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- Discussion Type
- discussion topiccobordism hypothesis
- Category Latest Changes
- Started by Urs
- Comments 57
- Last comment by Urs
- Last Active Jul 3rd 2022

added at cobordism hypothesis a pointer to

- Yonatan Harpaz,
*The Cobordism Hypothesis in Dimension 1*(arXiv:1210.0229)

where the case for $(\infty,1)$-categories is spelled out and proven in detail.

- Yonatan Harpaz,

- Discussion Type
- discussion topicChern-Weil homomorphism
- Category Latest Changes
- Started by Urs
- Comments 26
- Last comment by Urs
- Last Active Jul 3rd 2022

added to Chern-Weil homomorphism the description of the construction of the refined CW homomorphism by differential functions built using the universal connection as described by Hopkins-Singer.

- Discussion Type
- discussion topicGalois deformation ring
- Category Latest Changes
- Started by Anton Hilado
- Comments 6
- Last comment by Anton Hilado
- Last Active Jul 2nd 2022

- Discussion Type
- discussion topicdeformation theory
- Category Latest Changes
- Started by Anton Hilado
- Comments 1
- Last comment by Anton Hilado
- Last Active Jul 2nd 2022

- Discussion Type
- discussion topicBirch and Swinnerton-Dyer conjecture
- Category Latest Changes
- Started by Anton Hilado
- Comments 3
- Last comment by Anton Hilado
- Last Active Jul 2nd 2022

- Discussion Type
- discussion topicspecial values of L-functions
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Anton Hilado
- Last Active Jul 2nd 2022

Recorded the main idea at

*special values of L-functions*.

- Discussion Type
- discussion topicL-function
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Anton Hilado
- Last Active Jul 2nd 2022

have expanded the Idea-section at

*L-function*in an attempt to transport some actual idea. The main addition are these paragraphs:

The most canonically defined class of examples of L-functions are the

*Artin L-functions*defined for any Galois representation $\sigma \colon Gal \longrightarrow GL_n(\mathbb{C})$ as the Euler products of, essentially, characteristic polynomials of all the Frobenius homomorphisms acting via $\sigma$.Most other kinds of L-functions are such as to reproduces these Artin L-functions from more “arithmetic” data:

for 1-dimensional Galois representations $\sigma$ (hence for $n = 1$) Artin reciprocity produces for each $\sigma$ a Dirichlet character, or more generally a Hecke character $\chi$, and therefrom is built a

*Dirichlet L-function*or*Hecke L-function*$L_\chi$, respectively, which equals the corresponding Artin L-function $L_\sigma$;for general $n$-dimensional Galois representations $\sigma$ the conjecture of Langlands correspondence states that there is an automorphic representation $\pi$ corresponding to $\sigma$ and an

*automorphic L-function*$L_\pi$ built from that, which equalso the Artin L-function $L_\sigma$.

- Discussion Type
- discussion topicℓ-adic cohomology
- Category Latest Changes
- Started by Anton Hilado
- Comments 1
- Last comment by Anton Hilado
- Last Active Jul 2nd 2022

- Discussion Type
- discussion topicp-adic Hodge theory
- Category Latest Changes
- Started by adeelkh
- Comments 16
- Last comment by Anton Hilado
- Last Active Jul 2nd 2022

An anonymous user had created p-adic Hodge theory at some point. I added some references there on the approach of Beilinson and Bhatt.

- Discussion Type
- discussion topicsimplicial model for weak omega-categories
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jul 2nd 2022

Removed the following discussion:

Discussion on a previous version of this entry:

Mike: This term is kind of unfortunate;

*simplicial weak $\omega$-category*could also mean a simplicial object in weak $\omega$-categories. I don’t suppose we can do anything about that?Urs: my impression is that what Dominic Verity mainly wants to express with the term is “simplicial model for weak $\omega$-category”. Maybe we could/should use a longer phrase like that?

Mike: That would make me happier.

Urs: okay, I changed it. Let me know if this is good now.

*Toby*: But what about ’globular $\omega$-category’ and things like that? Doesn't ’simplicial $\omega$-category’ fit right into that framework? This page title sounds like an entire framework for defining $\omega$-category rather than a single $\omega$-category simplicially defined.Urs: i am open to suggestions – but notice that it does indeed seem to me that Dominic Verity wants to express “an entire framework for defining $\omega$-category”, namely the framework where one skips over the attempt to define $\omega$-categories and instead tries to find a characterization of what should be their nerves.

*Toby*: OK, that fits in with most of what's written here, but not the beginning**Simplicial models for weak $\omega$-categories**– sometimes called simplicial weak ∞-categories – are […] Maybe that was just poorly written, but it threw me off. Should it be A**simplicial model for weak $\omega$-categories**– which are then sometimes called simplicial weak ∞-categories – is […] or even A**simplicial model for weak $\omega$-categories**is […] and only later mention simplicial weak ∞-categories?Mike: You’re right that ’simplicial $\omega$-category’ it fits into ’globular $\omega$-category’ and ’opetopic $\omega$-category’ and so on. It seems more problematic in this case, though, since simplicial objects of random categories are a good deal more prevalent than globular ones and opetopic ones. But perhaps I should just live with it.

Urs: I have now expanded the entry text on this point, trying to make very clear to the reader what’s going on here.

*Toby*: Thanks, that's much clearer. And if Verity's definition is at weak complicial set, then we may not really need anything at simplicial weak ∞-category, so no need to offend Mike's sensibilities (^_^) either.

- Discussion Type
- discussion topicclass field theory
- Category Latest Changes
- Started by Anton Hilado
- Comments 2
- Last comment by Anton Hilado
- Last Active Jul 2nd 2022

- Discussion Type
- discussion topicprismatic cohomology
- Category Latest Changes
- Started by DavidRoberts
- Comments 25
- Last comment by nLab edit announcer
- Last Active Jul 2nd 2022

Stub to record today’s reference

- Bhargav Bhatt, Peter Scholze,
*Prisms and Prismatic Cohomology*, preprint (2019) arXiv:1905.08229

- Bhargav Bhatt, Peter Scholze,

- Discussion Type
- discussion topiclift
- Category Latest Changes
- Started by David_Corfield
- Comments 15
- Last comment by nLab edit announcer
- Last Active Jul 2nd 2022

Started lift.

weak factorization system has redirects from: lifting property, right lifting property, left lifting property, lifting problem, lifting problems.

Would it be better to have these redirect to lift?

- Discussion Type
- discussion topicbump function
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jul 2nd 2022

I have briefly created bump function in partiaL reply to this question

(I have otherwise replied to that question by email.)

- Discussion Type
- discussion topicIwasawa theory
- Category Latest Changes
- Started by Anton Hilado
- Comments 11
- Last comment by Urs
- Last Active Jul 2nd 2022

- Discussion Type
- discussion topicSiegel upper half-space
- Category Latest Changes
- Started by Anton Hilado
- Comments 2
- Last comment by Anton Hilado
- Last Active Jul 1st 2022

- Discussion Type
- discussion topicSiegel modular form
- Category Latest Changes
- Started by Anton Hilado
- Comments 1
- Last comment by Anton Hilado
- Last Active Jul 1st 2022

- Discussion Type
- discussion topicKapustin-Witten TQFT
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Anton Hilado
- Last Active Jul 1st 2022

stub for Kapustin-Witten TQFT in order to supply some links for this TP.SE question

- Discussion Type
- discussion topiccondensed set
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 66
- Last comment by David_Corfield
- Last Active Jul 1st 2022

- Discussion Type
- discussion topicstring structure
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by DavidRoberts
- Last Active Jun 30th 2022

touched string structure. Added some formal discussion, also polsihed layout and added references. But didn’t change the previous informal discussion.

- Discussion Type
- discussion topicHeinrich Tietze
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 30th 2022

- Discussion Type
- discussion topicTietze extension theorem
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Jun 30th 2022

created Tietze extension theorem

- Discussion Type
- discussion topiclist of theses in category theory
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active Jun 30th 2022

- Discussion Type
- discussion topicinteractions of images and pre-images with unions and intersections
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jun 30th 2022

At

*interactions of images and pre-images with unions and intersections*I have added pointer to Lawvere 69 and there at*Adjointness in Foundations*I added a bit more text and cross-references.

- Discussion Type
- discussion topicopen map
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 30th 2022

- Discussion Type
- discussion topicfan theorem
- Category Latest Changes
- Started by MarkSaving
- Comments 4
- Last comment by MarkSaving
- Last Active Jun 29th 2022

Change 1: Original page describes the fan theorem as requiring the bar to be decidable, claims that the “classical” fan theorem contradicts Brouwer’s continuity principle. The latter claim is not true; I corrected the error. I have stated the result as two separate theorems: the decidable fan theorem, about decidable bars, and the fan theorem, about bars in general.

Change 2: Slightly more information is provided about the relationship between the Fan Theorem and Bar Induction. Eventually, we should make a page about the latter.

Change 3: the section on equivalents to the fan theorem has been fixed somewhat. The section originally asserted that all of the statements provided were equivalent to the decidable fan theorem; in fact, some are equivalent to the decidable fan theorem and some to the full fan theorem.

- Discussion Type
- discussion topicclosed cover
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Jun 29th 2022

have started

*closed cover*, for the moment mainly in order to record references.

- Discussion Type
- discussion topicfinal functor
- Category Latest Changes
- Started by Tom Hirschowitz
- Comments 13
- Last comment by Hurkyl
- Last Active Jun 28th 2022

- Discussion Type
- discussion topic(n-connected, n-truncated) factorization system
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 27th 2022

- Discussion Type
- discussion topicidempotent monad
- Category Latest Changes
- Started by Todd_Trimble
- Comments 6
- Last comment by varkor
- Last Active Jun 27th 2022

I added more to idempotent monad, in particular fixing a mistake that had been on there a long time (on the associated idempotent monad). I had wanted to give an example that addresses Mike’s query box at the bottom, but before going further, I wanted to track down the reference of Joyal-Tierney, or perhaps have someone like Zoran fill in some material on classical descent theory for commutative algebras (he wrote an MO answer about this once) to illustrate the associated idempotent monad.

Some of this (condition 2 in the proposition in the section on algebras) was written as a preparatory step for a to-be-written nLab article on Day’s reflection theorem for symmetric monoidal closed categories, which came up in email with Harry and Ross Street.

- Discussion Type
- discussion topiccd-structure
- Category Latest Changes
- Started by adeelkh
- Comments 9
- Last comment by DavidRoberts
- Last Active Jun 27th 2022

Started the page cd-structure.

- Discussion Type
- discussion topicheterotic line bundle
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active Jun 27th 2022

- Discussion Type
- discussion topicmodel structure for orthogonal spectra
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jun 27th 2022

I am working on an entry

*model structure on orthogonal spectra*. So far it contains a detailed construction of the symmetric monoidal strict model structure, and then a detailed proof of the stable model structure.(I think its complete, but towards the end the expositional aspects need more polishing, i.e. more cross-links ect. But not today.)

This follows the writeup that I had started at

*Model categories of diagram spectra*, but (besides being more complete and more polished by now) it works around the issue that I ran into there, by defining the weak equivalences to be the stable weak homotopy equivalences ($\pi_\ast$-isos) right away. This means that the proof still verbatim gives a proof also of the stable model structure on sequential sequential spectra and on excisive functors, but not on symmetric spectra.

- Discussion Type
- discussion topicPeano curve
- Category Latest Changes
- Started by Todd_Trimble
- Comments 6
- Last comment by Urs
- Last Active Jun 26th 2022

The usual notion of Peano curve involves continuous images of the unit interval, not the whole real line (which could be considered as well, of course).

So I made some adjustments and stated some relevant facts at Peano curve, with a few pointers to proofs and to literature.

- Discussion Type
- discussion topichomotopy equalizer
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Urs
- Last Active Jun 26th 2022

- Discussion Type
- discussion topiccompletely decomposable topology > history
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 26th 2022

- Discussion Type
- discussion topiccovering
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Jun 26th 2022

- Discussion Type
- discussion topicNarciso Román-Roy
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Jun 26th 2022

- Discussion Type
- discussion topicGuiseppe Peano > history
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 25th 2022

am clearing this old misspelled entry in favor of the duplicate but correctly spelled

*Giuseppe Peano*

- Discussion Type
- discussion topick-tuply groupal n-groupoid
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jun 25th 2022

The reference provided today on the CatTheory mailing list

- A.R. Garzón, J.G. Miranda,
*Serre homotopy theory in subcategories of simplicial groups*Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123

I have added to k-tuply groupal n-groupoid, and also to n-group and infinity-group

- A.R. Garzón, J.G. Miranda,

- Discussion Type
- discussion topicpointed homotopy type
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 25th 2022

Hi Dmitri, I see you just made “pointed homotopy class” redirect here. But should it not redirect to “homotopy class”, since it’s about maps, not about spaces? I’ll add a line there to highlight the pointed case.

- Discussion Type
- discussion topicBartel Leendert van der Waerden
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 25th 2022

Bartel Leendert van der Waerden was a mathematician at the universities of Groningen, Leipzig, and Zürich.

He got his PhD degree in 1926 at the University of Amsterdam, advised by Hendrick de Vries.

## Selected works

Moderne Algebra. Teil I, Die Grundlehren der mathematischen Wissenschaften, vol. 33, Berlin, New York: Springer-Verlag, 1930, ISBN 978-3-540-56799-8

Moderne Algebra. Teil II, Die Grundlehren der mathematischen Wissenschaften, vol. 34, Springer-Verlag, 1931, ISBN 978-3-540-56801-8

Based on lectures by Emil Artin and Emmy Noether, it was the first textbook using the modern approach to algebra.

- Discussion Type
- discussion topicModerne Algebra
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 25th 2022

Created:

A textbook by Bartel Leendert van der Waerden.

Moderne Algebra. Teil I, Die Grundlehren der mathematischen Wissenschaften, vol. 33, Berlin, New York: Springer-Verlag, 1930, ISBN 978-3-540-56799-8

Moderne Algebra. Teil II, Die Grundlehren der mathematischen Wissenschaften, vol. 34, Springer-Verlag, 1931, ISBN 978-3-540-56801-8

Based on lectures by Emil Artin and Emmy Noether, it was the first textbook using the modern approach to abstract algebra.

- Discussion Type
- discussion topicvector space
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Dmitri Pavlov
- Last Active Jun 25th 2022

added pointer to:

- Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Chapter 8 of:
*Symmetry*(2021) $[$pdf$]$

- Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Chapter 8 of:

- Discussion Type
- discussion topicGiuseppe Peano
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 25th 2022

- Discussion Type
- discussion topiccategorical model of dependent types
- Category Latest Changes
- Started by Mike Shulman
- Comments 44
- Last comment by Urs
- Last Active Jun 25th 2022

Created categorical model of dependent types, describing the various different ways to strictify category theory to match type theory and their interrelatedness. I wasn’t sure what to name this page — or even whether it should be part of some other page — but I like having all these closely related structures described in the same place.

- Discussion Type
- discussion topicLucas Williams
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 25th 2022

- Discussion Type
- discussion topicconfiguration space of points
- Category Latest Changes
- Started by Tim_Porter
- Comments 91
- Last comment by Urs
- Last Active Jun 25th 2022

I fixed a link to a pdf file that was giving a general page, and not the file!

- Discussion Type
- discussion topicgroup
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Jun 25th 2022

reformatted the entry group a little, expanded the Examples-section a little and then pasted in the group-related “counterexamples” from counterexamples in algebra. Mainly to indicate how I think this latter entry should eventually be used to improve the entries that it refers to.

- Discussion Type
- discussion topicfinite group
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Jun 25th 2022

had added to

*finite group*two classical references, Atiyah on group cohomology of finite groups, and Milnor on free actions of finite groups on $n$-spheres.What I’d really like to know eventually is the degree-3 group cohomology with coefficients in $U(1)$ for the finite subgroups of $SO(3)$.

- Discussion Type
- discussion topicfinitely generated group
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 25th 2022

some minimum, to connect to

*Cayley graph*and*geometric group theory*

- Discussion Type
- discussion topiccovering space
- Category Latest Changes
- Started by Urs
- Comments 67
- Last comment by Urs
- Last Active Jun 25th 2022

I added to covering space a section In terms of homotopy fibers that explains the universal covering space as the homotopy fiber/principal oo-bundle classified by the cocycle that is the constant path inclusion of topological groupoids.

To fit this into the entry, I added some new sections and restructured slightly. Todd and David should please have a look.

What I just added is essentially what David Roberts says in various query boxes, notably in what is currently the last query box. Back then we talked about the "Roberts-Schreiber construction" or whatnot, but really what this is is just the standard way to compute homotopy fibers in the oo-category of oo-groupoids.

I suspect that Todd's bar construction described there can similarly be understood as being nothing but another way to compute the more abstractly defined homotopy pullback in concrete terms. I'll have to think about this, though. But probably Tim Porter or Mike Shulman will immediately recognize this as the relevant bar construction of homotopy pullbacks in homotopy coherent category theory.

- Discussion Type
- discussion topicOnsager–Machlup function
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 25th 2022

An editor signing as “Anonymous” had written material on the

*Onsager-Machlup function*into the*Sandbox*(rev 2542) and then*renamed the Sandbox*to “Onsager-Machlup”.(This way we discover that the Sandbox can be renamed, and that renaming it does not produce an edit announcement on the nForum – both of which shouldn’t be the case. I am talking with the technical team about it.)

I have re-named back to

*Sandbox*and put square brackets on the words “Onsager-Machlup function”. If you are Anonymous and want to create an entry with this title, please click on that grayish link that you now see in the*Sandbox*and proceed.

- Discussion Type
- discussion topicgeometric quantization
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Urs
- Last Active Jun 25th 2022

started an Examples-section at geometric quantization

- Discussion Type
- discussion topicGauss-Manin connection
- Category Latest Changes
- Started by zskoda
- Comments 7
- Last comment by Urs
- Last Active Jun 24th 2022

New stub, Gauss-Manin connection.

- Discussion Type
- discussion topicinfinity-group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 5
- Last comment by Urs
- Last Active Jun 24th 2022

- Discussion Type
- discussion topictensor product of Banach spaces
- Category Latest Changes
- Started by DavidRoberts
- Comments 9
- Last comment by Tom Mainiero
- Last Active Jun 24th 2022

Added links to related tensor products, including the new page inductive tensor product (under construction)

- Discussion Type
- discussion topicDavid L. Rector
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Jun 24th 2022

Added:

David Lee Rector is a mathematician at UC Irvine.

He got his PhD degree in 1966 from MIT, advised by Daniel M. Kan.

## Related entries

- Discussion Type
- discussion topicEilenberg-Mac Lane space
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Jun 24th 2022

Added doi and pointer to relevant sections to

Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 6 of

*Algebraic topology from a homotopical viewpoint*, Springer (2002) (toc pdf, doi:10.1007/b97586)(EM-spaces are constructed in section 6, the cohomology theory they represent is discussed in section 7.1, and its equivalence to singular cohomology is Corollary 12.1.20)

- Discussion Type
- discussion topicCharles Rezk
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 24th 2022

Charles Waldo Rezk is a mathematician at the University of Illinois Urbana–Champaign.

He got his PhD degree in 1996 at MIT, advised by Michael J. Hopkins.

His PhD students include Nathaniel Stapleton and Nima Rasekh.