added pointer to

- Paul Balmer,
*A guide to tensor-triangular classification*, in: Haynes Miller (ed.)*Handbook of Homotopy Theory*(arXiv:1912.08963)

added pointer to the recent

- Kent B. Vashaw,
*Balmer spectra and Drinfeld centers*(arXiv:2010.11287)

added pointer to

Paul Balmer,

*The spectrum of prime ideals in tensor triangulated categories*. J. Reine Angew. Math., 588:149–168, 2005 (arXiv:math/0409360)Paul Balmer,

*Spectra, spectra, spectra—tensor triangular spectra versus Zariski spectra of endomorphism rings*, Algebr. Geom. Topol., 10(3):1521–1563, 2010 (pdf)

(which have been listed at *Paul Balmer* all along, but were missing here, strangely)

and to the recent:

- Kent B. Vashaw,
*Balmer spectra and Drinfeld centers*(arXiv:2010.11287)

Let $G$ be a finite group acting smoothly on a smooth manifold $X$. Then the fixed loci $X^H$, for $H \subset G$ are smooth manifolds, and hence forming fixed-locus wise vector spaces of smooth differential $n$-forms (for any $n \in \mathbb{N}$) yields a functor

$\array{ G Orbits &\longrightarrow& VectorSpaces \\ G/H &\mapsto& \Omega^n_{dR}\big( X^H \big) }$**Question**: In which generality are these functors injective objects, and what’s the proof?

I can show this to be so for the case that $G$ is: of order 4 or cyclic of prime order.

From the proof of these special cases it is pretty clear how the general case will work, and I suppose I can prove any number of further special cases by a case-by-case analysis; but it remains unclear to me how to formulate the fully general proof.

A first inkling of how to approach this issue may be gleaned from the text offered as proof to Prop. 4.3 in Triantafillou 82 (where it’s PL dR forms instead of smooth forms, but the combinatorial part of the argument is the same), but I don’t see how these hints are more than that.

]]>added cross-link with *fundamental theorem of finitely generated abelian groups*

Ah, thanks.

]]>added graphics showing the orbit categories of the first five cyclic groups (here)

]]>This page had, besides its minimum content, somewhat weird formatting overhead. I have deleted that now, including the multiple `category:`

-declarations

When you spot a silly typo anywhere on the nLab , please fix it.

]]>Corrected the reference: it’s Corollary 21.6 in the actual second edition.

]]>Dealing with the empty/semi-deleted pages (484 of these semi-deleted is not a tiny number) seems an easy and sensible thing to do.

Having categories :people and :references also seems very reasonable.

But having categories: reference(https://ncatlab.org/nlab/all_pages/reference),

references (https://ncatlab.org/nlab/all_pages/references) and https://ncatlab.org/nlab/all_pages/Paper%20References seems silly. or rather an oversight.

On the other hand making a difference between "mathematical theories" and "mathematical objects" is one that's a bit more consequential. It helps to disambiguate between Topology (Capitalized, as the name of the theory) and any old topology on a set or a space. (of course proof theorists might want to say that mathematical theories are mathematical objects too, but the oversimplification that they are not seems worth doing). ]]>

Replace “Dually” by “Similarly” in the Salamander Lemma (this is not a “dual” statement in the category-theoretic sense, which context is appropriate here).

Mark S Davis

]]>Simply the definition, as found in “Combinatorics of coxeter groups” by Bjorner and Brenti.

Anonymous

]]>Included the description of n-truncated objects in the category of presheaves being the presheaves taking values in n-truncated spaces.

]]>brief `category: people`

-entry for hyperlinking references at *equivariant de Rham cohomology*

finally added publication data for:

- Jaap Kalkman,
*BRST Model for Equivariant Cohomology and Representatives for the Equivariant Thom Class*, Comm. Math. Phys. Volume 153, Number 3 (1993), 447-463. (euclid:1104252784)

Regarding our `> history`

suffixes:

The Instiki wiki software that the $n$Lab is running on does not offer a mechanism to users to delete pages. The closest there is to deleting a page is to clear its content and remove all links pointing to it.
But sometimes a page needs to be removed that (and because) it has a duplicate, or there may be some other reason that we must still retain links to a page with that same title, just not that particular page. In this case the quasi-deletion of the page must go along with its title being changed to *something* else. We adopted the habit of adding that `> history`

suffix (or at least some of us do. Maybe it was Toby’s original idea and then I followed it.)

Regarding the `categories`

-functionality:

Myself, I have been using the `categories`

-functionality in precisely two cases only: for `category: people`

and `category: reference`

. Others here have experimented with other uses, which is what you probably see, but none of that has become established usage, as far as I am aware and concerned.

Instead, the would-be role of the `categories`

-tag is played by the more sophisticated floating context menus, that I am putting in the top right of most entries (see the top-right of most any entry for an example, e.g. *topos*). If you want to do some kind of data-banking with “categories” of $n$Lab entries, then it’s these floating context-menus that you should go for.

Also, it seems there are 435 pages with “> history” at the end of their title, and having checked a few they largely seem to be blank pages last edited in 2009. For example,

- https://ncatlab.org/nlab/history/Bertrand+Toen+%3E+history
- https://ncatlab.org/nlab/show/A-infinity%20algebra%20%3E%20history
- https://ncatlab.org/nlab/show/A.%20J.%20Tolland%20%3E%20history

Is there a particular role that they play?

Brendan Fong

]]>added pointer to

- Mark P. Hertzberg, Mudit Jain,
*Explanation for why the Early Universe was Stable and Dominated by the Standard Model*, JCAP 2020 (arXiv:1911.04648)

Hi Richard,

Thanks so much for offering some time to work on this. If it’s not too late, something along the following lines would be helpful for us (and hopefully the whole community):

- Add an additional interface in the nLab somewhere to assign page-categories to meta-categories.
- Change the page_categories page to be organised by meta-category, with an additional “unassigned” section for page-categories that have not been assigned to a meta-category.
- Create the five meta-categories Theory, Mathematical Object, Reference, Person, and Miscellaneous, and assign/merge/rename/delete existing page-categories to these according to this spreadsheet.

Would this be possible? We’re flexible about the details, so please let us know if you have any suggestions or concerns. Please also let us know how we can help.

The main goal, from the point of view of using the nLab as a source for generating a formal ontology and doing text analysis, is to have tags that separate the pages that concern mathematics directly (eg. mathematical objects), those that concern cultural aspects of category theory (eg. people), and those that concern the functioning of the nLab itself (eg. meta, empty, or drafts)

Valeria can perhaps provide more detail, but as I understand it the suggested five meta-categories are motivated by simple examination of the ways the page-categories have been used. One perhaps undesirable property is that the “person” and “reference” meta-categories function like page-categories: it would make sense to assign pages directly to them. (Eg. the only page-category assigned to the “person” meta-category is itself called “person”, and it has more than 3000 entries.) On the other hand, the “theory” meta-category has about 25 constituent page-categories, each with ~5 to 50 entries.

Brendan Fong

]]>Delete extraneous “all” in last sentence of “far diagonal identifications” paragraph.

Mark S Davis

]]>$FinRel$ can’t be isomorphic to $Mat(Bool)$, since the latter is skeletal (and has countable many objects), but the former is not skeletal, and is usually large (eg in ZFC). So I’ve replaced the “isomorphic” by “equivalent”.

]]>made more of the keywords be hyperlinked;

made the pointer to the references be hyperlinked;

and did some minor polishing of the formatting,

such as extending hyperlinks over the full plural form of a term (which looks much better and works due to the magic of redirects),

and such as replacing

```
:
```

with

```
\colon
```

in math-mode (since otherwise it comes out ugly, for some reason)

]]>Added information about bimonoids in Rel and FinRel.

]]>