In articles by Balmer I see “tensor monoidal category” to be explained as a triangulated category equipped with a symmetric monoidal structure such that tensor product with any object “is an exact functor”, but I don’t see where he is specific about what “exact functor” is meant to mean. Maybe I am just not looking in the right article.

Clearly one wants it to mean “preserving exact triangles” in some evident sense. One place where this is made precise is in def. A.2.1 (p.106) of Hovey-Palmieri-Strickland’s “Axiomatic stable homotopy theory” (pdf).

However, these authors do not use the terminology “tensor triangulated” but say “symmetric monoidal compatible with the triangulation”. On the other hand, Balmer cites them as a reference for “tensor triangulated categories” (e.g. page 2 of his “The spectrum of prime ideals in tensor triangulated categories” ).

My question is: may I assume that “tensor triangulated category” is used synonymously with Hovey-Palmieri-Strickland’s “symmetric monoidal comaptible with the triangulation”?

]]>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)

starting something, on

- Laura Scull,
*A model category structure for equivariant algebraic models*, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)

some minimum, for completeness of the list at *D4*

added

more hyperlinks (and some whitespace) to the paragraph on maximal tori.

the statement that smooth actions of compact Lie groups on smooth manifolds are proper

added to *orbit category* a remark on what the name refers to (since I saw sonebody wondering about that)

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

-declarations

I would like to suggest that the page of page-categories of the nLab https://ncatlab.org/nlab/page_categories should have a more strict structure.

I wanted to propose something along the lines of, each page_category should be either:

1. a mathematical object (all the theorems, definitions and constructions go there)

2. a person or people (some 3163 pages--this is mostly ok already),

3. books and papers, (have 3 overlapping pages, it seems: https://ncatlab.org/nlab/all_pages/Paper%20References, https://ncatlab.org/nlab/all_pages/reference and https://ncatlab.org/nlab/all_pages/references) - some 200 pages, perhaps?

4. fields of mathematics (Wikipedia has 72 of these) AND

5. MISC, for whatever else people wanted to have, like jokes, or things difficult to classify.

This would help when relating the nLab to WikiData using the property https://www.wikidata.org/wiki/Property:P4215

Does this sound sensible to you? There are only 60 page_categories, so these we could do by hand very easily, if you think this is a good idea.

Maybe this has been discussed in the nForum before, maybe it was discarded because of traditional rules that I don't know about. If so, I would like to know the rationale. Thanks! ]]>

Under definition 1 of salamander lemma, I fixed a mistake in the definition of $A_\Box$ where there was a direct sum of two submodules, where there needed to be a sum (i.e., join) instead.

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

Anonymous

]]>In the section “In terms of truncations” I have added a few more cross links (both between the definitions in that section as well as to the respective items in HTT).

]]>brief `category: people`

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

created *equivariant de Rham cohomology* with a brief note on the Cartan model.

(I seem to remember that we had discussion of this in the general context of Lie algebroids elsewhere already, several years back. But now I cannot find it….)

]]>added a brief historical comment to *Higgs field* and added the historical references

Added a page about the category FinRel of finite sets and relations, and some of its properties.

]]>Added to bimonoid the fact that the category of modules over a bimonoid is monoidal.

]]>started universal coefficient theorem

]]>added a cool reference by Brian Conrad to [[cohomology]], which was mentioned at MathOverflow

]]>am starting some minimum here. Have been trying to read up on this topic. This will likely become huge towards beginning of next year

]]>I have briefly recorded the equivalence of FinSet${}^{op}$ with finite Booplean algebras at *FinSet – Properties – Opposite category*. Then I linked to this from various related entries, such as *finite set*, *power set*, *Stone duality*, *opposite category*.

(I thought we long had that information on the $n$Lab, but it seems we didn’t)

]]>some minimum

]]>Came across this categorification of pro-object.

]]>Redirect for bicategorical localization

]]>started an entry on the *Borel construction*, indicating its relation to the nerve of the action groupoid.