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

• In preparation for week296 I corrected the definition of dagger-compact category, since it was missing some coherence laws. The most convenient way to include these was to add a page containing Selinger’s definition of symmetric monoidal dagger-category . This in turn forced me to add pages containing definitions of associator, unitor, braiding and unitary morphism. Some of my links between these pages are afflicted by the difficulty of getting daggers to appear in names of pages. Maybe a lab elf can improve them.

Hmm, html links didn’t work here, so I’m trying textile.

• I have moved the personal data on Eberhard Zeidler from QFT entry to his own new-created entry.

• with the URL code for Relay Station, useful for linking from $n$lab

• I made a first draft of a page about unbounded operators, the battle plan contains some basic definitions, explanation of some subtleties of domain issues and what it means to be affiliated to a von Neumann algebra. Right now, only the rigged Hilbert space page refers to it.

• eom Springer Online Encyclopaedia of Mathematics. Needed an entry, I am sick of typing it in full and do not like to make it incomplete. So why not using eom as a link whenever I need to quote a particular article from eom, thus citations will be shorter with the same online functionalilty.

• I added some examples to Gray-category, including also a non-example which has fooled several people.

• The nCafé is currently haunted by a bug that prevents any comments from being posted. This should eventually go away, hopefully. For the time being I post my comment in reply to the entry Division Algebras and Supersymmetry II here:

Thanks, John and John for these results. This is very pleasing.

The 3-$\psi$s rule implies that the Poincaré superalgebra has a nontrivial 3-cocycle when spacetime has dimension 3, 4, 6, or 10.

Similarly, the 4-$\psi$s rule implies that the Poincaré superalgebra has a nontrivial 4-cocycle when spacetime has dimension 4, 5, 7, or 11.

Very nice! That's what one would have hoped for.

Can you maybe see aspects of what makes these cocycles special compared to other cocycles that the Poincaré super Lie algebra has? What other cocycles that involve the spinors are there? Maybe there are a bunch of generic cocycles and then some special ones that depend on the dimension?

Is there any indication from the math to which extent $(3,4,6,10)$ and $(4,5,7,11)$ are the first two steps in a longer sequence of sequences? I might expect another sequence $(7,8,10,14)$ and $(11, 12, 14, 18)$ corresponding to the fivebrane and the ninebrane. In other words, what happens when you look at $n \times n$-matrices with values in a division algebra for values of $n$ larger than 2 and 4?

Here a general comment related to the short exact sequences of higher Lie algebras that you mention:

properly speaking what matters is that these sequences are $(\infty,1)$-categorical exact, namely are fibration sequences/fiber sequences in an $(\infty,1)$-category of $L_\infty$-algebras.

The cocycle itself is a morphism of $L_\infty$-algebras

$\mu : \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R}$

and the extension it classifies is the homotopy fiber of this

$\mathfrak{superstring}(n+1,1) \to \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R} \,.$

Forming in turn the homotopy fiber of that extension yields the loop space object of $b^2 \mathbb{R}$ and thereby the fibration sequence

$b \mathbb{R} \to \mathfrak{superstring}(n+1,1) \to \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R} \,.$

The fact that using the evident representatives of the equivalence classes of these objects the first three terms here also form an exact sequence of chain complexes is conceptually really a coicidence of little intrinsic meaning.

One way to demonstrate that we really have an $\infty$-exact sequence here is to declare that the $(\infty,1)$-category of $L_\infty$-algebras is that presented by the standard modelstructure on dg-algebras on $dgAlg^{op}$. In there we can show that $b \mathbb{R} \to \mathfrak{superstring} \to \mathfrak{siso}$ is homotopy exact by observing that this is almost a fibrant diagram, in that the second morphism is a fibration, the first object is fibrant and the other two objects are almost fibrant: their Chevalley-Eilenberg algebras are almost Sullivan algebras in that they are quasi-free. The only failure of fibrancy is that they don't obey the filtration property. But one can pass to a weakly equivalent fibrant replacement for $\mathfrak{siso}$ and do the analog for $\mathfrak{superstring}$ without really changing the nature of the problem, given how simple $b \mathbb{R}$ is. Then we see that the sequence is indeed also homotopy-exact.

This kind of discussion may not be relevant for the purposes of your article, but it does become relevant when one starts doing for instance higher gauge theory with these objects.

Here some further trivial comments on the article:

• Might it be a good idea to mention the name "Fierz" somewhere?

• page 3, below the first displayed math: The superstring Lie 2-superalgebra is [an] extension of

• p. 4: the bracket of spinors defines [a] Lie superalgebra structure

• p. 6, almost last line: this [is] equivalent to the fact

• p. 13 this spinor identity also play[s] an important role in

• p. 14: recall this [is] the component of the vector

• I tried at locally presentable category to incorporate the upshot of the query box discussions into the text, then moved the query boxes to the bottom

• 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.

• Why I can not have this

1. Introduction 3
1.1 Categories and generalizations . . . . . . . . . . . . . . . . . 3
1.2. Basic idea of descent . . . . . . . . . . . . . . . . . . . . . . 5
2. From noncommutative spaces to categories 5
2.1. Idea of space and of noncommutative space . . . . . . . . . 5
2.2. Gel’fand-Naimark . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3. Nonaffine schemes and gluing of quasicoherent sheaves . . . 6
2.4. Noncommutative generalizations of QcohX . . . . . . . . . . 6
2.5. Abelian versus 1-categories . . . . . . . . . . . . . . . . . . 7

but instead I have automatic numbers like 1,2,3, 5, 8 (I know why, but how to avoid it??). I do NOT want nlab to make it like word, I want my numbering to stay it is, and if possible keeping the paragraphs. Putting > for quotation did not help!