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Anonymous

• I gave regular cardinal its own page.

Because I am envisioning readers who know the basic concept of a cardinal, but might forget what “regular” means when they learn, say, about locally representable category. Formerly the Lab would just have pointed them to a long entry cardinal on cardinals in general, where the one-line definition they would be looking for was hidden somewhere. Now instead the link goes to a page where the definition is the first sentence.

Looks better to me, but let me know what you think.

• I reorganized linearly distributive category by moving the long block of history down to the bottom, adding an “Idea” section and a description of how $*$-autonomous categories give rise to linearly distributive ones and linearly distributive ones give rise to polycategories. I also cross-linked the page better with polycategory and star-autonomous category.

• I have added pointer to

Their Prop. 7.2.2 is verbatim the characterization that BCMMS made the definition of “bundle gerbe module” a month and a half later (except that LU focus on open covers instead of more general surjective submersions, but that’s not an actual restriction and in any case not the core of the definition).

which essentially recovers Lupercio & Uribe’s Def. 7.2.1.

From the arXiv timestamps I gather that it must have been an intense couple of weeks for all these auhtors in spring 2001. But Lupercio & Uribe came out first, by a fair margin. And in equivariant generality, right away…

• starting some minimum

• I have been working on the entry twisted bundle.

Apart from more literature, etc. I have started typing something like a first-principles discussion: first a general abstract definition from twisted cohomology in any cohesive $\infty$-topos, then unwinding this in special cases to obtain the traditional cocycle formulas found in the literature.

Needs more polishing here and there, but I have to pause now.

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• starting something, on the kind of theorems originating with

Nothing to be seen here yet, but I need to save. (Am not sold on the entry title, except that “topology” is not really the right term here.)

• Albrecht Bertram, Stable Maps and Gromov-Witten Invariants, School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 (pdf)
• 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.

• Several recent updates to literature at philosophy, the latest being

• Mikhail Gromov, Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2., pdf; Structures, Learning and Ergosystems: Chapters 1-4, 6 (2011) pdf

which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…

• Have added a bunch of references to this entry.

Question: What precisely can one say about the relation between the topological space underlying the Hilbert scheme of points of $\mathbb{C}$ and/or $\mathbb{C}^2$, and the Fulton-MacPherson compactification of the corresponding configuration spaces of points?

There is commentary in just this direction on p. 189 of:

but it remains unclear to me what exactly the statement is, in the end.

• added to Eckmann-Hilton argument the formal proposition formulated in any 2-category.

BTW, doesn’t anyone have a gif with the nice picture proof?

• A stub.

• Just heard a nice talk by Simon Henry about measure theory set up in Boolean topos theory (his main result is to identify Tomita-Takesaki-Connes’ canonical outer automorphisms on $W^\ast$-algebras in the topos language really nicely…).

I have to rush to the dinner now. But to remind myself, I have added cross-links between Boolean topos and measurable space and for the moment pointed to

• Matthew Jackson, A sheaf-theoretic approach to measure theory, 2006 (pdf)

for more. Simon Henry’s thesis will be out soon.

Have to rush now…

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• Correct the characterization of nerves of groupoids.

• Added a section on terminology.

• I added two characterisations of weak homotopy equivalences to model structure on simplicial sets.

For the record, I found the inductive characterisation in Cisinski’s book [Les préfaisceaux comme modèles des types d’homotopie, Corollaire 2.1.20], but I feel like I’ve seen something like it elsewhere. The characterisation in terms of internal homs comes from Joyal and Tierney [Notes on simplicial homotopy theory], but they take it as a definition.

• I’ve added to Eilenberg-Moore category an explicit definition of EM objects in a 2-category and some other universal properties of EM categories, including Linton’s construction of the EM category as a subcategory of the presheaves on the Kleisli category.

Question: can anyone tell me what Street–Walters mean when they say that this construction (and their generalised one, in a 2-category with a Yoneda structure) exhibits the EM category as the ‘category of sheaves for a certain generalised topology on’ the Kleisli category?

• Added stub with definition and example.

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• a category:reference-entry for the upcoming book

• Created a stub with a definition and an example.

• I added to biadjunction the statement and some references for the fact that any incoherent one can be improved to a coherent one.

• This is a brief description of the construction that started appearing in category-theoretic accounts of deep learning and game theory. It appeared first in Backprop As Functor (https://arxiv.org/abs/1711.10455) in a specialised form, but has slowly been generalised and became a cornerstone of approaches unifying deep learning and game theory (Towards Foundations of categorical Cybernetics, https://arxiv.org/abs/2105.06332), (Categorical Foundations of Gradient-based Learning, https://arxiv.org/abs/2103.01931).

Our group here in Glasgow is using this quite heavily, so since I couldn’t find any related constructions on the nLab I decided to add it. This is also my first submission. I’ve read the “HowTo” page, followed the instructions, and I hope everything looks okay.

There’s quite a few interesting properties of Para, and eventually I hope to add them (most notably, it’s an Para is an oplax colimit of a functor BM -> Cat, where B is the delooping of a monoidal category M).

A notable thing to mention is that I’ve added some animated GIF’s of this construction. Animating categorical concepts is something I’ve been using as a pedagogical tool quite a bit (more here https://www.brunogavranovic.com/posts/2021-03-03-Towards-Categorical-Foundations-Of-Neural-Networks.html) and it seems to be a useful tool getting the idea across with less friction. If it renders well (it seems to) and is okay with you, I might add more to the Optics section, and to the neural networks section (I’m hoping to get some time to add our results there).

Bruno Gavranović

• For a change, I added some actual text to this category:people-entry, highlighting a little the content and relevance of (parts of) the research.

• Discussion of the formulas for the standard characteristic forms has been missing in various entries (e.g. at Chern class at characteristic form, etc.). Since there is little point in discussing the Chern forms independently from the Pontrjagin forms etc. I am now making it a stand-alone section to be !include-ed into relevant entries, to have it all in one place.

Not done yet, though, but it’s a start.