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

• I’ve inserted some proofs of statements made at Heyting algebra, particular on the “regular element” left adjoint to the full inclusion $Bool \to Heyt$.

The proof that $L \to L_{\neg\neg}$ preserves implication seemed harder than I was expecting it to be. Or maybe my proof is a clumsy one? If anyone knows a shorter route to this result, I’d be interested.

• by chance I noticed that two days ago somebody created an entry Circuitoids. I am not sure what to do about it…

• at 2-adjunction I would like to list a bunch of 2-category theoretic analogs of standard facts about ordinary adjunctions. Such as: a right adjoint is a full and faithful 2-functor precisely if the counit of the 2-adjunction is an equivalence, etc.

But I haven’t really thought deeply about 2-adjunctions myself yet. Is there some reference where we could take such a list of properties from?

• I wanted to add some stuff about completely distributive lattices, when I got annoyed by the fact that few of the entries on lattices, frames, etc, carried a table of contents, and that I kept being surprised by which related entries already existed and which not.

Just a start. Please feel free to expand.

• I've made some small additions to the article on algebraic lattices (including fixing the languishing typo in the introduction, "An alegbraic lattice is...").
• I stumbled across a nice reference while looking for something else, so added it to axiom of choice so I might read it all later.

• I have added a little bit to spine.

(Will maybe write out the proof of the proposition there in a little while.)

• I have worked a bit on 2-congruence.

The main addition is that I started an Examples-section, where I started writing out an explicit proof (little exercise in unwinding the definitions) of the statement:

The 2-category of 2-congruences in $Grpd$ is equivalent to that of small categories.

One should write out more. But it is getting late for me now. I should continue another day.

• Added more material to Boolean algebra, particularly the principle of duality and the connection to Boolean rings, and a wee bit of material on Stone duality.

Stone duality deserves greater expansion, bringing out the dualities via ambimorphic (ahem, schizophrenic) structures on the 2-element set, and mentioning the connection to Chu spaces. Another day, another dollar.

• it had annoyed me for a long time that we had now dedicated entry for Giraud theorem. I have now created one, so that the redirects no longer simply go to Grothendieck topos. But then, I didn’t have the energy to add more than a few pointers, for the moment.

• I noticed a link to page for atomic Boolean algebra , so I made the page. It's pretty simple. People may wish to add examples or something.
• I have created a stub for Bob Thomason as there were frequent references to the Thomason model category structure in the nLab but no link to an entry on the person and his work. The St. Andrews biography is good, giving good details on his work. I have linked to it but perhaps we need some more comment in the nLab entry.

• I added some random meanings to order, then noted that they’re not entirely unrelated, since these orders can be ordered.

• added to Kan lift the def of absolute Kan lifts, and some more examples.

Btw, in all of the sources I’ve read about this, Kan lifts are simply called (left, right) liftings. What do you think about renaming this to liftings? will it conflict with some other kind of “lifting”?

• I have added more to higher generation by subgroups. As I said on another thread, this material feels as if there should be a nPOV / categorified version that could be quite interesting, so any thoughts would be welcome.

• New entry universal epimorphism redirectinig also universal monomorphism. It is not among those variants listed in epimorphism. We also do not list absolute epimorphism (epimorphism which stays epimorphism after applying any functor to it). Every split epimorphism stays split after applying a functor hence it is absolute, but is there a counterexample of an absolute epimorphism which is not in fact split ?

By the way, here is an archived version of the old query from strict epimorphism

David Roberts: I’m interested in a bicategorical version of this. You haven’t happened to have done this Mike?

Mike Shulman: Not more than can be extracted from 2-congruence (michaelshulman) and regular 2-category (michaelshulman). What is there called an “eso” is the bicategorical version of a strong epi (which agrees with an extremal epi in the presence of pullbacks), and what is there called “the quotient of a 2-congruence” is the bicategorical version of a regular epi. I’ve never thought about the bicategorical version of a strict epi; since strict epis agree with regular epis in the presence of finite limits I’ve never really had occasion to care about them independently.

• A graduate student at Johns Hopkins who is being supervised by Jack Morava, named Jon(athan) Beardsley, wrote a short article Bousfield Lattice. More on this in a moment.

• I created locally regular category and added a corresponding section to allegory.

Edit: removed some complaints that were due to it being too late at night and my brain not working correctly.

• I am experimenting with a notion of Heisenberg Lie $n$-algebras, for all $n \in \mathbb{N}$.

I have made an experimental note on this here in the entry Heisenberg Lie algebra.

It’s explicitly marked as “experimental”. If it turns out to be a bad idea, I’ll remove it again. Please try to shoot it down to see if I can rescue it! :-)

I mean, the definition in itself is elementary and very simple. The question is if this is “the right notion” to consider. The reasoning here is:

by the arguments as mentioned on the nCafé here we may feel sure that Chris Rogers’s notion of Poisson Lie n-algebra is correct. (Not that there were any particular doubts, but the fact that we can derive it from very general abstract homotopy theoretic constructions reinforces belief in it.)

But the ordinary Heisenberg Lie algebra is just the sub-Lie algebra of the Poisson Lie algebra on the constant and the linear functions. Therefore it makes sense to look at the sub-Lie $n$-algebra of the Poisson Lie $n$-alhebra on the constant and linear differential forms That’s what my experimental definition does.

• Added some relevant bits to connected limit, fiber product, and pushout. I wanted to record the result at connected limit that functors preserve connected limits iff they preserve wide pullbacks, which may be a slightly surprising result if one has never seen it before.

• I have been expanding and polishing the entry Heisenberg group.

This had existed in bad shape for quite a while, but now it’s maybe getting into better shape.

I tried to spend some sentences on issues which I find are rarely highlighted appropriately in the literature. So there is discussion now of the fact that

• there are different Lie groups for a given Heisenberg Lie algebra,

• and the appearance of an “$i$” in $[q,p] = i$ may be all understood as not picking the simply conncted ones of these;

I also added remarks on the relation to Poisson brackets, and symplectomorphisms.

In this context: either I am dreaming, or there is a mistake in the Wikipedia entry Poisson bracket - Lie algebra.

There it says that the Poisson bracket is the Lie algebra of the group of symplectomorphisms. But instead, it is the Lie algebra of a central extension of the group of Hamiltonian symplectomorphisms.

• I have started an entry canonical extension.

• I am fiddling with an entry table - models for (infinity,1)-operads meant to allow to see 10+ different model categories and their main Quillen equivalences at one glance.

I guess there are better ways to typeset this. (Volunteers please feel free to lend a hand!) But for the time being I’ll settle with what I have so far.

• Added a section arithmetic D-modules. This is the optimal theory for p-adic cohomology of varieties over finite fields, since it has the six operations. This section is complementary to rigid cohomology.

• In

http://ncatlab.org/nlab/show/Lie+2-algebra, at

“… the differential respects the brackets: for all $x \in g_0$ and $h \in g_1$ we have

$\delta [x,h]=[x,\delta h]$

…”

is wrong. The equation should be:

$\delta \alpha(x,h) = [x,\delta h]$

Since I don’t know if I have the right to change an nLab entry,I post this here as an suggestion.

• Created a new entry on "special $\Delta$-spaces". Does anyone know of a better name?

The entry is poorly edited since I'm not fluent with the iTex syntax.
• I made some much-needed corrections at simplicial complex, directed mostly at errors which had been introduced by yours truly. I also created quasi-topological space (the notion due to Spanier).

I haven’t thought this through, but regarding the process of turning a simplicial complex into a simplicial set, the usual sequence of words seems to involve putting a non-canonical ordering on the set of vertices and then getting ordered simplices from that. But is there anything “wrong” with taking the composite

$SimpComp \hookrightarrow Set^{Fin_{+}^{op}} \stackrel{Set^{i^{op}}}{\to} Set^{\Delta^{op}}$

where the inclusion is the realization of simplicial complexes as concrete presheaves on nonempty finite sets, and the second arrow is pulling back along the forgetful functor $i$ from nonempty totally ordered finite sets to nonempty finite sets? This looks much more canonical.

• I hope Urs doesn’t mind my inserting a not-too-serious but nevertheless amusing example at symplectomorphism.

• Someone set up lawvere theory, but did not add anything to it. They had previously done an edit to FinSet. The new page has a redirect from Lawvere+theory, so I don’t see what 88.104.160.245 is doing. Can someone check the edit at [[FinSet]. It looks as if the person knows some things and so has added a bit, but it is so long since I knew that stuff well so I cannot tell if it is a valid edit or not.

• I have created a stub on Volodin. I have been unable to find out more on him. Can anyone help?

• Started presentation of a category by generators and relations. This is probably an evil definition (there was an old discussion on this in the context of quotient category), and there is perhaps a more modern way to do this, so feel free to change the entry. I used “quotient category” as in CWM and mentioned that this is not the definition in the nLab.

• I have added some material into Vietoris complex relating to links with algebraic K-theory and some conjectures of Kapranov and Saito.

• The last two days Stephan Spahn was visiting me, and we chatted a lot about étaleness in cohesive ∞-toposes.

We found proofs that

• for every notion of infinitesimal cohesive neighbourhood

$i : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$

the total space projections of locally constant $\infty$-stacks are formally étale;

• the formally étale morphisms with respect to any choice of infinitesimal cohesion satisfy all the axioms of axiomatic open maps (or rather their $\infty$-version, of course).

(These are to be written up. Requires plenty of 3d iterated $\infty$-pullback diagrams which are hard to typeset).

Recall – from synthetic differential infinity-groupoid – that for the infinitesimal cohesive neighbourhood

$i :$ Smooth∞Grpd $\hookrightarrow$ SynthDiff∞Grpd

the axiomatically formally étale morphisms between smooth manifolds are precisely the étale maps in the traditional sense.

Motivated by all this, I finally see, I think, what the correct definition of cohesive étale ∞-groupoid is:

simply: $X \in \mathbf{H}$ is an étale cohesive $\infty$-groupoid if it admits an atlas $X_0 \to X$ by a formally étale morphism in $\mathbf{H}$.

I have spelled out the proof now here that with this definition a Lie groupoid $\mathcal{G}$ is an étale groupoid in the traditional sense, precisely if it is cohesively étale when regarded as an object of the infinitesimal cohesive neighbourhood $Smooth \infty Grpd \hookrightarrow SynthDiff \infty Grpd$.

I hope to further expand on all this with Stephan. But I may be absorbed with other things. Next week I am in Goettingen, busy with a seminar on $L_\infty$-connections.

• I found in the disambiguation page basis an entry basis of a vector space missing, and so I created one. I have then cross-linked this with relevant existing entries, such as basis theorem. Also created a brief entry orthogonal basis in order to get rid of grayish links is some entries.

• In context of size issues on admissible structures (in the sense of DAG V) I wondered which closure properties (e.g. obviously its closed under pullbacks along relatively k-compact morphisms) the class of relatively k-compact (for a regular cardinal k) morphisms in a (∞,1)-category satisfies. Is there any reference concerning this?

• I needed an entry that lists references on twisted K-homology, so I created one. This made me notice that we currently lack an entry K-homology. I can try to create a stub for that a little later…

• the link to the picture in the entry Charles Wells is broken. Does anyone know how to fix it or have an alternative picture?

• Perhaps we need a page on Jean-Louis Koszul, and possibly an agreement on terminology / names for entries. Ben has created a page called Koszul, but usually single names like that would be used for the ’person’ page of that mathematician, so … The term in in any case (as adjective) is also used in various other contexts e.g. for operads, so possibly there needs to be some rationalisation. My thought would be to combine a page on Koszul (and the Wikipedia (English) page on him is poor, and includes some very poor translation from the French ) with a certain amount of disambiguation, however I am not an expert on things ’Koszul’ and this may not be the most efficient way to go.

• I created orbifold groupoids with some classes of groupoids whose elements I like to think of as ‘‘orbifold groupoids‘‘. It would be nice to have a discussion of the interelation of these classes there, too.