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Thanks to a Guest comment here I looked at this page for essentially the first time, and realized that it’s one more thing that’s naturally done in linear constructive mathematics (“complemented subsets” or “disjoint pairs” are the elements of the linear powerset).
One question: this page says that the disjoint pairs form a “Boolean rig”, but that doesn’t seem right to me. A Boolean rig would, I presume, lack a negation operation entirely; but here we do have an involutive “negation” even though it’s not the “additive inverse”. I would say that the disjoint pairs form a De Morgan algebra, and in fact more generally a *-autonomous lattice. Am I misinterpreting the intended meaning of “Boolean rig”?
Also, what is the “Handbook of Constructive Analysis” referred to (as a graylink from Bishop \& Bridges)? I can’t find it on google.
I created this article so that links from Cheng space point to an actual article on the nLab.
The entry monomorphism used to start off saying that a monomorphism is an epimorphism in the opposite category…
I have polished and expanded the text now, trying to make it look more like an actual exposition and explanation. I have also expanded a little the Examples-section, and similarly at epimorphism.
These weird kind of entries date from the early days of the nLab, when none of us saw yet what the nLab would once be. Back then it was fun to proceed this way, now it feels awkward.
I hereby pose a challenge to the nForum community:
I challenge you to each pick one entry on a basic topic (nothing fancy), go to the corresponding nLab entry and give it a gentle introductory Idea-section, make sure that the basic motivating examples are mentioned in the order in which the newbie needs to see them, and that the key facts are stated as nicely discernible propositions, best with proof or at least with some helpful pointer, in short, to make the entry a useful read for those readers who would profit from reading it, especially those who do not know the nPOV yet, but might be guided to learn and appreciate it.
added this pointer:
Added a mentioning of the term logos at the beginning of Heyting category.
wrote an entry Deligne’s theorem on tensor categories on the statement that every regular tensor category is equivalent to representations of a supergroup. Added brief paragraphs pointing to this to superalgebra and supersymmetry, added cross-links to Tannaka duality, Doplicher-Roberts reconstruction etc. Also created a disambiguation page Deligne’s theorem
added pointer to today’s
a bare list of references, to be !include-ed into the References-sections of relevant entries (such as at anyons and at topological order)
The list means to bring out the wide-spread consideration, in theoretical articles, of anyons whose positions in real space vary on a torus (or even higher genus surfaces) instead of a plane – an assumption that is necessary for many of the intended theoretical conclusions to be valid, but rather dubious as an assumption about actual physical systems (away from simulation).
The preprint by Gaiotto & Johnson-Freyd at the end is one of the few places that I am aware of where this assumption is questioned, and I included a couple of paragraphs of quote.
(This all in preparation for an article pointing out that anyonic states can in principle be localized also in more abstract spaces than “position space”, some of which are naturally toroidal, such as the case of reciprocal momentum space for which I took the liberty of pointing to our existing 2206.13563.)
Needed to refer within the circle of pages in algebraic geometry
A notion in algebraic geometry.
I just noticed and noted that Gabriella Böhm wrote a book, on generalizations in Hopf world,
Added a reference to today’s
added pointer to:
Add reference to marked 2-limit.
I created hypermonoid, polishing the comments I made in the hypermonoid thread into an article. The last subsection of the article mentions a general technique for constructing hypermonoids which ought to immediately suggest further examples to a quantum group specialist like Zoran, but I am not such a specialist. I also inserted some shameless self-promotion under References.
Removed the sentence
“If X has decidable equality, then the negation of equality is a (in fact the unique) tight apartness on X, and any function from X to any set Y (with any tight apartness on Y) must be strongly extensional.”
because is not true. Assuming WLPO, Cantor space has decidable equality but the negation of equality is still not the tight apartness relation on Cantor space.
Anonymouse
Created doctrinal adjunction. The page could probably use some examples and/or fleshing out.
created directed homotopy type theory
added a Properties-section to pullback
Where does this concept come from? The page lists no relevant references, nor can I find any search results for “quadrable cospan”. Furthermore, the “Note on terminology” mentions the terminology “carrable”, which means something different as far as I can tell (and certainly in the cited references): namely, a morphism along which all pullbacks are admitted.
Fixed/clarified the notation in the definition of local objects in a model category. Added references.
Also added references to (infinity,1)-categorical hom-space in that context.
added pointer to Sorokin 01
There is a small error in the current proof that the category of endofunctors on a Q-category is a Q-category. I am going to correct it as soon as I find my way through the notation (I used it different on the paper). It now reads
The (CR⊣CL)-unit is the dual Cη of the original counit η
Cη:IdCA→CL∘CR=CLRand the counit is the dual of the original unit
Cε:CR∘CL=CRL→IdCˉA.
The wrong thing is that CL∘CR=CRL, not CLR and that is why the unit and counit got interchanged; they should not get interchanged, but CL and CR should. I am going to sort this out. Thus Cη where η is unit goes Cη:IdCA→CRL.
Edit: the correct version is now below.
Added to BF-theory the reference that right now I am believing is the earliest one:
Gary Horowitz, Exactly soluable diffeomorphism invariant theories Commun. Math. Phys. 125, 417-437 (1989)
But maybe I am wrong. Does anyone have an earlier one? I saw pointers to A. Schwarz articles from the late 70s, but I am not sure if he really considered BF as such.
I added a little bit to maximal ideal (first, a first-order definition good for commutative rings, and second a remark on the notion of scheme, adding to what Urs wrote about closed points).
The second bit is almost a question to myself: I don’t feel I really grok the notion of scheme (why it’s this and not something slightly different that’s the natural definition, the Tao if you like). In particular, it’s where fields – simple objects in the category of commutative rings – make their entrance in the notion of covering by affine opens that I don’t feel I really understand.
Added:
A survey of various notions between unital rings and nonunital rings:
changed higher algebra - contents to algebra - contents in context sidebar
Anonymouse
Trivial edit to start discussion.
What’s happening at the start here? We have both tropical rig and semiring defined. The latter is given with the extension by {∞}. Is this just duplication with a mistake?
At semiring having given 4 definitions, it says
The nLab uses the second definition to define a semiring, and the fourth definition to define a rig. The first and third are then called nonunital semirings and nonunital rigs respectively.
Do we really have this as a policy?