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

• In the nLab article on the universal enveloping algebra, the section describing the Hopf algebra structure originally stated that “the coproduct $\Delta: U L \to U(L \coprod L)\cong U L\otimes UL$ is induced by the diagonal map $L \to L \coprod L$.”

I assume that this is a mistake, and I have since changed the coproduct $\coprod$ to a product $\times$. However, I don’t know a great deal about Hopf algebras, so please correct me if I’ve made a mistake here.

• I have created an entry spectral symmetric algebra with some basics, and with pointers to Strickland-Turner’s Hopf ring spectra and Charles Rezk’s power operations.

In particular I have added amplification that even the case that comes out farily trivial in ordinary algebra, namely $Sym_R R$ is interesting here in stable homotopy theory, and similarly $Sym_R (\Sigma^n R)$.

I am wondering about the following:

In view of the discussion at spectral super scheme, then for $R$ an even periodic ring spectrum, the superpoint over $R$ has to be

$R^{0 \vert 1} \;=\; Spec(Sym_R \Sigma R) \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\mathbb{R}^n} \right)_+ \right) \,.$

This of course is just the base change/extension of scalars under Spec of the “absolute superpoint”

$\mathbb{S}^{0\vert 1} \simeq Spec(Sym_{\mathbb{S}} (\Sigma \mathbb{S}))$

(which might deserve this notation even though the sphere spectrum is of course not even periodic).

This looks like a plausible answer to the quest that David C. and myself were on in another thread, to find a plausible candidate in spectral geometry of the ordinary superpoint $\mathbb{R}^{0 \vert 1}$, regarded as the base of the brane bouquet.

• also created axiom UIP, just for completeness. But the entry still needs some reference or else some further details.

• I added some references to convex space and began a new entry on homomorphism.

It would be great to see the article on convex spaces continue... it sort of trails off now. I've tried to enlist Tobias Fritz.
• back in “The point of pointless topology” Peter Johnstone suggested that localic homotopy theory ought to be developed:

So far, relatively little work has been done on specific applications of locale theory in contexts like these; so it is perhaps appropriate to conclude this article by mentioning some areas which (in the writer’s opinion, at least) seem ripe for study in this way. One is homotopy theory: the work of Joyal, Fourman and Hyland [15] shows that in a constructive context it may be necessary to regard the real Une as a (nonspatial) locale, at least if we wish to retain the Heine-Borel theorem that its closed bounded subsets are compact. So there is scope for developing the basic ideas of homotopy theory for locales, starting from the localic notion of the unit interval; when interpreted in the two contexts mentioned above, it should yield results in the “Ex-homotopy theory” and “equivariant homotopy theory” that have been studied in recent years by James [27, 28]

Has anything been done in this direction?

• Some stuff that Zoran wrote on recollement reminded me that I had been long meaning to write Artin gluing, which I’ve done, starting in a kind of pedestrian way (just with topological spaces). Somewhere in the section on the topos case I mention a result to be found in the Elephant which I couldn’t quite find; if you know where it is, please let me know.

• at separation axiom I have expanded the Idea section here, trying to make it more introductory and expository.

• 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.

• A message to Mike:

Hi Mike,

I hear that in Swansea you ended by talking about things related to elementary $\infty$-toposes. I didn’t get a chance to see anyone’s notes yets. Do you have electronic notes to share?

• In line with the “pages named after theorems” philosophy, I’ve created toposes are extensive, including in particular the (somewhat hard to track down) constructive proof that a cocomplete elementary topos is infinitary extensive.

• at sober space the only class of examples mentioned are Hausdorff spaces. What’s a good class of non-Hausdorff sober spaces to add to the list?

• I've created a new article entitled algebra for a C-C bimodule, a straightforward concept encapsulating both algebras and coalgebras for endofunctors, as well as further generalities besides. There's surely a better name than using "C-C bimodule" (replacing it with "endoprofunctor", perhaps? Although I actually find that less preferable...), for someone to propose or let me know already exists, as the case may be.

(I've also made some small edits to the articles on algebras and coalgebras for endofunctors; in particular, the former had forgotten to define the morphisms of such algebras)

• At closed subspace, I added some material on the 14 operations derivable from closures and complements. For no particularly great reason except that it’s a curiosity I’d never bothered to work through until now.

• I realized that we had a stub entry “configuration space” with the physics concept, and a stub entry “Fadell’s configuration space” with the maths concept, and no interrelation between them, also without any examples. So I created a disambiguation page

and then

but I also left

separate for the moment, thinking that in principle the term in matematics may be understood more generally, too. But maybe something should be merged here.

I added the example of the unordered configuration space of $\mathbb{R}^\infty$ as a model for the classifying space for the symmetric group to the relevant entries. But otherwise they do remain stubby, alas.

• I finally gave spectral super-scheme an entry, briefly stating the idea.

This goes back to the observation highlighted in Rezk 09, section 2. There is some further support for the idea that a good definition of supergeometry in the spectrally derived/$E_\infty$ context is nothing but $E_\infty$-geometry over even periodic ring spectra. I might add some of them later.

Thanks to Charles Rezk for discussion (already a while back).

• Quickly generated free topos. Mostly references.

• the evident proposal for the definition of $(\infty,1)$-categoical images here (which implies in particular a notion of images of of $n$-functors between $n$-groupoids).