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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory 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 finite foundations functional-analysis functor 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 itex k-theory lie lie-theory limit limits linear linear-algebra locale localization logic 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-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 science 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 topicMay recognition theorem
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Aug 3rd 2020

- Discussion Type
- discussion topicpregroup grammar
- Category Latest Changes
- Started by alexis.toumi
- Comments 9
- Last comment by nLab edit announcer
- Last Active Aug 3rd 2020

- Discussion Type
- discussion topicEk-Algebras
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Aug 3rd 2020

started extracting stuff from Ek-Algebras

added the statement of the main result and created a new subsection at k-tuply monoidal n-category to link to it

renamed the previous entry E-k operad into little cubes operad

- Discussion Type
- discussion topicTate sphere
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Richard Williamson
- Last Active Aug 3rd 2020

am giving this an entry of its own, split off from

*motivic homotopy theory*.Nothing much here yet, just a bare minimum so far

- Discussion Type
- discussion topicétale cohomology
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Richard Williamson
- Last Active Aug 3rd 2020

created stub for etale cohomology

- Discussion Type
- discussion topicsimplicial homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Aug 3rd 2020

I felt we were lacking an entry titled

*simplicial homotopy theory*that usefully collects the relevant entries that we do have. So I am starting one.

- Discussion Type
- discussion topicspectral algebraic geometry
- Category Latest Changes
- Started by Théo de Oliveira S.
- Comments 1
- Last comment by Théo de Oliveira S.
- Last Active Aug 3rd 2020

- Discussion Type
- discussion topicderived algebraic geometry
- Category Latest Changes
- Started by adeelkh
- Comments 73
- Last comment by Urs
- Last Active Aug 3rd 2020

I added a link to a MathOverflow answer which I found helpful in understanding the connection between the two senses of "derived algebraic geometry". I wonder if there's anything more that can be said about this?

Also I'm wondering whether the first sense, i.e. the study of derived categories of coherent sheaves, is really a common use of the term "derived algebraic geometry"; I've always seen it in the second sense.

And perhaps we should mention

*homotopical*algebraic geometry as well?

- Discussion Type
- discussion topicBTZ black hole
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Aug 3rd 2020

- Discussion Type
- discussion topicshape theory
- Category Latest Changes
- Started by Urs
- Comments 61
- Last comment by Urs
- Last Active Aug 2nd 2020

added to shape theory a section on how strong shape equivalence of paracompact spaces is detected by oo-stacks on these spaces

By the way: I have a question on the secion titled "Abstract shape theory". I can't understand the first sentence there. It looks like this might have been broken in some editing process. Can anyone fix this paragraph and maybe expand on it?

- Discussion Type
- discussion topicrewriting
- Category Latest Changes
- Started by David_Corfield
- Comments 4
- Last comment by Tim_Porter
- Last Active Aug 2nd 2020

I thought to add

- Amar Hadzihasanovic,
*Diagrammatic sets and rewriting in weak higher categories*, (arXiv:2007.14505)

- Amar Hadzihasanovic,

- Discussion Type
- discussion topicgeometrization conjecture
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Aug 2nd 2020

stub for

*geometrization conjecture*

- Discussion Type
- discussion topicorbifold
- Category Latest Changes
- Started by Urs
- Comments 41
- Last comment by Urs
- Last Active Aug 2nd 2020

I am moving the following old query box exchange from orbifold to here.

old query box discussion:

I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely

**does not**view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?

Urs Schreiber: please, go ahead. It would be appreciated.

end of old query box discussion

- Discussion Type
- discussion topicheterotic string theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Aug 2nd 2020

added pointer to today’s

- Andrea Fontanella, Tomas Ortin,
*On the supersymmetric solutions of the Heterotic Superstring effective action*(arxiv:1910.08496)

- Andrea Fontanella, Tomas Ortin,

- Discussion Type
- discussion topicPontrjagin dual
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jul 31st 2020

- Discussion Type
- discussion topicshort map
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by Daniel Luckhardt
- Last Active Jul 31st 2020

- Discussion Type
- discussion topicAmar Hadzihasanovic
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 31st 2020

- Discussion Type
- discussion topicdouble category
- Category Latest Changes
- Started by John Baez
- Comments 15
- Last comment by AlexanderCampbell
- Last Active Jul 30th 2020

- I added more info on pseudo double categories and double bicategories to double category. I also simplified the picture of a square, which had been bristling with scary unnecessary detail. There's a slight blemish in the left vertical arrow, which I can't see how to fix.

- Discussion Type
- discussion topicreverse category
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active Jul 30th 2020

- Discussion Type
- discussion topicMorita equivalence
- Category Latest Changes
- Started by Thomas Holder
- Comments 12
- Last comment by David_Corfield
- Last Active Jul 30th 2020

This is intended to continue the issues discussed in the Lafforgue thread!

I have added an idea section to Morita equivalence where I sketch what I perceive to be the overarching pattern stressing in particular the two completion processes involved. I worked with ’hyphens’ there but judging from a look in Street’s quantum group book the pattern can be spelled out exactly at a bicategorical level.

I might occasionally add further material on the Morita theory for algebraic theories where especially the book by Adamek-Rosicky-Vitale (pdf-draft) contains a general 2-categorical theorem for algebraic theories.

Another thing that always intrigued me is the connection with shape theory where there is a result from Betti that the endomorphism module involved in ring Morita theory occurs as the shape category of a ring morphism in the sense of Bourn-Cordier. Another thing worth mentioning on the page is that the Cauchy completion of a ring in the enriched sense is actually its cat of modules (this is in Borceux-Dejean) - this brings out the parallel between Morita for cats and rings.

- Discussion Type
- discussion topicDay convolution
- Category Latest Changes
- Started by Harry Gindi
- Comments 51
- Last comment by Mike Shulman
- Last Active Jul 29th 2020

See Day convolution

I started writing up the actual theorem from Day’s paper “On closed categories of functors”, regarding an extension of the “usual” Day convolution. He identifies an equivalence of categories between biclosed monoidal structures on the presheaf category $V^{A^{op}}$ and what are called pro-monoidal structures on A (with appropriate notions of morphisms between them) (“pro-monoidal” structures were originally called “pre-monoidal”, but in the second paper in the series, he changed the name to “pro-monoidal” (probably because they are equivalent to monoidal structures on the category of “pro-objects”, that is to say, presheaves)).

This is quite a bit stronger than the version that was up on the lab, and it is very powerful. For instance, it allows us to seamlessly extend the Crans-Gray tensor product from strict ω-categories to cellular sets (such that the reflector and Θ-nerve functors are strong monoidal). This is the key ingredient to defining lax constructions for ω-quasicategories, and in particular, it’s an important step towards the higher Grothendieck construction, which makes use of lax cones constructed using the Crans-Gray tensor product.

- Discussion Type
- discussion topicAlex Eskin
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 28th 2020

brief

`category: people`

-entry for hyperlinking references at*pillowcase orbifold*

- Discussion Type
- discussion topicpillowcase orbifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 28th 2020

- Discussion Type
- discussion topicRiemannian orbifold
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active Jul 28th 2020

- Discussion Type
- discussion topicGUT
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Jul 28th 2020

**Edit to**: GUT by Urs Schreiber at 2018-04-01 01:21:13 UTC.**Author comments**:added pointer to textbook account

- Discussion Type
- discussion topicmodel structure on dg-categories
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jul 28th 2020

created model structure on dg-categories

- Discussion Type
- discussion topicCompLF/HOAS
- Category Latest Changes
- Started by atmacen
- Comments 4
- Last comment by atmacen
- Last Active Jul 28th 2020

- Discussion Type
- discussion topicFréchet spaces is not banach space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Richard Williamson
- Last Active Jul 27th 2020

- Discussion Type
- discussion topicAtiyah Lie groupoid
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Jul 27th 2020

at

*Atiyah Lie groupoid*was this old query box discussion, which hereby I am moving from there to here:+– {: .query} What is all of this $diag$ stuff? I don't understand either $(P \times P)/_{diag} G$ or $(P_x \times P_x)_{diag} G$. —Toby

David Roberts: It’s to do with the diagonal action of $G$ on $P\times P$ as opposed to the antidiagonal (if $G$ is abelian) or the action on only one factor. I agree that it’s a bad notation.

*Toby*: How well do you think it works now, with the notation suppressed and a note added in words? (For what it's worth, the diagonal action seems to me the only obvious thing to do here, although admittedly the others that you mention do exist.)*Todd*: I personally believe it works well. A small note is that this construction can also be regarded as a tensor product, regarding the first factor $P$ as a right $G$-module and the second a left module, where the actions are related by $g p = p g^{-1}$.*Toby*: H'm, maybe we should write diagonal action if there's something interesting to say about it. =–

- Discussion Type
- discussion topicrecollement
- Category Latest Changes
- Started by zskoda
- Comments 24
- Last comment by jonsterling
- Last Active Jul 27th 2020

I have created the entry recollement. Adjointness, cohesiveness etc. lovers should be interested.