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- Discussion Type
- discussion topicinter-universal Teichmüller theory
- Category Latest Changes
- Started by Urs
- Comments 140
- Last comment by Richard Williamson
- Last Active 4 hours ago

added to

*inter-universal Teichmüller theory*a pointer to the recent note- Yamashita,
*FAQ on ‘Inter-Universality’*(pdf)

(Though after reading I am not sure if that note helps so much.)

- Yamashita,

- Discussion Type
- discussion topiclinearly distributive category
- Category Latest Changes
- Started by Mike Shulman
- Comments 32
- Last comment by Todd_Trimble
- Last Active 8 hours ago

I reorganized linearly distributive category by moving the long block of history down to the bottom, adding an “Idea” section and a description of how $*$-autonomous categories give rise to linearly distributive ones and linearly distributive ones give rise to polycategories. I also cross-linked the page better with polycategory and star-autonomous category.

- Discussion Type
- discussion topicsketch
- Category Latest Changes
- Started by David_Corfield
- Comments 9
- Last comment by nLab edit announcer
- Last Active 15 hours ago

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 topiclittle cubes operad
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active 19 hours ago

added statement of and references to the weak equivalence with the Fulton-MacPherson operad (here)

added the same statement also at

*Fulton-MacPherson operad*

- Discussion Type
- discussion topicDynkin diagram
- Category Latest Changes
- Started by Grant_Bradley
- Comments 6
- Last comment by Grant_Bradley
- Last Active 20 hours ago

- Discussion Type
- discussion topicE8 manifold
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by Tim_Porter
- Last Active 23 hours ago

Added to fix a dead link.

- Discussion Type
- discussion topicsimplicial principal bundle
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Guest
- Last Active 23 hours ago

am splitting off simplicial principal bundle from simplicial group

- Discussion Type
- discussion topicthick subcategory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 12
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicsimplicial complex
- Category Latest Changes
- Started by Todd_Trimble
- Comments 36
- Last comment by Urs
- Last Active 1 day ago

Made some edits and additions at simplicial complex.

- Discussion Type
- discussion topichomotopy of rational maps
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active 2 days ago

starting something, on the kind of theorems originating with

- Graeme Segal,
*The topology of spaces of rational functions*, Acta Math. Volume 143 (1979), 39-72 (euclid:1485890033)

Nothing to be seen here yet, but I need to save. (Am not sold on the entry title, except that “topology” is not really the right term here.)

- Graeme Segal,

- Discussion Type
- discussion topicenumerative geometry
- Category Latest Changes
- Started by Tim_Porter
- Comments 6
- Last comment by Urs
- Last Active 2 days ago

There is an empty page entitled Towards an enumerative geometry of the moduli space of curves.

- Discussion Type
- discussion topictriangulation theorems -- references
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 2 days ago

a bare subsection with a list of references, to be

`!include`

-ed into the References-section of relevant enties, such as at*triangulation*and at*triangulation theorem*, for ease of synchronizing

- Discussion Type
- discussion topictriangulation theorem
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicTibor Radó
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category:people`

-entry for hyperlinking references at*triangulation theorem*and*Riemann surface*

- Discussion Type
- discussion topicStewart S. Cairns
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category:people`

-entry for hyperlinking references at*triangulation theorem*

- Discussion Type
- discussion topicHellmuth Kneser
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category:people`

-entry for hyperlinking references at*topological manifold*

- Discussion Type
- discussion topicEdwin E. Moise
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category:people`

-entry for hyperlinking references at*triangulation theorem*and*3-manifold*

- Discussion Type
- discussion topicCiprian Manolescu
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

brief

`category:people`

-entry for hyperlinking references at*triangulation theorem*

- Discussion Type
- discussion topic2-algebraic geometry
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Théo de Oliveira S.
- Last Active 3 days ago

On the occasion of Brandenburg 14 I have – finally – created an entry

*2-algebraic geometry*.We had

*almost*created that a few times before, only that we never did. Maybe the closest we came in the section*Derived algebraic geometry – Relation to noncommutative geometry*.I have tried to do some minimum cross-linking, with 2-ring, etc. But one might want to do more.

- Discussion Type
- discussion topicprogramming language
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 3 days ago

- Discussion Type
- discussion topicJørgen Tornehave
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

brief

`category:people`

-entry for hyperlinking references at*infinity loop space*and*E-infinity ring spectrum*

- Discussion Type
- discussion topictwisted K-theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 3 days ago

am slowly starting to add some genuine content to twisted K-theory

- Discussion Type
- discussion topicDavid Brook
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

brief

`category:people`

-entry for hyperlinking references at*higher twisted K-theory*

- Discussion Type
- discussion topicSwiss cheese operad
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by RodMcGuire
- Last Active 4 days ago

Created:

## Idea

The Swiss cheese operad is an analogue of the little disks operad, where disks are replaced by half-disks, which contain both ordinary disks in their interior, as well as half-disks positioned at the flat boundary.

This structure can be organized into an operad in the category of modules over the little disks operad.

## References

Alexander Voronov,

*The Swiss-Cheese Operad*, arXiv:math/9807037.Najib Idrissi,

*Swiss-Cheese operad and Drinfeld center*, arXiv:1507.06844.

- Discussion Type
- discussion topicregular cardinal
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by Dmitri Pavlov
- Last Active 4 days ago

I gave regular cardinal its own page.

Because I am envisioning readers who know the basic concept of a cardinal, but might forget what “regular” means when they learn, say, about locally representable category. Formerly the Lab would just have pointed them to a long entry cardinal on cardinals in general, where the one-line definition they would be looking for was hidden somewhere. Now instead the link goes to a page where the definition is the first sentence.

Looks better to me, but let me know what you think.

- Discussion Type
- discussion topicmodule over an operad
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active 4 days ago

Created:

## Definition

An operad is a monoid in the monoidal category of symmetric sequences equipped with the substitution product.

A

**module over an operad**is just a right module over this monoid.Right modules are very different from left modules, the latter are essentially algebras over an operad.

## Related concepts

## References

V. A. Smirnov. ON THE COCHAIN COMPLEX OF TOPOLOGICAL SPACES. Mathematics of the USSR-Sbornik 43:1 (1982), 133–144. doi.

Martin Markl,

*Models for operads*, Comm. Algebra 24 (1996), no. 4, 1471–1500. arXiv:hep-th/9411208v1.

- Discussion Type
- discussion topicbundle gerbe module
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 4 days ago

I have added pointer to

- Ernesto Lupercio, Bernardo Uribe, Section 7.2 of:
*Gerbes over Orbifolds and Twisted K-theory*, Comm. Math. Phys. 245(3): 449-489. (arXiv:math/0105039, doi:10.1007/s00220-003-1035-x)

Their Prop. 7.2.2 is verbatim the characterization that BCMMS made the definition of “bundle gerbe module” a month and a half later (except that LU focus on open covers instead of more general surjective submersions, but that’s not an actual restriction and in any case not the core of the definition).

Also added pointer to

- Marco Mackaay,
*A note on the holonomy of connections in twisted bundles*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 44 (2003) no. 1, pp. 39-62. (arXiv:math/0106019, numdam:CTGDC_2003__44_1_39_0)

which essentially recovers Lupercio & Uribe’s Def. 7.2.1.

From the arXiv timestamps I gather that it must have been an intense couple of weeks for all these auhtors in spring 2001. But Lupercio & Uribe came out first, by a fair margin. And in equivariant generality, right away…

- Ernesto Lupercio, Bernardo Uribe, Section 7.2 of:

- Discussion Type
- discussion topicorbifold K-theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topictwisted bundle
- Category Latest Changes
- Started by Urs
- Comments 29
- Last comment by Urs
- Last Active 4 days ago

I have been working on the entry twisted bundle.

Apart from more literature, etc. I have started typing something like a first-principles discussion: first a general abstract definition from twisted cohomology in any cohesive $\infty$-topos, then unwinding this in special cases to obtain the traditional cocycle formulas found in the literature.

Needs more polishing here and there, but I have to pause now.

- Discussion Type
- discussion topictwisted Chern character
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 4 days ago

added pointer to

- Daniel Freed, Michael Hopkins, Constantin Teleman, Section 2 of:
*Twisted equivariant K-theory with complex coefficients*, Journal of Topology, Volume 1, Issue 1, 2007 (arXiv:math/0206257, doi:10.1112/jtopol/jtm001)

- Daniel Freed, Michael Hopkins, Constantin Teleman, Section 2 of:

- Discussion Type
- discussion topicfunctors and comma categories
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active 5 days ago

- Discussion Type
- discussion topicmapping cylinder
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicBruno Gavranović
- Category Latest Changes
- Started by bgavran
- Comments 2
- Last comment by Todd_Trimble
- Last Active 5 days ago

- Discussion Type
- discussion topicAlbrecht Bertram
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

brief

`category:people`

-entry for hyperlinking references at*Gromov-Witten theory*

- Discussion Type
- discussion topicGromov-Witten invariants
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

added pointer to:

- Albrecht Bertram,
*Stable Maps and Gromov-Witten Invariants*, School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 (pdf)

- Albrecht Bertram,

- Discussion Type
- discussion topicrational map
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 5 days ago

added this pointer:

- János Kollár, Section 3 of:
*Algebraic hypersurfaces*, Bull. Amer. Math. Soc. 56 (2019), 543-568 (arXiv:1810.02861, doi:10.1090/bull/1663)

- János Kollár, Section 3 of:

- Discussion Type
- discussion topicalgebraic geometry
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

added pointer to:

- Igor Shafarevich,
*Basic Algebraic Geometry 1 – Varieties in Projective Space*, Springer 1977, 1994, 2013 (pdf, doi:10.1007/978-3-642-57908-0)

- Igor Shafarevich,

- Discussion Type
- discussion topicshape via cohesive path ∞-groupoid
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Urs
- Last Active 6 days ago

I have added pointer to the arXiv copy to the item

- Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov,
*Classifying spaces of infinity-sheaves*(arXiv:1912.10544)

- Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov,

- Discussion Type
- discussion topicMorse theory
- Category Latest Changes
- Started by David_Corfield
- Comments 5
- Last comment by nLab edit announcer
- Last Active 6 days ago

Added some references at Morse theory.

- Discussion Type
- discussion topicShigefumi Mori
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

brief

`category:people`

-entry for hyperlinking references at*birational geometry*

- Discussion Type
- discussion topicJános Kollár
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

brief

`category:people`

-entry for hyperlinking references at*Hilbert scheme*and*birational geometry*

- Discussion Type
- discussion topicHilbert scheme
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active 6 days ago

Have added a bunch of references to this entry.

**Question:**What precisely can one say about the relation between the topological space underlying the Hilbert scheme of points of $\mathbb{C}$ and/or $\mathbb{C}^2$, and the Fulton-MacPherson compactification of the corresponding configuration spaces of points?There is commentary in just this direction on p. 189 of:

- William Fulton, Robert MacPherson,
*A compactification of configuration spaces*, Ann. of Math. (2), 139(1):183–225, 1994 (jstor:2946631)

but it remains unclear to me what exactly the statement is, in the end.

- William Fulton, Robert MacPherson,

- Discussion Type
- discussion topicmodular theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active 6 days ago

- Discussion Type
- discussion topicMartin A. Guest
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

brief

`category:people`

-entry for hyperlinking references at*homotopy of rational maps*

- Discussion Type
- discussion topicEckmann-Hilton argument
- Category Latest Changes
- Started by Urs
- Comments 72
- Last comment by Urs
- Last Active 6 days ago

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 topicKohhei Yamaguchi
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

brief

`category:people`

-entry for hyperlinking references at*homotopy of rational maps*

- Discussion Type
- discussion topicMichal Adamaszek
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

brief

`category:people`

-entry for hyperlinking references at*homotopy of rational maps*

- Discussion Type
- discussion topicdifferential K-theory
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 6 days ago

I have expanded the list of references at

*differential K-theory*with a few comments thrown in.

- Discussion Type
- discussion topicmeasure theory
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by David_Corfield
- Last Active 6 days ago

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 topicFurstenberg-Zimmer structure theorem
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by Urs
- Last Active 6 days ago

- Discussion Type
- discussion topicmeasure theory and Boolean toposes
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by David_Corfield
- Last Active 7 days ago

Just heard a nice talk by Simon Henry about measure theory set up in Boolean topos theory (his main result is to identify Tomita-Takesaki-Connes’ canonical outer automorphisms on $W^\ast$-algebras in the topos language really nicely…).

I have to rush to the dinner now. But to remind myself, I have added cross-links between

*Boolean topos*and*measurable space*and for the moment pointed to- Matthew Jackson,
*A sheaf-theoretic approach to measure theory*, 2006 (pdf)

for more. Simon Henry’s thesis will be out soon.

Have to rush now…

- Matthew Jackson,

- Discussion Type
- discussion topicAsgar Jamneshan
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicTerence Tao
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active 7 days ago

- Discussion Type
- discussion topicover-topos
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by nLab edit announcer
- Last Active 7 days ago

created over-topos

- Discussion Type
- discussion topicLie groupoid
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active 7 days ago

added a list of examples to Lie groupoid

- Discussion Type
- discussion topicnerve
- Category Latest Changes
- Started by Mike Shulman
- Comments 25
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicinfinity-groupoid
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 26
- Last comment by Urs
- Last Active Jul 27th 2021

- Discussion Type
- discussion topicKan complex
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Jul 27th 2021

added a sentence to the Idea-section at Kan complex

- Discussion Type
- discussion topicclassical model structure on simplicial sets -- references
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 27th 2021

this is a bare list of (commented) references, to be

`!include`

-ed as a subsection into the References-section of relevant entries (notably at*model structure on simplicial sest*and at*classical model structure on simplicial sets*, from whose joint content this list here is compiled), for ease of synchronizing across entries

- Discussion Type
- discussion topicmodel structure on simplicial sets
- Category Latest Changes
- Started by Zhen Lin
- Comments 2
- Last comment by Urs
- Last Active Jul 27th 2021

I added two characterisations of weak homotopy equivalences to model structure on simplicial sets.

For the record, I found the inductive characterisation in Cisinski’s book [

*Les préfaisceaux comme modèles des types d’homotopie*, Corollaire 2.1.20], but I feel like I’ve seen something like it elsewhere. The characterisation in terms of internal homs comes from Joyal and Tierney [*Notes on simplicial homotopy theory*], but they take it as a*definition*.