I wanted to be able to point to *[[expectation value]]* without the link being broken. So I added a sentence there, but nothing more for the moment.

I started a bare minimum at *quantum probability* (redirecting *noncommutative probability space* etc.)

Some entries have long been secretly referencing such an entry, and I have cross-linked accordingly, for instance from *von Neumann algebra* and *quantum computing*.

I had the feeling somewhere we already had a detailed account of probability theory dually in terms of von NNeumann algebras, but if we do I didn’t find it(?)

]]>I gave *diffiety* more of an Idea-section

Created multiplicative disjunction.

]]>It would be nice to finish the description of the theorem at GNS construction, if someone has the head for doing that. :-)

]]>Does anyone agree that one should document the “Feynman Categories” in the sense of Kaufmann and Ward in the nLab?

It puzzles me a little that despite extensive publications, extending back to 2013, and a talk, and despite the systematizing aspirations of Kaufmann–Ward’s work, the concept of Feynman category seems to yield zero search hits, both in the nLab, and in the nForum. Is this simply hazard?

]]>Few words added at Catalan number.

]]>microformal morphism a la Theodore Voronov.

]]>Stub for Morse potential.

]]>New stub Weyl functional calculus redirecting also Weyl quantization. I would like to see ref.

- Lars Hörmander,
*The weyl calculus of pseudo-differential operators*, Comm. Pure Appl. Math.**32**, 3, 359–443, May 1979, doi,

but have no access to it (can anybody help?). I also added a sentence at Idea section of functional calculus reflecting that the previous definition there is not fitting functional calculi in the context of quantization, including Weyl’s case. One should do this generality discussion more carefully. the previous definition said that the functional calculus needs to be a homomorphism (from ordinary functions to operator functions). This is true for the functional calculus described in the entry, but not for the wider usage of the phrase like in Weyl functional calculus. Maybe we can resolve this in a better way.

]]>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’ll be using the “Feynman slash notation” “$A\!\!\!/$” a little, for the Clifford algebra element associated with a given vector $A$ in the given inner product space.

I forget if there is some native code to get this notation. Is there? Presently I am using the hack

```
A\!\!\!/
```

]]>
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.

Guys (maybe Urs knows), how do you export the TeX of a page which is mainly consisting of !includes of other pages ? I mean I want to make a TeX file of a sequence of about 10 or so nLab pages. If I take source of one at the time, then I have to merge the preambles carefully what is not very feasible. But if I just export the file which has the list of includes than the !include lines are not expanded! Can we force expansion of the source in the output somehow ? This would be very painstaiking…

Example: zadarmat1test1onefile (zoranskoda)

]]>As of recently the redirecting behaviour of the nLab software has changed, and it is breaking our conventions.

For instance

And interestingly:

etc.

I suppose it makes sense for us if accents are disregarded in entry identification, that could be useful. But we do need that lower case is distinguished from upper case, or else many existing entries will be hidden.

]]>added a minimum of words at *commutative operad* (the entry remains a stub)

I’ve been entering corrections into the article theory of algebraically closed fields in response to a chat room discussion, but see that the \underbrace command doesn’t work as expected (see the Definition section). What’s the right way to write what is obviously wanted here?

]]>added the case of dgc superalgebras (here) and expanded the list of examples accordingly

]]>needed to be able to point to *duality in physics*, so I created an entry. For the moment just a glorified redirect.

I moved [[(n,k)-transformation]] to [[transfor]], as seemed to be agreed upon by those who spoke up in the discussion there.

]]>Blute-Cockett-Seely-Trimble describes a string diagram / circuit diagram / proof net calculus for linearly distributive categories, which is significantly complicated by the presence of units; as discussed in section 2.3 of the paper, some of the unit and counit links have to be “attached” to other strings to prevent diagrams that should be distinct from getting identified. However, the example given there depends on the fact that the units $\top$ and $\bot$ for the tensor and cotensor products are different; as noted therein, if $\top$ in the example were replaced by $\bot$ then the two problematic diagrams *would* represent the same morphism. This makes me wonder, if we have a linearly distributive category that happens to satisfy $\top=\bot$, then does the whole string diagram calculus work without these extra attachments?

This would be especially convenient because any closed monoidal category $(C,\otimes,\top)$ can be embedded by a closed functor into a linearly distributive (indeed $\ast$-autonomous) category in which $\top=\bot$, namely $Chu(C,\top)$. So we could soundly use linearly distributive string diagrams to reason about closed monoidal categories, without the need for the clunky “boxes” that are sometimes used to deal with internal-homs.

]]>I have expanded the Idea section at *state on a star-algebra* and added a bunch of references.

The entry used to be called “state on an operator algebra”, but I renamed it (keeping the redirect) because part of the whole point of the definition is that it makes sense without necessarily having represented the “abstract” star-algebra as a C*-algebra of linear operators.

]]>added to *string theory FAQ* two new paragraphs:

Prompted by the MO discussion

]]>The category of topological spaces fully faithfully embeds into the category of simplicial objects of the category of filters, where “the category of filters” means the full subcategory of topological spaces consisting of spaces such that any superset of a non-empty open subset is open (details below).

The embedding is essentially the definition of a topological space in terms of neighbourhood systems, e.g. (Bourbaki, General Topology, $I\S1.2,Ax.(V_I)-(V_IV)$).

Do you know any references this embedding or the category of simplicial objects in the category of filters, perhaps under another name? I was not able to find any; nlab does not seem to have an entry for either the category of filters or its simplicial category.

I sketch the construction below and a couple of open questions it leads to. More details appear in a research proposal [1], along with a number of questions. Little attempt is made to answer these questions, as probably they are already well-known.

The embedding $t:Top\longrightarrow sFilt$ into the category $sFilt$ of simplicial objects in the category $Filt$ of filters:

set-wise a space X is sent to $(|X|, |X|\times |X|,|X|\times |X|\times |X|,....)$,

where $|X|$ is the set of points of $X$

connecting maps are coordinate maps,

filters on $|X|^n$ are defined as follows:

a subset $U$ of $|X|$ is big iff $U=X$

a subset $U$ of $|X|\times |X|$ is big iff each point $x$ of $X$ has an open neighbourhood $U_x$ such that $\{x\}\times U_x \subset U$.

for $n \ge 3$, the filter on $|X|^n$ is the coarsest filter such that all coordinate maps $|X|^n-\longrightarrow |X|\times |X|$ are continuous wrt the topology on $|X|\times |X|$ defined above.

There is a similar but simpler embedding of metric spaces with uniformly continuous maps: a subset of $|X|^n$ is big iff it contains an $\epsilon$-neighbourhood of the diagonal for some $\epsilon$.

Two open questions:

Is there a model structure on the larger category compatible with a model structure on topological spaces?

Topological spaces, metric (uniform) spaces and filters all “live” in the same larger category, and this allows to use language of category theory to talk about compactness, precompactness, completeness, and equicontinuity. Can one reformulate and prove Arzela-Ascoli theorem category-theoretically?

Here is an example of use of the language.

A sequence $f_i:X\longrightarrow M$ of functions from a topological space $X$ to a metric space $M$ is

**equicontinuous**iff the following morphism is well-defined:$i(\{\omega\}) \times t(X) \longrightarrow m(M)$ $(i,x) \mapsto f_i(x)$

A sequence $f_i:X\longrightarrow M$ of functions from a metric space $X$ to a metric space $M$ is

**uniformly equicontinuous**iff the following morphism is well-defined:$i(\{\omega\}) \times m(X) \longrightarrow m(M)$ $(i,x) \mapsto f_i(x)$

A sequence $f_i:X\longrightarrow M$ of functions from a metric space $X$ to a metric space $M$ is

**uniformly Cauchy**iff the following morphism is well-defined:$E(\omega_{cofinite}) \times m(X) \longrightarrow m(M)$ $(i,x) \mapsto f_i(x)$

Here $t:Top\longrightarrow sFilt$, $m:UniformSpaces\longrightarrow sFilt$ are the two embeddings mentioned above into the category $sFilt$ of simplicial objects in the category $Filt$ of filters;

$\omega_{cofinite}$ is the filter of cofinite subsets of $\omega$; $\{\omega\}$ is the filter on $\omega$ with only one big subset $\omega$ itself;

the functor $i:Filt\longrightarrow sFilt$, $F\mapsto (F,F,F,...)$ with connecting maps being always identity, the functor $E:Filt\longrightarrow sFilt$, $F\mapsto (F,F\times F,F\times F\times F,...)$ sends a filter into the sequence of its Cartesian powers with coordinate maps.

References:

[1] Topological and metric spaces as full subcategories of the category of simplicial objects of the category of filters. A draft of a research proposal

]]>I’ve tried on two pages and am told after submitting an edit that the page is locked because I’m editing. The change aren’t saved.

]]>