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- Discussion Type
- discussion topicSandbox/lindist
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 50 minutes ago

- Discussion Type
- discussion topicSwiss cheese operad
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by RodMcGuire
- Last Active 13 hours 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 13 hours 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 topiclinearly distributive category
- Category Latest Changes
- Started by Mike Shulman
- Comments 26
- Last comment by valeriadepaiva
- Last Active 13 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 topicmodule over an operad
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active 14 hours 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 18 hours 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 19 hours ago

- Discussion Type
- discussion topictwisted bundle
- Category Latest Changes
- Started by Urs
- Comments 29
- Last comment by Urs
- Last Active 19 hours 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 21 hours 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 1 day ago

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

- Discussion Type
- discussion topicBruno Gavranović
- Category Latest Changes
- Started by bgavran
- Comments 2
- Last comment by Todd_Trimble
- Last Active 1 day ago

- Discussion Type
- discussion topichomotopy of rational maps
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active 1 day 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 topicAlbrecht Bertram
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day 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 1 day 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 topicHomePage
- Category Latest Changes
- Started by Urs
- Comments 70
- Last comment by dgroyals1
- Last Active 1 day ago

I have added at

*HomePage*in the section*Discussion*a new sentence with a new link:If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.

I had completly forgotton about that page

*Welcome to the nForum (nlabmeta)*. I re-doscivered it only after my recent related comment here.

- Discussion Type
- discussion topicrational map
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 1 day 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 1 day 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 topicphilosophy
- Category Latest Changes
- Started by zskoda
- Comments 10
- Last comment by curban
- Last Active 1 day ago

Several recent updates to literature at philosophy, the latest being

- Mikhail Gromov,
*Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2.*, pdf;*Structures, Learning and Ergosystems: Chapters 1-4, 6*(2011) pdf

which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…

- Mikhail Gromov,

- Discussion Type
- discussion topicshape via cohesive path ∞-groupoid
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Urs
- Last Active 2 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 2 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 2 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 2 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 2 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 2 days ago

- Discussion Type
- discussion topicMartin A. Guest
- 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*homotopy of rational maps*

- Discussion Type
- discussion topicEckmann-Hilton argument
- Category Latest Changes
- Started by Urs
- Comments 72
- Last comment by Urs
- Last Active 2 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 2 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 2 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 2 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 2 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 2 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 2 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 2 days ago

- Discussion Type
- discussion topicTerence Tao
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active 2 days ago

- Discussion Type
- discussion topicover-topos
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by nLab edit announcer
- Last Active 3 days ago

created over-topos

- Discussion Type
- discussion topicLie groupoid
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active 3 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 3 days ago

- Discussion Type
- discussion topicinfinity-groupoid
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 26
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicKan complex
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active 3 days ago

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 3 days ago

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 3 days ago

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*.

- Discussion Type
- discussion topicEilenberg-Moore category
- Category Latest Changes
- Started by FinnLawler
- Comments 24
- Last comment by Tim_Porter
- Last Active 4 days ago

I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.

Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?

- Discussion Type
- discussion topicHořava-Witten theory
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by alexchandel
- Last Active 4 days ago

stub for Hořava-Witten theory

- Discussion Type
- discussion topicBousfield localization of spectra
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active 4 days ago

gave

*Bousfield localization of spectra*a more informative Idea-section

- Discussion Type
- discussion topicrationalization
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active 4 days ago

created rationalization even though it overlaps with the material at rational topological space

- Discussion Type
- discussion topicself-adjoint
- Category Latest Changes
- Started by varkor
- Comments 7
- Last comment by varkor
- Last Active 4 days ago

- Discussion Type
- discussion topicself-adjoint functor
- Category Latest Changes
- Started by varkor
- Comments 2
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicStable categories and structured ring spectra
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active 4 days ago

- Discussion Type
- discussion topicelliptic genera as partition functions -- references
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 4 days ago

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 topicadjoint string
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active 4 days ago

- Discussion Type
- discussion topicambidextrous adjunction
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by varkor
- Last Active 4 days ago

started a stub for

*ambidextrous adjunction*, but not much there yet

- Discussion Type
- discussion topicbiadjoint pair
- Category Latest Changes
- Started by varkor
- Comments 3
- Last comment by varkor
- Last Active 4 days ago

- Discussion Type
- discussion topicbiadjunction
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by varkor
- Last Active 4 days ago

I added to biadjunction the statement and some references for the fact that any incoherent one can be improved to a coherent one.

- Discussion Type
- discussion topicsmashing localization
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

am expanding out this reference item at several places:

- Tilman Bauer,
*Bousfield localization and the Hasse square*(2011) (pdf), chapter 6 in: Christopher Douglas, John Francis, André Henriques, Michael Hill (eds.),*Topological Modular Forms*, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)

- Tilman Bauer,

- Discussion Type
- discussion topiccomonad
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active 5 days ago

- Discussion Type
- discussion topicpara construction
- Category Latest Changes
- Started by nLab edit announcer
- Comments 14
- Last comment by rongmin
- Last Active 5 days ago

This is a brief description of the construction that started appearing in category-theoretic accounts of deep learning and game theory. It appeared first in Backprop As Functor (https://arxiv.org/abs/1711.10455) in a specialised form, but has slowly been generalised and became a cornerstone of approaches unifying deep learning and game theory (Towards Foundations of categorical Cybernetics, https://arxiv.org/abs/2105.06332), (Categorical Foundations of Gradient-based Learning, https://arxiv.org/abs/2103.01931).

Our group here in Glasgow is using this quite heavily, so since I couldn’t find any related constructions on the nLab I decided to add it. This is also my first submission. I’ve read the “HowTo” page, followed the instructions, and I hope everything looks okay.

There’s quite a few interesting properties of Para, and eventually I hope to add them (most notably, it’s an Para is an oplax colimit of a functor BM -> Cat, where B is the delooping of a monoidal category M).

A notable thing to mention is that I’ve added some animated GIF’s of this construction. Animating categorical concepts is something I’ve been using as a pedagogical tool quite a bit (more here https://www.brunogavranovic.com/posts/2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html) and it seems to be a useful tool getting the idea across with less friction. If it renders well (it seems to) and is okay with you, I might add more to the Optics section, and to the neural networks section (I’m hoping to get some time to add our results there).

Bruno Gavranović

- Discussion Type
- discussion topicAdrian Norbert Schellekens
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 5 days ago

- Discussion Type
- discussion topicChern- and Pontrjagin forms -- section
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 5 days ago

Discussion of the formulas for the standard characteristic forms has been missing in various entries (e.g. at

*Chern class*at*characteristic form*, etc.). Since there is little point in discussing the Chern forms independently from the Pontrjagin forms etc. I am now making it a stand-alone section to be`!include`

-ed into relevant entries, to have it all in one place.Not done yet, though, but it’s a start.

- Discussion Type
- discussion topictwisted de Rham cohomology
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 5 days ago

added pointer to the original article:

- Ryan Rohm, Edward Witten, around (23) and appendix of:
*The antisymmetric tensor field in superstring theory*, Annals of Physics Volume 170, Issue 2, September 1986, Pages 454-489 (doi:10.1016/0003-4916(86)90099-0)

- Ryan Rohm, Edward Witten, around (23) and appendix of: