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

• added the remark (here) that acyclic Kan fibrations are (degreewise) surjective

• Giving this its own entry (split off from cosmos) for ease of hyperlinking to this particular notion (as announced in the thread there).

It’s essentially just material copied over for now, but wording adjusted for flow and clarity (I hope) but I have added a more pertinent Idea-section and started a section on Examples (just a beginning, so far).

• a stub, trying to bring in infrastructure for discussion of the finite subgroups of $O(5)$

• collecting some references on the argument for

• exactly 24 D7-branes in F-theory on K3

• exactly 24 NS5-branes in HET-theory on K3

(this used to be a remark at F-theory, now copied over and expanded here)

and taking the liberty of adding a comment relating this, via “Hypothesis H”, to the third stable homotopy group of spheres seen under Pontryagin’s theorem.

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

Antonin Delpeuch

• now I have finally the time to come back to this, as announced, and so I am now starting an entry:

So far there is just some literature collected. I now plan to extract the essence of Seely’s artice into the entry in some technical detail.

• Since the page geometry of physics – categories and toposes did not save anymore, due to rendering timeouts caused by its size, I have to decompose it, hereby, into sub-pages that are saved and then re-!included separately.

With our new announcement system this means, for better or worse, that I will now have to “announce” these subsections separately. Please bear with me.

• Added a little content here, having given another talk this week on Friedman, slides.

• I was looking to correct “Issac Newton” in the first line, but it send me an error about the table of contents. I don’t know how to fix that.

• I added a short abstract description of the Cartesian fibration over the interval, and commented that the section describes a construction with additional strictness properties in the quasi-category model.

• following public demand, I created an entry titled

(If you don’t like this title, please have a look first to see if it makes sense afterwards. If still not, I won’t be dogamitic about this and am open for suggestions for other titles.)

• For completeness I have added pointer to

though there should really be some accompanying discussion of how this form of the statement is related to the usual one in terms of presheaves.

• I worked on synthetic differential geometry:

I rearranged slightly and then expanded the "Idea" section, trying to give a more comprehensive discussion and more links to related entries. Also added more (and briefly commented) references. Much more about references can probably be said, I have only a vague idea of the "prehistory" of the subject, before it became enshrined in the textbooks by Kock, Lavendhomme and Moerdijk-Reyes.

Also, does anyone have an electronic copy of that famous 1967 lecture by Lawvere on "categorical dynamics"? It would be nice to have an entry on that, as it seems to be a most visionary and influential text. If I understand right it gave birth to topos theory, to synthetic differential geometry and all that just as a spin-off of a more ambitious program to formalize physics. If I am not mistaken, we are currently at a point where finally also that last bit is finding a full implmenetation as a research program.

• I have added to monoidal model category statement and proof (here) of the basic statement:

Let $(\mathcal{C}, \otimes)$ be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$. If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then 2) the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor

$\gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.$

The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

• fixed the statement of Example 5.2 (this example) by restricting it to $\mathcal{C} = sSet$

• 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.

• Merry Christmas Wishes
https://www.wishesonoccasion.com/merry-christmas-wishes-quotes/
<a href="https://www.wishesonoccasion.com/merry-christmas-wishes-quotes/">Merry Christmas Wishes</a> it is a time of year when some feel sickened, or very lonely.
So, find a nice Christmas quote to distribute some joy and humor around you because that’s Christmas too!

• created Tannaka duality

with a short proof of the duality for the category of permutation representations of a group, using the Yoneda lemma three or four times in a row and nothing else.

either I am mixed up (in which case we'll roll back), or I guess this is the way that it's usually done in the literature? I haven't really checked. Sorry, I just needed that quickly as a lemma for my discussion at homotopy group of an infinity-stack

• Added reference to “infinity-categories for the working mathematician”, the book in progress by R-V.

• a stub entry, for the moment just to satisfy requested links at meros

• brief category:people-entry for satisfying a requested hyperlink at meros.

• Explained at mapping cone how the mapping cone is model for a homotopy cofiber. In fact I used that to define and motivate the mapping cone.

Then I moved the example in Top to the top of the list, as that is the archetypical example.

• 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?

• I am wondering about the following:

Let $Singularities$ denote the global orbit category of finite groups, i.e. simply the full sub-$2$-category of all $\infty$-groupoids on those of the form $\ast \!\sslash\! G$ for $G$ a finite group.

Regarded as an $\infty$-site with trivial coverage, this is a cohesive $\infty$-site. Therefore, given any $\infty$-topos $\mathbf{H}_{\subset}$ we obtain a new $\infty$-topos

$\mathbf{H} \;\coloneqq\; PSh_\infty(Singularities, \mathbf{H}_{\subset})$

which has the following properties:

1. for each finite group $G$ there is the usual $\ast \!\sslash\! G \in \mathbf{H}$, but in addition there is an object to be denoted $\prec^G \in \mathbf{H}$ – to be thought of as the the “generic $G$-orbi-singularity”

(namely that arising as the image of the corresponding object in $Singularities$ under the Yoneda-embedding and passing along the inverse terminal geometric morphism of $\mathbf{H}_{\subset}$ )

2. it carries an adjoint triple of modalities

$\lt \;\;\dashv\;\; \subset \;\;\dashv\;\; \prec$

$singular \dashv smooth \dashv orbisingular$
3. such that (at least when $\mathbf{H}_{\subset}$ is itself cohesive):

1. $\lt(\prec^G) \simeq \ast$

(“the purely singular aspect of an orbi-singularity is a plain quotient of a point, hence a point”)

2. $\subset(\prec^G) \simeq \ast \!\sslash\! G$

(“the purely smooth aspect of an orbi-singularity is a homotopy quotient of a point)

3. $\prec(\prec^G) \simeq \prec^G$

(“an orbi-singularity is purely orbi-singular”)

$\,$

I am wondering about the converse:

Suppose an $\infty$-topos $\mathbf{H}$ is such that these three conditions hold (the first one without its parenthetical remark).

Can we conclude that $\mathbf{H}$ is of the form $PSh_\infty(Singularities, \mathbf{H}_{\subset})$?

If not, which axioms could be added to make it work?

• A stub for M-theory. What’s supposed to be so mysterious about it? Is it that people don’t even know what form it would take?

• added to the people-entry Edward Witten a paragraph Fields medal work with a commented list of articles that according to Atiyah won Witten the Fields medal in 1990.

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

• A stub. Hopefully those more knowledgeable will add some content.

• I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.

• just a stub for the moment, in order to make links work

• starting something, not done yet but need to save

• Jim Gates, Yangrui Hu, S.-N. Hazel Mak, Adinkra Foundation of Component Decomposition and the Scan for Superconformal Multiplets in 11D, $\mathcal{N} = 1$ Superspace (arXiv:2002.08502)

• brief category:people-entry for hyperlinking references

• I have spelled out an argument (here) for the statement that in a local topos the full subcategory of concrete objects provides a factorization of $\Gamma \dashv coDisc$ as

$\Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AAA}}{\hookleftarrow} } Set$
• Seeing that “completion of a space” was redirecting both for

I am making a little disambiguation page here, to disentangle the links.

But no text here yet. And we are lacking entries on adic completion of spaces anyways.

• Added a section on terminology.

• Added a stub of definition, together with a reference to an ’easy’ characeterization of arrows

• added to shape theory a section on how strong shape equivalence of paracompact spaces is detected by oo-stacks on these spaces

By the way: I have a question on the secion titled "Abstract shape theory". I can't understand the first sentence there. It looks like this might have been broken in some editing process. Can anyone fix this paragraph and maybe expand on it?

• added pointer to conference this month: here

• stub entry, for the moment just to satisfy links

• added various references, notably on computation of graviton scattering amplitudes.

• Thought I’d begin this stub.

• To fix a grey link.

• As I’ve already said elsewhere, I’ve been working on this entry and trying to give a precise definition based on my hunches of what guys like Steenrod really meant by “a convenient category of topological spaces”. (I must immediately admit that I’ve never read his paper with that title. Of course, he meant specifically compactly generated Hausdorff spaces, but nowadays I think we can argue more generally.)

I also said elsewhere that my proposed axiom on closed and open subspaces might be up for discussion. The other axioms maybe not so much: dropping any of them would seem to be a deal-breaker for what an algebraic topologist might consider “convenient”. Or so I think.

• Updated Simona Paoli’s page.

• brief category:people-entry for hyperlinking references