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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• Created new page.

• Added link to Galois deformation ring.

• Added statement of the Birch and Swinnerton-Dyer conjecture.

• Made mention of the Galois action on l-adic cohomology.

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

• 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.

• Added section on the Hilbert class field.

• 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?

• Adding “idea” section.

• Added definition.

• Added a reference. May edit more later.

• Page created, but author did not leave any comments.

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

• brief category:people-entry for hyperlinking references

• Create a new page to keep record of PhD theses in category theory (with links to the documents where possible), particularly older ones that are harder to discover independently. At the moment, this is just a stub, but I plan to fill it out more when I have the chance.

• added (here) the example of local homeomorphisms

• 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.

• have started closed cover, for the moment mainly in order to record references.

• Added the property that final functors and discrete fibrations form an orthogonal factorisation system.

• For completeness, I have expanded out (here) the argument of the corollary that was sitting here, claiming that $n$-images are preserved by $\infty$-pullback.

• 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.

• starting something

• 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.

• 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.

• Created.

• some minimum

• Mentioned Grothendieck topologies.

• I added a link to the personal webpage of the researcher. The personal webpage is updated (annualy?) by Narciso himself.

Jordi Gaset

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

• 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

• 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.

• 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.

• 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.

• brief category:people-entry for hyperlinking references

• 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.

• brief category:people-entry for hyperlinking references

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

• 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.

• 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 $SO(3)$.

• 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 $X \to \Pi_1(X)$ 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.

• 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.

• Added the adjective grouplike to A_infty space as else I just get a monoid object instead of a group object.

Anonymous

• 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

• 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)