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

• I added to star-autonomous category a mention of “$\ast$-autonomous functors”.

• 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.

• Made a start at coordination. I’m unsure whether it’s worth spending too long on the intricate accounts of Schlick and Reichenbach, and then of whose makes best sense of Einstein’s proposals. Then there’s plenty of recent literature on the subject.

For me, it would probably only be worth expanding if we could thrash out an account of what the nPOV has to say on the subject. Urs has suggested we look at Bohrification. That sounds like the best lead. Reading through the Bohr topos entry, however, it seemed to me that little is said there about how to integrate that with other parts of the synthetic QFT story. There’s the idea of the ’fifth axiom’, but shouldn’t we expect these quantum phase spaces to have appeared earlier as part of the quantization process. Or do we see it merely as way to interpret our way back from the weird quantum world to something as classical as possible so as to be able to relate theory to the recordings of our classical instruments?

• I have expanded a bit at Serre-Swan theorem: gave it an actual Idea-section, mentioned more variants (over general ringed spaces, in higher geometry) and added more references.

• edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$-categories

• I have started an entry on shuffles. It is meant to be an ’elementary introduction’ so there will be room for deeper exploration of them in follow-on entries.

• Jonas Frey has raised the question of the notation $[n]$ in the entry for simplex category. I would go along with his choice of notation as it is the one I use myself. (I was surprised to see another convention being used.)

• I wrote the article distribution. I'm by no means an expert though. I left open a section "Applications" in case someone would like to add some, or if not I'll try to fill this in soon.

• For ease of linking to from various entries, and in order to have all the relevant material in one place, I am creating an entry

Presently this contains

1. an Idea-section,

2. some preliminaries to set the scene,

3. the statement and proof for the case of compactly supported distributions, taken from what I had just writted into the entry compactly supported distribution,

4. the informal statement for general distributions, so far just with a pointer to Kock-Reyes 04,

5. a section “Applications”, so far with

1. some comments on the relevance in pQFT;

2. some vague pointer to Lawvere-Kock’s generalization to a more general theory of “extensive quantity”

both of which deserve to be expanded.

Eventually I want to have more details on the page, but I’ll leave it at that for the time being. Please feel invited to join in.

I’ll go now and add pointers to this page from “distribution” and from other pages that mention the fact.

• added statement and proof that compactly supported distributions are equivalently the smooth linear functionals: here

(in the sense of either diffeological spaces, or smooth sets, or formal smooth sets/Cahiers topos).

• At Fréchet space I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of $\mathbb{R}^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{R}^n$. And I touched the description of this example in the main text, now here.

• At field (physics) I am beginning to write an actual introduction to the topic, now in a new section titled “A first idea of quantum fields”.

This means to introduce the concept with precise detail, but in a simple context (trivial and bosonic field bundles over Minkowski spacetime, perturbatively quantized) that allows to get a quick idea of the idea of the concept of (quantum) fields as such, without being distracted by other details.

So far I made it up to the derivation of the EOMs. Discussion of (deformation) quantization is to follow (maybe by tonight, depending on how much trouble I have with the trains) and I plan to sprinkle in the detailed example from scalar field in parallel with the abstract discussion.

• I wrote something at meaning explanation, but I didn’t add any links to it yet because I’m hoping to get some feedback from type theorists as to its correctness (or lack thereof).

• Created a stub tangent bundle categories as a link target to be disambiguated from tangent categories (with a hatnote at the latter). What I’m calling “tangent bundle categories” here are usually called just “tangent categories”, but that clashes with our page tangent category, so I invented a variation. Better suggestions are welcome.

• Somebody named Adam left a comment box a while ago at premonoidal category saying that naturality of the associator requires three naturality squares. I believe that this is true when phrased explicitly in terms of one-variable functors, but the slick approach using the “funny tensor product” allows us to rephrase it as a single natural transformation between functors $C\otimes C\otimes C\to C$. I’ve edited the page accordingly. I also added the motivating example (the Kleisli category of a strong monad) and a link to sesquicategory.

There is a comment on the page that “It may be possible to weaken the above make $(Cat,\otimes)$ a symmetric monoidal 2-category, in which a monoid object is precisely a premonoidal category”. However, the Power-Robinson paper says that “We remark that $(C \otimes -) : Cat \to Cat$ is not a 2-functor,” which seems to throw some cold water on the obvious approach to that idea. Was the thought to define a different 2-categorical structure on $Cat$ than the usual one, e.g. using unnatural transformations? It seems that at least one would still have to explicitly require centrality of the coherence isomorphisms.

• added to Grothendieck construction a section Adjoints to the Grothendieck construction

There I talk about the left adjoint to the Grothendieck construction the way it is traditionally written in the literature, and then make a remark on how one can look at this from a slightly different perspective, which then is the perspective that seamlessly leads over to Lurie's realization of the (oo,1)-Grothendieck construction.

There is a CLAIM there which is maybe not entirely obvious, but straightforward to check. I'll provide the proof later.

• 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…

• Started bornological set. Some people call it a bornological space, but that conflicts with the terminology in functional analysis which refers to a locally convex TVS with a suitable “bounded = continuous” property. I quickly wrote that uniformly continuous maps between metric spaces induce bounded maps, but I’ll recheck when I have a free moment.

• Needed to be able to point to contractible chain complex and discovered that we didn’t have an entry for that, so I quickly created one.

• Created Frobenius pseudomonoid (i.e. $\ast$-autonomous pseudomonoid).

• I gave Koszul complex and Idea-section and stated two key Properties in citable form (but without proof), one of them the statement that a sufficient condition for the Koszul complex to be a resolution of $R/(x_1, \cdots, x_n)$ is that $R$ is Noetherian, the $x_i$ are in the Jacobson radical, and the cohomology in degree -1 vanishes.

Finally I stated the special case of this (here) where $R$ is a formal power series algebra over a field and the elements $x_i$ are formal power series with vanishing constant term.

(I have added the relevant facts as citable numbered examples at Noetherian ring and at Jacobson radical.)

This happens to be the case that one need in BV-formalism in field theory. I am writing this out now at A first idea of quantum field theory (here).

• I have added a brief paragraph to the entry Myles Tierney. Can others please check (especially Todd, of course) if we want to add more? Myles was a leading category theorist and will be missed.