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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

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- Discussion Type
- discussion topicfiltered group
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

- Discussion Type
- discussion topicrational number
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

- Discussion Type
- discussion topicJohan Alm
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

- Discussion Type
- discussion topiccomplete Hopf algebra
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

- Discussion Type
- discussion topicMalcev Lie algebra
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

- Discussion Type
- discussion topicprinciple of equivalence
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Keith Harbaugh
- Last Active Mar 28th 2021

at

*principle of equivalence*I have restructured the Examples-section: added new subsections in “In physics” on gauge transformations and on general covariance (just pointers so far, no text), and then I moved the section that used to be called “In quantum mechanics” to “Examples-In category theory” and renamed it to “In the definition of $\dagger$-categories” (for that is really what these paragraphs discuss, not any notion of equivalence in quantum mechanics, the application of $\dagger$-categories in that context notwithstanding)

- Discussion Type
- discussion topicMalcev group
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

- Discussion Type
- discussion topic(infinity,1)Prof
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

- Discussion Type
- discussion topic(infinity,1)-profunctor
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

- Discussion Type
- discussion topiccontraction
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 28th 2021

Added related concepts:

- Discussion Type
- discussion topiccomplete augmented algebra
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Mar 27th 2021

- Discussion Type
- discussion topicuniversal vector bundle
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Mar 27th 2021

- Discussion Type
- discussion topicnilpotent completion of spectra
- Category Latest Changes
- Started by Charles Rezk
- Comments 1
- Last comment by Charles Rezk
- Last Active Mar 27th 2021

- Discussion Type
- discussion topicequalizer
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Mar 27th 2021

added to equalizer statement and proof that a category has equalizers if it has pullbcks and products

- Discussion Type
- discussion topiccoequalizer
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Dmitri Pavlov
- Last Active Mar 27th 2021

I have added to

*coequalizer*basic statements about its relation to pushouts.In the course of this I brought the whole entry into better shape.

- Discussion Type
- discussion topicpolynomial monad
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by David_Corfield
- Last Active Mar 27th 2021

added to

*polynomial monad*the article by Batanin-Berger on homotopy theory of algebras over polynomial monads.

- Discussion Type
- discussion topicdependent linear type theory
- Category Latest Changes
- Started by Urs
- Comments 74
- Last comment by David_Corfield
- Last Active Mar 27th 2021

Following discussion in some other threads, I thought one should make it explicit and so I created an entry

Currently this contains some (hopefully) evident remarks of what “dependent linear type theory” reasonably should be at least, namely a hyperdoctrine with values in linear type theories.

The entry keeps saying “should”. I’d ask readers to please either point to previous proposals for what “linear dependent type theory” is/should be, or criticise or else further expand/refine what hopefully are the obvious definitions.

This is hopefully uncontroversial and should be regarded an obvious triviality. But it seems it might be one of those hidden trivialities which deserve to be highlighted a bit more. I am getting the impression that there is a big story hiding here.

Thanks for whatever input you might have.

- Discussion Type
- discussion topicspin structure
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Mar 27th 2021

quick note at

*spin structure*on the characterization*over Kähler manifolds*

- Discussion Type
- discussion topicGuillemin-Sternberg geometric quantization conjecture
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Mar 27th 2021

started

*Guillemin-Sternberg geometric quantization conjecture*So far just the brief Idea and a few commented references.

- Discussion Type
- discussion topicinternalization
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Urs
- Last Active Mar 27th 2021

started to add to internalization a list of links to examples. Probably we have much more.

- Discussion Type
- discussion topicscheme
- Category Latest Changes
- Started by Richard Williamson
- Comments 13
- Last comment by David_Corfield
- Last Active Mar 27th 2021

I found the definition of a scheme to be slightly unclear/insufficiently precise at one point, so I have tweaked things slightly, and added more details. Indeed, it is quite common to find a formulation similar to ’every point has an open neighbourhood isomorphic to an affine scheme’, whereas I think it important to be clear that one does not have the freedom to choose the sheaf of rings on the local neighbourhood, it must be the restriction of the structure sheaf on $X$.

- Discussion Type
- discussion topiclax (∞,1)-colimit
- Category Latest Changes
- Started by Hurkyl
- Comments 7
- Last comment by Hurkyl
- Last Active Mar 27th 2021

I added the description of lax (co)limits of Cat-valued functors via (co)ends and ordinary (co)limits. I should probably flesh this out more.

I’ve adopted the convention on twisted arrows at twisted arrow category, which is opposite of that in GNN.

In the case of ordinary 2-category, when the diagram category is a 1-category, is the expression of lax (co)limits via ordinary weighted (co)limits really as simple as taking the weights $C_{\bullet/}$ or $C_{/\bullet}$? I can’t find a reference that spells that out clearly; if there really is such a simple description it should be put on the lax (co)limit page.

- Discussion Type
- discussion topicČech homology
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Mar 26th 2021

I have started an entry for Čech homology.

- Discussion Type
- discussion topicdomain theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Mar 26th 2021

added pointer to:

- Tom de Jong, Martín Hötzel Escardó,
*Domain Theory in Constructive and Predicative Univalent Foundations*, in:*29th EACSL Annual Conference on Computer Science Logic*, CSL 2021, LIPIcs proceedings 183, 2021 (doi:10.4230/LIPIcs.CSL.2021.28, arXiv:2008.01422)

- Tom de Jong, Martín Hötzel Escardó,

- Discussion Type
- discussion topicenriched category
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by David_Corfield
- Last Active Mar 26th 2021

I added two recent examples of enriched categories: tangent bundle categories and Lawvere theories.

- Discussion Type
- discussion topicJohn Cartmell
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Mar 26th 2021

- Discussion Type
- discussion topicgeneralized algebraic theory
- Category Latest Changes
- Started by Kevin Watkins
- Comments 4
- Last comment by David_Corfield
- Last Active Mar 26th 2021

I’m having trouble understanding the relationship between what is described in the page generalized algebraic theory and what is in the referenced article by Cartmell.

For example, the nLab page seems to contemplate three levels of symbols—“supersort”, “sort”, and “operation”—while if I understand correctly, Cartmell’s GATs only have symbols at two levels, for sorts and operations. Also, I would only expect a GAT’s sort symbols to be applied to terms, not types as the nLab page seems to contemplate. (The nLab page speaks of “derived operations in the theory of sorts” rather than types, but I believe the same concept is intended.) My intuition is that the world of GATs in Cartmell’s sense more or less corresponds to a certain sublanguage of LF (rather than $F_\omega$), so there shouldn’t be anything like a symbol that is applied to arguments that are types and yields a type.

- Discussion Type
- discussion topicinternal category
- Category Latest Changes
- Started by Urs
- Comments 65
- Last comment by David_Corfield
- Last Active Mar 26th 2021

I edited the formatting of internal category a bit and added a link to internal infinity-groupoid

it looks like the first query box discussion there has been resolved. Maybe we can remove that box now?

- Discussion Type
- discussion topicexceptional field theory
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Mar 26th 2021

started some minimum at

*exceptional field theory*(the formulation of 11d supergravity that makes the exceptional U-duality symmetry manifest)

- Discussion Type
- discussion topicGuillaume Bossard
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 26th 2021

- Discussion Type
- discussion topicsphere fiber bundle
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 25th 2021

added the statement that

The stable tangent bundle of a unit sphere bundle $S(\mathcal{V})$ in a real vector bundle $\mathcal{V} \overset{p}{\longrightarrow} M$ (Example \ref{UnitSphereBundles}) over a smooth manifold $M$ is isomorphic to the pullback of the direct sum of the stable tangent bundle of the base manifold with that vector bundle:

$T^{stab} S(\mathcal{V}) \; \simeq \; S(p)^\ast \big( T^{stab} M \oplus_M \mathcal{V} \big) \,.$Still need to add a more canonical reference and/or a proof.

- Discussion Type
- discussion topiccategory over an operad
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Mar 25th 2021

- Discussion Type
- discussion topicChristine Escher
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2021

brief

`category:people`

-entry for hyperlinking references at*sphere-fiber bundle*and*exotic 7-sphere*

- Discussion Type
- discussion topicDiarmuid Crowley
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2021

brief

`category:people`

-entry for hyperlinking references at*sphere-fiber bundle*and*exotic 7-sphere*

- Discussion Type
- discussion topicHurewicz fibration
- Category Latest Changes
- Started by Harry Gindi
- Comments 8
- Last comment by Urs
- Last Active Mar 25th 2021

I dropped a query box over at Hurewicz fibration about a small difference of definition between the Lab and May's Concise Topology (in May, one only needs to have the RLP with respect to inclusions of CGWH spaces (into their cylinder objects at 0), while on the Lab, it says it must hold for all topological spaces.

By the way, I like that page a lot, since it doesn't have an excessively long idea section (actually, there isn't any idea section, but I like that). It's written so it's easy to find the definition without wading through all of the clutter.

I was wondering if it would be possible to swap the idea section and definition section around in a few articles and see if it makes them more readable. There's nothing more irritating than skipping the idea section only to find that the definition references the concepts introduced there. A definition should be readable entirely without having to read the idea section.

- Discussion Type
- discussion topicMatthieu Sozeau
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2021

brief

`category:people`

-entry for hyperlinking references at*opetopic type theory*

- Discussion Type
- discussion topicAntoine Allioux
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2021

brief

`category:people`

-entry for hyperlinking references at*opetopic type theory*

- Discussion Type
- discussion topicopetopic type theory
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by Urs
- Last Active Mar 25th 2021

Created

*opetopic type theory*with a bit of explanation based on what I understood based on what Eric Finster explained and demonstrated to me today.This is the most remarkable thing.

I have added pointers to his talk slides and to his online opetopic type system, but I am afraid unguided exposition to either does not reveal at all the utmost profoundness of what Eric made me see when he explained and demonstrated OTT to me on his notebook. I hope he finds time and a way to communicate this insight.

- Discussion Type
- discussion topicIgor Tyutin
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Mar 24th 2021

- Discussion Type
- discussion topicBRST complex
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Mar 24th 2021

at BRST complex I started writing out a detailed example to supplement this reply over at TP:SE. I am not done yet, but I have to interrupt now.

- Discussion Type
- discussion topicBRST
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Mar 24th 2021

- Discussion Type
- discussion topicnumerable fiber bundle
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by nLab edit announcer
- Last Active Mar 24th 2021

- Discussion Type
- discussion topicChern-Weil theory
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Mar 24th 2021

tried to polish a bit the matrial at Chern-Weil theory.

(not that there is much, yet, but still)

- Discussion Type
- discussion topicdifferential geometry
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Urs
- Last Active Mar 24th 2021

have expanded the single sentence at differential geometry to something like a paragraph, indicating how differential geometry is the “higher geometry modeled on the pre-geometry $\mathcal{G} = CartSp$”

- Discussion Type
- discussion topicuniversal fibration of (infinity,1)-categories
- Category Latest Changes
- Started by Hurkyl
- Comments 5
- Last comment by Hurkyl
- Last Active Mar 24th 2021

- Discussion Type
- discussion topicexotic 7-sphere
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 24th 2021

am splitting off this entry from

*exotic smooth strcuture*, in order to facilitate linking to specifically the case of exotic 7-sphere.Accordingly, so far the bulk of the entry is just copied over from the corresponding section at

*exotic smooth structure*,But I also added a new paragraph,

and that is what motivated me to split this off. Namely it occured to me that from the point of view of M-theory on 8-manifolds, Milnor’s classical construction of exotic smooth 7-spheres as boundaries of 8-manifolds is very particularly the construction of near horizon limits of black M2-brane spacetimes in the context of M-theory on 8-manifolds.

This must be known in the literature, and I’d like to collect what is known about it. So far I found a brief comment in this direction, in section 3.2 of

- David Morrison, M. Ronen Plesser, section 3.2 of
*Non-Spherical Horizons, I*, Adv.Theor.Math.Phys.3:1-81, 1999 (arXiv:hep-th/9810201)

Will be adding more as I find more.

- David Morrison, M. Ronen Plesser, section 3.2 of

- Discussion Type
- discussion topicexotic smooth structure
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active Mar 24th 2021

recorded two references to physics application at exotic smooth structure

(no time for anything else)

- Discussion Type
- discussion topictopological groupoid
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 24th 2021

- Discussion Type
- discussion topicelliptic genera as partition functions -- references
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 24th 2021

a bare list of references, to be

`!include`

-ed into relevant entries (such as*Witten genus*,*M5-brane elliptic genus*but also inside*elliptic cohomology – references*) – for ease of harmonizing lists of references

- Discussion Type
- discussion topicreconstruction theorem
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 24th 2021

added hyperlinks to some more of the keywords (such as

*Giraud theorem*).I see lots of room to clean up this old entry, but will leave it at that for the moment.

- Discussion Type
- discussion topicDoplicher-Roberts reconstruction theorem
- Category Latest Changes
- Started by John Baez
- Comments 2
- Last comment by Urs
- Last Active Mar 24th 2021

- Discussion Type
- discussion topicextended topological quantum field theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Dmitri Pavlov
- Last Active Mar 23rd 2021

added pointer to:

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

- Lukas Müller,

- Discussion Type
- discussion topicHandbook of Categorical Algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Mar 23rd 2021

- Discussion Type
- discussion topichomotopy lifting property
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by jim_stasheff
- Last Active Mar 23rd 2021

expanded homotopy lifting property

- Discussion Type
- discussion topicM5-brane
- Category Latest Changes
- Started by Urs
- Comments 21
- Last comment by Urs
- Last Active Mar 23rd 2021

I have added to

*M5-brane*a fairly detailed discussion of the issue with the fractional quadratic form on differential cohomology for the dual 7d-Chern-Simons theory action (from Witten (1996) with help of Hopkins-Singer (2005)).In the new section

*Conformal blocks and 7d Chern-Simons dual*.

- Discussion Type
- discussion topicNew Spaces for Mathematics and Physics
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Ulrik
- Last Active Mar 23rd 2021

I have started a category:reference page

such as to be able to point to it for reference, e.g. from Kontsevich 15 etc.

- Discussion Type
- discussion topicEmily Nardoni
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 23rd 2021

- Discussion Type
- discussion topicIbrahima Bah
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 23rd 2021

- Discussion Type
- discussion topichigher category theory and physics
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Mar 23rd 2021

I have been further working on the entry higher category theory and physics. There is still a huge gap between the current state of the entry and the situation that I am hoping to eventually reach, but at least now I have a version that I no longer feel ashamed of.

Here is what i did:

Partitioned the entry in two pieces: 1. “Survey”, and 2. “More details”.

The survey bit is supposed to give a quick idea of what the set of the scene of fundamental physics is. It starts with a kind of creation story of physics from $\infty$-topos theory, which – I think – serves to provide a solid route from just the general abstract concept of space and process to the existence and nature of all $\sigma$-model quantum field theories of “$\infty$-Chern-Simons theory”-type (which includes quite a few) and moreover – by invoking the “holographic principle of higher category theory” – all their boundary theories, which includes all classical phase space physics.

The Survey-bit continues with indicating the formalization of the result of quantizing all these to full extended quantum field theories. It ends with a section meant to indicate what is and what is not yet known about the quantization step itself. This is currently the largest gap in the mathematical (and necessarily higher categorical) formalization of physics: we have a fairly good idea of the mathematics that describes geometric background structure for physics and a fairly good idea of the axioms satisfied by the quantum theories obtained from these, but the step which takes the former to the latter is not yet well understood.

The “More details”-bit is stubby. I mainly added one fairly long subsection on the topic of “Gauge theory”, where I roughly follow the historical route that eventually led to the understanding that gauge fields are modeled by cocycles in higher (nonabelian) differential cohomology.

Apart from this I added more references and some cross-links.

I know that the entry is still very imperfect. If you feel like pointing out all the stuff that is still missing, consider adding at least some keywords directly into the entry.

- Discussion Type
- discussion topicinternal logic
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Mar 23rd 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.