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- Discussion Type
- discussion topicMaterial set theory
- Category Latest Changes
- Started by TobyBartels
- Comments 80
- Last comment by Mike Shulman
- Last Active 3 hours ago

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

- Discussion Type
- discussion topicHopf algebra structure of universal enveloping algebra
- Category Latest Changes
- Started by Bartek
- Comments 1
- Last comment by Bartek
- Last Active 8 hours ago

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.

- Discussion Type
- discussion topicspectral symmetric algebra
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by David_Corfield
- Last Active 20 hours ago

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.

- Discussion Type
- discussion topicaxiom UIP
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Mike Shulman
- Last Active 20 hours ago

also created

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

- Discussion Type
- discussion topiclaw of double negation
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by NikolajK
- Last Active 21 hours ago

created

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

- Discussion Type
- discussion topicADE classification
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 22 hours ago

started something at

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

- Discussion Type
- discussion topicconvex space, homomorphism
- Category Latest Changes
- Started by John Baez
- Comments 13
- Last comment by RodMcGuire
- Last Active 1 day ago

- 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.

- Discussion Type
- discussion topichigher homotopy van Kampen theorem
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Mike Shulman
- Last Active 2 days ago

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

- Discussion Type
- discussion topiclocalic homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Mike Shulman
- Last Active 2 days ago

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?

- Discussion Type
- discussion topicArtin gluing
- Category Latest Changes
- Started by Todd_Trimble
- Comments 9
- Last comment by Mike Shulman
- Last Active 2 days ago

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.

- Discussion Type
- discussion topicseparation axiom
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

at

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

- Discussion Type
- discussion topicHausdorff implies sober
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by NikolajK
- Last Active 3 days ago

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.

- Discussion Type
- discussion topicdifferential form
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active 3 days ago

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

- Discussion Type
- discussion topicbornological set
- Category Latest Changes
- Started by Todd_Trimble
- Comments 27
- Last comment by DavidRoberts
- Last Active 3 days ago

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.

- Discussion Type
- discussion topicelementary infinity-topos
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by Mike Shulman
- Last Active 5 days ago

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?

- Discussion Type
- discussion topictoposes are extensive
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active 5 days ago

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.

- Discussion Type
- discussion topiclogical functor
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Mike Shulman
- Last Active 5 days ago

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)

- Discussion Type
- discussion topicsober space
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by IngoBlechschmidt
- Last Active 5 days ago

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?

- Discussion Type
- discussion topicBousfield localization of model categories
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 5 days ago

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

- Discussion Type
- discussion topicalgebra for a C-C bimodule
- Category Latest Changes
- Started by SridharRamesh
- Comments 20
- Last comment by Mike Shulman
- Last Active 5 days ago

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)

- Discussion Type
- discussion topicclosed subspace
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by Urs
- Last Active 6 days ago

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.

- Discussion Type
- discussion topicPointless topology
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Urs
- Last Active 6 days ago

Urs started pointless topology, and I continued it.

- Discussion Type
- discussion topicHausdorff topological space
- Category Latest Changes
- Started by Urs
- Comments 18
- Last comment by Urs
- Last Active 6 days ago

added to

*Hausdorff topological space*a brief paragraph*Beyond topological spaces*

- Discussion Type
- discussion topicconfiguration space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

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.

- Discussion Type
- discussion topicrelative Postnikov tower and n-images
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 6 days ago

I have added to

*Postnikov tower*paragraphs on the*relative*version, (*definition*and*construction*in simplicial sets).I also added the remark that the relative Postnikov tower is the tower given by the (n-connected, n-truncated) factorization system as $n$ varies, hence is the tower of

*n-images*of a map in $\infty Grpd$. And linked back from these entries.

- Discussion Type
- discussion topicFrobenius morphism
- Category Latest Changes
- Started by Stephan A Spahn
- Comments 11
- Last comment by Urs
- Last Active 6 days ago

I changed the name of the page Frobenius map to Frobenius morphism and added the descriptions à la Demazure to it.

- Discussion Type
- discussion topicspectral supergeometry
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 7 days ago

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).

- Discussion Type
- discussion topichigher inductive type
- Category Latest Changes
- Started by Urs
- Comments 61
- Last comment by Mike Shulman
- Last Active 7 days ago

added pointer to

- Steve Awodey, Nicola Gambino, Kristina Sojakova,
*Homotopy-initial algebras in type theory*(arXiv:1504.05531)

to

*initial algebra over an endofunctor*,*higher inductive type*and*W-type*- Steve Awodey, Nicola Gambino, Kristina Sojakova,

- Discussion Type
- discussion topicfree topos
- Category Latest Changes
- Started by spitters
- Comments 17
- Last comment by Mike Shulman
- Last Active 7 days ago

Quickly generated free topos. Mostly references.

- Discussion Type
- discussion topicimage
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Urs
- Last Active 7 days ago