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

]]>Created linear bicategory.

]]>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?

]]>This MO question seemed interesting to me, and I wondered if anyone here had a reaction.

]]>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

]]>Created local colimit.

]]>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.)

]]>Andrew Stacey: would you tell us what the word ’heuristic’ means? I’ve seen you complain in various places at MathOverflow about how people use this word (e.g., here, in the comments). Frankly, I’ve been frustrated by your not elaborating further (your justification being that MO is not suited for such discussions), and now I avoid writing the word in a place where I think you might be reading, not wishing to bring upon myself your public displeasure (even though I think I know how to use the word correctly, according to what I’ve read).

I hope you think the nForum *is* suitable for such discussions!

I find myself agreeing with QQJ: “Actually the principal definition from the Concise Oxford English dictionary is (adjective) ‘enabling a person to discover or learn something for themselves’. Plenty much broad scope there to zero your aagghhument.” And indeed, my Compact Unabridged OED (2nd edition) gives simply “serving to find out or discover” as their definition (a).

The “rule of thumb” definition mentioned by Per Vognsen is supported and elaborated on by Wikipedia:

Heuristic; Greek: “Εὑρίσκω”, “find” or “discover”) refers to experience-based techniques for problem solving, learning, and discovery that gives a solution which is not guaranteed to be optimal. Where the exhaustive search is impractical, heuristic methods are used to speed up the process of finding a satisfactory solution via mental shortcuts to ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, stereotyping, or common sense.

In more precise terms, heuristics are strategies using readily accessible, though loosely applicable, information to control problem solving in human beings and machines.

and it seems to me Per did use the word correctly according to that range of meanings. The ways in which I would use the term would also fall within this range.

These applications also seem supported by the OED. Under definition (b), OED cites the IBM Journal of Research and Development (1958), which is fairly precise as illustrative quotations go:

For the moment… we shall consider that a heuristic method (or a heuristic, to use the noun form) is a procedure that may lead us by a short cut to the goal we seek or it may lead us down a blind alley.

What do you say? Concrete examples of what you have in mind would be wonderful.

]]>The linear logic wiki describes several translations of classical logic into linear logic, which translate the classical implication $A\to B$ as the linear $!? A' \multimap ?B'$ or $?(!A' \multimap B')$ or $!(A' \multimap ?B')$, or $!A' \multimap ?!B'$ (where $A'$ and $B'$ translate $A$ and $B$ respectively). But wikipedia claims that it can be translated as $!A' \multimap ?B'$, which is none of these. Is wikipedia wrong?

]]>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

an Idea-section,

some preliminaries to set the scene,

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

*compactly supported distribution*,the informal statement for general distributions, so far just with a pointer to Kock-Reyes 04,

a section “Applications”, so far with

some comments on the relevance in pQFT;

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.

Someone must have already studied the Chu construction $Chu(Cat,Set)$ on the cartesian closed monoidal category $Cat$ with dualizing object $Set\in Cat$. But “$Chu(Cat,Set)$” is kind of hard to seach for, and right now I can’t find anything about it. Does anyone know a reference?

]]>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.

added to convenient vector space a Properties-section mentioning their embedding into the Cahiers topos, and added the reference by Kock where this is proven.

]]>Wouldn’t it be better if you remove the underlines for the words with hyperlinks? (at least within the paragraphs)

I think this is one of the major css design issues, which actually slows you down while reading. A color code is adequate in my opinion.

As it is, it looks a bit crowded. Check for example, compared to a wiki page

]]>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.

]]>I am struggling with “500 Internal Server Errors” that appear when saving and/or displaying the notes the I am writing.

I have been trying hard to determine what exact line causes the error, but I don’t recognize any systematics.

I was suspecting that it has to do with equation references as in

```
(eq:EquationName)
```

but I can’t isolate this as the source of the problem.

First I get the errors upon saving (after hitting “submit”) but after a while they also affect content that was previously already saved succesfully.

For instance right now on my sytem just asking the *Sandbox* to display produces a “500 Internal Server Error”.

I added two recent examples of enriched categories: tangent bundle categories and Lawvere theories.

]]>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.

]]>