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

• I added a clearer “The idea” section for Adams operation, and changed the word “functorial” to “natural” in a number of places, because while various sources do say the Adams operations are functorial, they must really be natural transformations from the functor $K: Top \to AbGp$ to itself.

• Aleks Kissinger has contacted me about his aims to start a collection of nLab entries on quantum information from the point of view of the Bob Coecke school.

Being very much delighted about this offer, I created a template entry quantum information for his convenience.

• In response to an email sent to Urs by Andrew Marshall, a slight amendment to good open cover was made in the proof that paracompact manifolds admit good open covers.

• Well I made a start, basing the entry on Urs’s FOM comment.

• Created, with so far just an overview of all the possibilities.

• felt the desire to have an entry on the general idea (if any) of synthetic mathematics, cross-linking with the relevant examples-entries.

This has much room for being further expanded, of course.

• created at internal logic an Examples-subsection and spelled out at Internal logic in Set how by turning the abstract-nonsense crank on the topos Set, one does reproduce the standard logic.

• Some important references added, including recent geometric study

• Leonid O. Chekhov, Marta Mazzocco, Vladimir N. Rubtsov, Painlevé monodromy manifolds, decorated character varieties, and cluster algebras, IMRN 24 (2017) 7639–7691 doi

and a section of papers studying the mathematics of Painlevé equations and of Fuchsian $j_\Gamma$-functions from model theoretic perspective (strongly minimal structures), including a new paper in Annals 2020.

• adjusted the formatting of the formulas a little, for readability. While I was at it, I added a one-sentence Idea-section, for completeness.

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• Have added to HowTo a description for how to label equations

In the course of this I restructured the section “How to make links to subsections of a page” by giving it a few descriptively-titled subsections.

• I corrected a couple og microscopic typos at k-ary factorization system, and then I noticed that something is unclear in the definition: first of all the family of factorization system is asked to be strong (= uniqueness of solution to any lifting problem) or weak (existence, no uniqueness)? And when the definition says

$M_1 \subseteq \dots \subseteq M_{\kappa-1}$ whenever this is meaningful (equivalently, $E_{k-1} \subseteq\dots\subseteq E_{1}$)

what does it precisely mean? Are we asking that right classes be nested?

Thirdly, it is my humble opinion that saying

A discrete category has a (necessarily unique) $(-1)$-ary factorisation system.

is formally incorrect: discrete categories are groupoids where the only arrows are identities, so this is a particular kind of 0-ary factorization system.

Instead, negative thinking suggests that (-1)-ary factorization systems live in non-unital categories, and detect precisely the case where the class of isomorphisms is empty (recall that in a WFS $(L,R)$ the intersection $L\cap R$ consists of all isomorphisms; if in a 0-ary factorization system we had $L=R=L\cap R=Iso(\mathbf C)$, morally in a (-1)-ary system the intersection has to be empty, giving a category without identities -i.e. a particular kind of “plot”, in the jargon of this paper which I finally convinced my friend Salvatore to put on the arXiv-, and more precisely an associative, “strongly nonunital” plot).

This leads to another question: how can be the notion of (W)FS be extended to Mitchell’s semicategories (with empty or partially defined identity function)?

• 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?

• These two related principals are not the only axioms with this name. A more set-theoretic “axiom of multiple choice” also exists (which, as far as I can tell, is unrelated to these axioms). I added the set theoretic axiom and gave some references where it is used this way.

• I am starting something at six operations.

(Do we already have an nLab page on this? I seemed to remember something, but can’t find it.)

• I added a bit to the section on the ultrafilter monad in ultrafilter. This could stand to be fleshed out still more. The immediate reason for my editing here was to put down the notion of “compact Hausdorff object” (which is used in a remark at BoolAlg).

• Reorganized and added to the list of types of numbers.

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