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- Discussion Type
- discussion topicgeometric realization of simplicial topological spaces
- Category Latest Changes
- Started by Urs
- Comments 71
- Last comment by Urs
- Last Active Jun 25th 2021

started an entry geometric realization of simplicial topological spaces.

I decided this is a topic big enough to justify splitting it off from geometric realization (of simplicial sets).

But not much there yet. I just wanted to record for the moment that this realization too, does preserve pullbacks.

- Discussion Type
- discussion topicsemi-simplicial topological space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 25th 2021

- Discussion Type
- discussion topicsemi-simplicial object
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Jun 25th 2021

have split off

*semi-simplicial object*from*semi-simplicial set*.

- Discussion Type
- discussion topicgraded modality
- Category Latest Changes
- Started by David_Corfield
- Comments 9
- Last comment by David_Corfield
- Last Active Jun 25th 2021

At our workshop yesterday we heard Dominic Orchard speak about “Graded Modal Logic for Linear Dependent Type Theory”/ So I’ve started graded modality.

Now any connection to Licata-Shulman-Riley?

- Discussion Type
- discussion topicequivariant elliptic cohomology
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by Urs
- Last Active Jun 25th 2021

stub for

*equivariant elliptic cohomology*, for the moment just to record the references given there

- Discussion Type
- discussion topicrelative category
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by Hurkyl
- Last Active Jun 25th 2021

- Discussion Type
- discussion topicequivariant differential topology
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Jun 25th 2021

- Discussion Type
- discussion topichadron
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 25th 2021

added graphics of the light hadron masses from Fodor-Hoelbling 12

- Discussion Type
- discussion topicbisimplicial set
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Tim_Porter
- Last Active Jun 25th 2021

I have split off bisimplicial set and bisimplicial group from bisimplicial object

- Discussion Type
- discussion topicinternal logic
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by Tim_Porter
- Last Active Jun 25th 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.

- Discussion Type
- discussion topicanalytic monad
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 9
- Last comment by Sam Staton
- Last Active Jun 24th 2021

Created:

## Idea

Analytic monads are monads on Set that correspond to operads in Set.

## Definition

More precisely, an operad $O$ in Set induced a monad $T$ on Set:

$T(S)=\coprod_{n\ge0} O_n \times_{\Sigma_n} S^n.$Such a monad $T$ is equipped with a canonical weakly cartesian natural transformation to the moand $Sym$ arising from the commutative operad.

## Properties

A theorem of Joyal \cite{Joyal} states that there is a monoidal equivalence between the monoidal category of endofunctors $Set\to Set$ that admits a weakly cartesian natural transformation to $Sym$ and the monoidal category of species, i.e., symmetric sequences in Set with the substitution product.

In particular, the category of analytic monads on Set is equivalent to the category of operads in Set.

## The colored case

The correspondence carries over to colored operads (with a set of colors $C$) if we use the slice category $Set/C$ instead of Set.

## The nonsymmetric case

A similar correspondence can be established for nonsymmetric case, except that we must include the data of a transformation to $Sym$, which is no longer unique.

## The homotopical case

The correspondence generalizes to (∞,1)-categories, with some statements becoming more elegant. See Gepner–Haugseng–Kock \cite{GHK}.

## Related concepts

## References

André Joyal,

*Foncteurs analytiques et espèces de structures*, Combinatoire énumérative (Montréal/Québec, 1985), Lecture Notes in Mathematics 1234 (1986), 126-159. doi.Mark Weber,

*Generic morphisms, parametric representations and weakly Cartesian monads*, Theory Appl. Categ. 13 (2004), 191–234.David Gepner, Rune Haugseng, Joachim Kock,

*∞-Operads as Analytic Monads*, arXiv:1712.06469.

- Discussion Type
- discussion topicbraid group
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active Jun 24th 2021

the entry

*braid group*said what a braid is, but forgot to say what the braid group is; I added in a sentence, right at the beginning (and fixed some other minor things).

- Discussion Type
- discussion topicworldline formalism
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jun 24th 2021

created

*worldline formalism*to go with this Physics.SE answer

- Discussion Type
- discussion topicM-category
- Category Latest Changes
- Started by Mike Shulman
- Comments 8
- Last comment by Todd_Trimble
- Last Active Jun 23rd 2021

I’ve added Peter May’s Galois theory example to M-category in a section “Applications”.

- Discussion Type
- discussion topicHigher Operads, Higher Categories
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 23rd 2021

Created:

A book by Tom Leinster.

London Mathematical Society Lecture Note Series 298 (2004).

Cambridge University Press.

ISBN 0 521 53215 9.

- Discussion Type
- discussion topicOrbifolds in Mathematics and Physics
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 23rd 2021

- Discussion Type
- discussion topicprecision experiment
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jun 23rd 2021

- Discussion Type
- discussion topictopological G-space
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Jun 23rd 2021

added pointer to Bredon 72. Will add this pointer also to various related entries on equivariant homotopy theory

- Discussion Type
- discussion topicSchwinger effect
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jun 23rd 2021

- Discussion Type
- discussion topicLie algebra cohomology
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jun 23rd 2021

started Lie algebra cohomology,

(for the moment mainly to record that reference on super Lie algebra cocycles)

- Discussion Type
- discussion topicmonad
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Dmitri Pavlov
- Last Active Jun 22nd 2021

following Zoran’s suggestion I added to the beginning of the Idea-section at monad a few sentences on the general idea, leading then over to the Idea with respect to algebraic theories that used to be the only idea given there.

Also added a brief stub-subsection on monads in arbitrary 2-categories. This entry deserves a bit more atention.

- Discussion Type
- discussion topicLawvere theory
- Category Latest Changes
- Started by Urs
- Comments 44
- Last comment by Dmitri Pavlov
- Last Active Jun 22nd 2021

started a Properties-section at Lawvere theory with some basic propositions.

Would be thankful if some experts looked over this.

Also added the example of the theory of sets. (A longer list of examples would be good!) And added the canonical reference.

- Discussion Type
- discussion topiclocally cartesian closed (infinity,1)-category
- Category Latest Changes
- Started by Urs
- Comments 46
- Last comment by Urs
- Last Active Jun 22nd 2021

- Discussion Type
- discussion topiclocally cartesian closed model category
- Category Latest Changes
- Started by Mike Shulman
- Comments 15
- Last comment by Urs
- Last Active Jun 22nd 2021

I’ve added a definition to locally cartesian closed model category, although I’m open to debate about whether this is the right definition. This definition is more or less exactly what one needs to interpret dependent products in type theory with function extensionality (I plan to add a proof of this). But it’s certainly less obviously correct from a pure model-categorical viewpoint. For one thing, it doesn’t imply that we have a cartesian closed model category, which one would naively expect a notion of “locally cartesian closed model category” to do.

- Discussion Type
- discussion topicadjoint (infinity,1)-functor
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by Urs
- Last Active Jun 22nd 2021

polished and expanded adjoint (infinity,1)-functor

- Discussion Type
- discussion topiccompletion of a ring
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Jun 22nd 2021

it seems we were lacking an entry

*completion of a ring*. I started a bare minimum and cross-linked with*p-adic integers*and*localization of a ring*. Wanted to do more, but am being interrupted now.

- Discussion Type
- discussion topicAtiyah-Segal completion theorem
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jun 22nd 2021

stub for

*Atiyah-Segal completion theorem*, for the moment just to record a reference

- Discussion Type
- discussion topicrepresentation ring
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 22nd 2021

added to

*representation ring*a brief remark on its relation to the equivariant K-theory of the point (very stubby still)

- Discussion Type
- discussion topicMochizuki's proof of abc
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active Jun 22nd 2021

- Discussion Type
- discussion topicMochizuki's corollary 3.12
- Category Latest Changes
- Started by nLab edit announcer
- Comments 10
- Last comment by nLab edit announcer
- Last Active Jun 21st 2021

- Discussion Type
- discussion topicinitial Θ-data
- Category Latest Changes
- Started by Richard Williamson
- Comments 45
- Last comment by Guest
- Last Active Jun 21st 2021

Am making a start on trying to understand something of Mochizuki’s IUTT papers. I do not hold out any promises on how far I am going to get, or how long it is going to take me! Even this very first definition is going to me a long time, I think, as I intend to try to fill out all details. All help will be appreciated!

- Discussion Type
- discussion topicSzpiro's conjecture
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Jun 21st 2021

- Discussion Type
- discussion topicHawaiian earring space > history
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Dmitri Pavlov
- Last Active Jun 21st 2021

I have changed the title of this article, as well as references to the object within it. Use of the term “Hawaiian Earring” is objected to by Hawaiian mathematicians. Please see these two threads, one by native Hawaiian and math PhD Dr. Marissa Loving, and the other by an expert on the Hawaiian Earring, Dr. Jeremy Brazas.

https://twitter.com/MarissaKawehi/status/1406244897611522049

https://twitter.com/jtbrazas/status/1406652385263501319

I have retitled the article “Shrinking wedge of circles”, which is the name used for this space in Hatcher’s “Algebraic Topology”. I have a retained a note in the body of the article that the space is sometimes referred to as the “Hawaiian earring space”.

This small change in name helps to make mathematics a more inclusive and just field, especially in consideration of the historical marginalization and exclusion of indigenous mathematicians. By taking this action, the nLab site can help to spread a change in language more widely, including on other math reference sites.

I hope that this change is readily accepted and approved by the nLab community. Thank you!

Justin Lanier

- Discussion Type
- discussion topicelliptic cohomology -- references
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 21st 2021

a bare list of references, to be

`!include`

-ed into relevant entries, such as at*elliptic cohomology*, but also at*equivariant elliptic cohomology*,*elliptic genus*,*Witten genus*etc.(in an attempt to clean up and harmonize the referencing across all these entries – still some way to go towards that goal, but it should be a start)

- Discussion Type
- discussion topiccomplex oriented cohomology theory
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by Urs
- Last Active Jun 21st 2021

added a tiny bit of basics to

*complex oriented cohomology theory*

- Discussion Type
- discussion topicgravitational wave
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Jun 21st 2021

just to make links work, I have started a minimum at

*gravitational wave*.

- Discussion Type
- discussion topicneural network
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Jun 21st 2021

Stub. For the moment just for providing a place to record this reference:

- Jean Thierry-Mieg,
*Connections between physics, mathematics and deep learning*, Letters in High Energy Physics, vol 2 no 3 (2019) (doi:10.31526/lhep.3.2019.110)

- Jean Thierry-Mieg,

- Discussion Type
- discussion topicpro-object
- Category Latest Changes
- Started by Tim_Porter
- Comments 29
- Last comment by Urs
- Last Active Jun 21st 2021

I understood that the old terminology was ’projective system’, and ’projective limit’ refereed to the limit of a projective system. Can anyone confirm that? if I am right the present entry is slightly incorrect, but this needs checking first before changing it.

- Discussion Type
- discussion topicTôhoku
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 3
- Last comment by DavidRoberts
- Last Active Jun 20th 2021

- Discussion Type
- discussion topicTim Hosgood
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 20th 2021

- Discussion Type
- discussion topicinternal category
- Category Latest Changes
- Started by Urs
- Comments 74
- Last comment by Urs
- Last Active Jun 20th 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 topicNitin Nitsure
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2021

brief

`category:people`

-entry for hyperlinking references at*FGA explained*and elsewhere

- Discussion Type
- discussion topicSteven L. Kleiman
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2021

- Discussion Type
- discussion topicLothar Göttsche
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2021

brief

`category:people`

-entry for hyperlinking references at*FGA explained*, and elsewhere

- Discussion Type
- discussion topicFGA
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2021

- Discussion Type
- discussion topicFGA explained
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by Urs
- Last Active Jun 20th 2021

I updated a link on FGA explained to one of the chapters, to point to the arXiv version (the one on constructing Hilbert and Quot schemes). I also added a link to the conference page itself, which has links to scans of lecture notes, as the direct lecture notes links seem to be broken.

- Discussion Type
- discussion topicEGA
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 5
- Last comment by Urs
- Last Active Jun 20th 2021

- Discussion Type
- discussion topicUnderstanding colimits in Set
- Category Latest Changes
- Started by andrew
- Comments 1
- Last comment by andrew
- Last Active Jun 20th 2021

- Discussion Type
- discussion topicabsorption category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 9
- Last comment by Todd_Trimble
- Last Active Jun 19th 2021

- Discussion Type
- discussion topicSpin(4)
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jun 19th 2021

- Discussion Type
- discussion topicrotation
- Category Latest Changes
- Started by zskoda
- Comments 12
- Last comment by Urs
- Last Active Jun 19th 2021

Wrote a minimum of substance to this entry. My interest is prompted by current thinking on $n$-dimensional analogues of Euler angles. There are several articles (mainly in quantum chemistry and mathematical physics literature) around 1969-1974, which introduce some analogues via calculational procedures. In my taste an elementary geometrical introduction using both intrinsic and extrinsic approach, in a spirit of Euler, will be more suggestive.

- Discussion Type
- discussion topicsimplicial object
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Jun 19th 2021

added to simplicial object a section on the canonical simplicial enrichment and tensoring of $D^{\Delta^{op}}$ for $D$ having colimits and limits.

- Discussion Type
- discussion topicsimplex category
- Category Latest Changes
- Started by FinnLawler
- Comments 23
- Last comment by Urs
- Last Active Jun 19th 2021

Some tidying up and additions at simplex category, in particular a section on its 2-categorical structure, and more on universal properties.

I’ve edited the definition to focus more on the augmented simplex category $\Delta_a$ instead of the ’topologists’ $\Delta$’, but I haven’t changed their names, because it seemed to me that that was the best way to keep everyone involved in the discussion at that page happy. (I also changed the ordinal sum functor from $+$ to $\oplus$, after Tim’s suggestion.)

- Discussion Type
- discussion topicdyadic rational module
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicdyadic rational algebra
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicGiry monad
- Category Latest Changes
- Started by zskoda
- Comments 57
- Last comment by Todd_Trimble
- Last Active Jun 18th 2021

Added a reference of Robert Furber, Bart Jacobs at Giry monad.

- Discussion Type
- discussion topicarchimedean group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 8
- Last comment by Todd_Trimble
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicEmery Thomas
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicDonald W. Anderson
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 18th 2021

- Discussion Type
- discussion topicGamma-space
- Category Latest Changes
- Started by Mike Shulman
- Comments 11
- Last comment by Dmitri Pavlov
- Last Active Jun 18th 2021

Created Gamma-space.