I have reorganised set theory and spun off material set theory.

]]>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.

]]>Given an $\infty$-topos, an object $X$, and a $f$ 1-monomorphism (i.e. (-1)-truncated) is the internal hom $[X,f]$ again a 1-mono?

And dually for $f$ 1-epimorphism (i.e. effective epimorphism) do we have some extra conditions such that $[f,X]$ is 1-mono?

]]>I was looking around a little for natural examples of Hopf ring spectra discussed in the literature. In Strickland-Turner 97 is discussed Hopf (semi-)(co-)ring structure on the extended power spectrum of the sphere spectrum, which they write $D S^0$.

Now the extended power spectrum of any spectrum $X$, that’s the direct sum over $n$ of the homotopy quotients of the $n$-fold smash powers of $X$ by the canonical symmetric group $\Sigma(n)$ action

$D X = \underset{n}{\vee} \left( X^{\wedge^n}/\Sigma(n) \right) \,.$I suppose this may be thought of as the spectral analog of the symmetric algebra construction where for $V$ a $k$-vector space we form $Sym_k(V)$. (If $k$ is of char 0 then we may form this from the tensor algebra by quotienting out the symmetric group action.) The analog of the ground field $k$ is now the sphere spectrum.

This should be the intuitive explanation of why there may be Hopf-like structure on these extended power spectra: They are like rings of functions on affine lines, and hence the additive group structure of the affine line induces a coproduct on its ring of functions. If I understand well, this matches with what Strickland-Turner have, where the coproduct $\Delta$ (later $\delta$) is induced from the diagonal $X \to X \vee X$ (towards the bottom of p. 2).

Interestingly now, while the ordinary symmetric algebra $Sym_k(k)$ of the ground ring itself is trivial, the extended power spectrum of the sphere spectrum itself is nontrivial. This is because $\mathbb{S}^{\wedge^n} \simeq \mathbb{S}$ but hence the homotopy quotient by $\Sigma_n$ contributes a copy of $\Sigma^\infty_+ B \Sigma(n)$ at each stage.

That makes me want to say that the $D S^0$ in Strickland-Turner is usefully thought of as the “polynomial ring” $Sym_{\mathbb{S}} \mathbb{S}$ being like functions on the “absolute spectral affine line”. Or something like this. Does this make sense?

]]>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.

created *law of double negation* with just the absolute minimum. Added a link from *double negation*, but nothing more.

started something at *ADE classification*, but am out of steam (and time) now.

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. ]]>

I added to higher homotopy van Kampen theorem the statement of the theorem by Jacob Lurie.

]]>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.

gave the statement an entry with a pointer to a proof: *Hausdorff implies sober*, then added pointer to this at *sober space* where it was claimed without proof or citation, and at *Hausdorff space* where it had previously not been mentioned yet.

I added a reference at differential form to the wiki-textbook Geometry of differential forms, written for a physics audience in mind.

]]>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.

]]>I have expanded logical functor by some stuff taken from the Elephant.

In the course of this I have created stubs for cartesian morphism, evaluation map and touched power object.

I have also done some editorial edits to topos (adding subsections and lead-ins)

]]>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?

finished typing part 1) and 2) of the proof of the existence theorem at [[Bousfield localization of model categories]]

]]>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 [[algebra for an endofunctor|algebras]] and [[coalgebra for an endofunctor|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.

]]>Urs started pointless topology, and I continued it.

]]>added to *Hausdorff topological space* a brief paragraph *Beyond topological spaces*