discussion topicSteve Awodey
- Category Latest Changes
- Started by nLab edit announcer
- Comments 7
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicDaniel Moskovich
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicadiabatic quantum computation
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicquantum lambda-calculus
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicquantum programming languages -- references
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Nov 19th 2022

a bare list of references, to be

`!includ`

-ed into the list of references of relevant entries, such as at*quantum computing*and*quantum programming*, for ease of updating and syncing

- Discussion Type
- discussion topicstructural ZFC
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

- Discussion Type
- discussion topictype of classes
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicclass
- Category Latest Changes
- Started by DavidRoberts
- Comments 26
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

Added reference to the paper

Paul Blain Levy,

*Formulating Categorical Concepts using Classes*, arXiv:1801.08528

- Discussion Type
- discussion topicclass object
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

- Discussion Type
- discussion topicdiscrete object classifier
- Category Latest Changes
- Started by nLab edit announcer
- Comments 11
- Last comment by nLab edit announcer
- Last Active Nov 19th 2022

- Discussion Type
- discussion topiclarge set
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 18th 2022

- Discussion Type
- discussion topicsmall set
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Nov 18th 2022

created small set

- Discussion Type
- discussion topicfoundations and philosophy
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 18th 2022

- Discussion Type
- discussion topicrelative monad
- Category Latest Changes
- Started by mattecapu
- Comments 10
- Last comment by J-B Vienney
- Last Active Nov 18th 2022

- Discussion Type
- discussion topiclocale of real numbers
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

Adding reference

- Graham Manuell,
*Uniform locales and their constructive aspects*, (arXiv:2106.00678)

as a construction of the locale of real numbers can be found in section 5.3 of that article

Anonymous

- Graham Manuell,

- Discussion Type
- discussion topicformal topology
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

- Discussion Type
- discussion topicreflection
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Nov 17th 2022

The entry used to be a plain disambiguation between

*reflective subcategory*and an empty list of other meaning of reflection.But since the only evident point that entries like

*rotation*,*translation*,*boost*, etc. could point to for related concepts is an entry called*reflection*. So maybe that should be the main meaning here, and alternative meaning be secondary.I slightly edited accordingly. But besides the different link structure now, there is still no substantial content here.

- Discussion Type
- discussion topiccocartesian coclosed category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

- Discussion Type
- discussion topicreflection at a hyperplane
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

- Discussion Type
- discussion topicSEAR
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

- Discussion Type
- discussion topicChristoph Dorn
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by David_Corfield
- Last Active Nov 17th 2022

brief category:people-entry for hyperlinking references at

*associative n-category*and*homotopy.io*

- Discussion Type
- discussion topicsimply sorted set theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

started page on a class of set theories, which include unsorted set theory, two-sorted set theory, and three-sorted set theory, and where membership is a relation. Usually contrasted with dependently sorted set theory, where membership is a typing judgment.

Anonymous

- Discussion Type
- discussion topicthree-sorted set theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

starting article on set theories with three sorts, an example of which is structural ZFC.

Anonymous

- Discussion Type
- discussion topiclocale
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

added to locale a section relation to toposes stating localic reflection

- Discussion Type
- discussion topicIngo Blechschmidt
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

adding reference

- Ingo Blechschmidt,
*Generalized spaces for constructive algebra*, (arXiv:2012.13850)

Anonymous

- Ingo Blechschmidt,

- Discussion Type
- discussion topicequivariant homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 32
- Last comment by nilesjohnson
- Last Active Nov 17th 2022

expanded the discussion at equivariant homotopy theory

expanded the statement of the classical Elmendorf theorem

added the statement of the general Elmendorf theorem in general model categories

added remarks on G-equivariant oo-stacks, as special cases of this

- Discussion Type
- discussion topicMichael Hill
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Nov 17th 2022

have added under “Selected writings” the Kervaire-saga (here and at he other author’s pages):

Michael Hill, Michael Hopkins, Douglas Ravenel,

*Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem*, New Mathematical Monographs, Cambridge University Press (2021) [doi:10.1017/9781108917278]Michael Hill, Michael Hopkins, Douglas Ravenel,

*On the non-existence of elements of Kervaire invariant one*, Annals of Mathematics**184**1 (2016)[doi:10.4007/annals.2016.184.1.1, arXiv:0908.3724, talk slides]Michael Hill, Michael Hopkins, Douglas Ravenel,

*The Arf-Kervaire problem in algebraic topology: Sketch of the proof*, Current Developments in Mathematics, 2010: 1-44 (2011) (HHRKervaire.pdf:file, doi:10.4310/CDM.2010.v2010.n1.a1)Michael Hill, Michael Hopkins, Douglas Ravenel,

*The Arf-Kervaire invariant problem in algebraic topology: introduction*(2016) [pdf]

- Discussion Type
- discussion topicneutral constructive mathematics
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

- Discussion Type
- discussion topicconstructive mathematics
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

I made “constructive logic” redirect to here (“constructive mathematics”) instead of to “intuitionistic mathematics”, as it used to

- Discussion Type
- discussion topicBas Spitters
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

Added reference

- Bas Spitters,
*Constructive algebraic integration theory*, Annals of Pure and Applied Logic, Volume 137, Issues 1–3, January 2006, Pages 380-390 (doi:10.1016/j.apal.2005.05.031)

Anonymous

- Bas Spitters,

- Discussion Type
- discussion topicintegral
- Category Latest Changes
- Started by zskoda
- Comments 12
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

Urs has added Euler integration prompted by Tom’s post at nCafe; I wanted to do that and will contribute soon. I noticed there is no entry integral in $n$Lab, but it redirects to integration. I personally think that integral as a mathematical object is a slightly more canonical name for a mathematical entry than integration, if the two are not kept separated. Second, the entry is written as an (incomplete) disambiguation entry and with a subdivision into measure approach versus few odd entries. I was taught long time ago by a couple of experts in probability and measure theory that a complete subordination to the concept of integral to a concept of measure is pedagogically harmful, and lacks some important insights. This has also to do with the choice of the title: integration points to a process, and the underlying process may involve measure. Integral is about an object which is usually some sort of functional, or operator, on distributions which are to be acted upon.

Thus I would like to rename the entry into integral (or to create a separate entry from integration) and make it into a real entry, the list of variants being just a section, unlike in the disambiguation only version. What do you think. Then I would add some real ideas about it.

- Discussion Type
- discussion topicIgor Křiž
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 17th 2022

adding accents to the page name, to make Dmitri’s requested links work at

*localic group*

- Discussion Type
- discussion topicprecompact space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topictotally bounded space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

moving section on precompact spaces under properties section instead of definition section

Anonymous

- Discussion Type
- discussion topicpreuniform locale
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

- Discussion Type
- discussion topicweak excluded middle
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

Created weak excluded middle, which is equivalent to the one of de Morgan’s laws that is not intuitionistically valid.

- Discussion Type
- discussion topicDe Morgan Heyting category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Nov 16th 2022

these are to De Morgan Heyting algebras what Heyting categories are to Heyting algebras and what Boolean categories are to Boolean algebras.

Anonymous