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

• I have added to homotopy group a very brief pointer to Mike’s HoTT formalization of $\pi_1(S^1)$.

Eventually I would like to have by default our $n$Lab entries be equipped with detailed pointers to which aspects have been formalized in HoTT (if they have), and in which .v-file precisely.

• I added some first statements about projective resolutions also to projective object.

(the toc is neither meant to already be complete nor to be optimally organized, please expand and polish as you see the need)

• Another new article: sequence space. I await the inevitable report that this term is also used for other things.

• New page: Banach coalgebra.

Hopefully you all know that $l^1$ is a Banach algebra under convolution, but did you know that $l^\infty$ is a Banach coalgebra under nvolution? (Actually, they are both Banach bialgebras!)

• created a little table: chains and cochains - table and included it into the relevant entries (some of which still deserve to be edited quite a bit).

• I have created a table relations - contents and added it as a floatic TOC to the relevant entries.

• I added a few observations under a new section “Results” at bornological set. Bornological sets form a quasitopos. I don’t have a good reference for the theorem of Schanuel.

Related is an observation which hadn’t occurred to me before: the category of sets equipped with a reflexive symmetric relation is a quasitopos. I’d like to return to this sometime in the context of thinking about morphisms of (simple) graphs.

• I have started an entry (∞,n)-category with adjoints, prompted by wanting to record these slides:

• Nick Rozenblyum, Manifolds, Higher Categories and Topological Field Theories, talk Northwestern University (2012) (pdf slides)

If anyone can say more about the result indicated there, I’d be most grateful for a comment.

Also, I seem to hear that at Luminy 2012 there was some extra talk, not appearing on the schedule (maybe by Nick Rozenblyum, but I am not sure) on something related to geometric quantization. If anyone has anything on that, I’d also be most grateful.

• I am starting a table of contents, to be included as a floating TOC for entries related to duality:

duality - contents

But it’s a bit rough for the time being. I haven’t decided yet how to best organize it and I am probably still lacking many items that deserve to be included. To be developed. All input is welcome.

• I started the article Z-infinity-module. Hopefully someone here can say something more interesting about them!

• I'm putting all the big duality theorems from measure theory at Riesz representation theorem. Only a couple are filled in so far, but I'm out of time for today.

• Heya. I haven’t actually made the necessary changes, but the various pages on dependent type theory make the statement that every DTT or MLTT is the internal logic of an LCCC and every LCCC is the categorical semantics of some DTT/MLTT. However, this is extremely confusing (it took me 2 or 3 hours to find a page where it was made completely clear), since it makes explicit use of super-strong extensionality (I think this is called beta-translation), that is to say, it is a theorem about extensional DTTs/MLTTs.

It’s not even totally clear to me that every intensional type theory actually has an (∞,1)-categorical semantics without the consideration of the univalence axiom. I would make this clearer, but I am really out of my depth with type theories, so I’m just alerting you to the fact that this is stated confusingly almost everywhere (the only place where it’s clear is in the page on identity types).

• Disambiguation: dual. Here I listed all of the pages on a kind of dual (but not a kind of duality, which is at duality).

• We ended up with two entries on Paul-André Melliès. One did not have the hyphen. I did a redirect to the older one which has the hyphenation.

• New page: positive cone, including the extended positive cone of a W*-module.

• Wrote Lambert W function. It was an excuse to record Joyal’s proof of Cayley’s theorem on the number of tree structures one can put on an $n$-element set (which is $n^{n-2}$).

• I’ve been inactive here for some months now; I hope this will significantly change soon.

I have written a stubby beginning of iterated monoidal category, with what is admittedly a conjectural definition that aims to be slick. I am curious whether anyone can help me with the following questions:

• Is the definition correct (i.e., does it unpack to the usual definition)? If so, is there a good reference for that fact?

• Assuming the definition is correct, it hinges on the notion of normal lax homomorphism (between pseudomonoids in a 2-category with 2-products). Why the normality?

In other words (again assuming throughout that the definition is correct), it would seem natural to consider the following type of iteration. Start with any 2-category with 2-products $C$, and form a new 2-category with 2-products $Mon(C)$ whose 0-cells are pseudomonoids in $C$, whose 1-cells are lax homomorphisms (with no normality condition, viz. the condition that the lax constraint connecting the units is an isomorphism), and whose 2-cells are lax transformations between lax homomorphisms. Then iterate $Mon(-)$, starting with $C = Cat$. Why isn’t this the “right” notion of iterated monoidal category, or in other words, why do Balteanu, Fiedorowicz, Schwänzel, and Vogt in essence replace $Mon(-)$ with $Mon_{norm}(-)$ (where all the units are forced to coincide up to isomorphism)?

Apologies if these are naive questions; I am not very familiar with the literature.

• I created a stub on excision, but this is just a link to the Wikipedia page for the moment.

• Concrete, abstract: group actions, groups; concrete categories, categories; Cartesian spaces, vector spaces; von Neumann algebras, $W^*$-alebras; material sets, structural sets; etc. At concrete structure.

• as some of you will have seen, I had spent part of the last week with attending talks at String-Math 2012 and posting some notes about these, to the $n$Café (here). For many of these notes I added material to existing $n$Lab entries (mostly just references) or created $n$Lab entries (mostly just stubs).

But since at the same time I was also finalizing the writup of an article as well as doing yet some other things, the whole undertaking was a bit time-pressured. As a result, I decided it would be too much to announce every single $n$Lab edit that I did here on the $n$Forum.

So I ask you for understaning that hereby I just collectively announce these edits here: those who care should please scan through the list of blue links here and see if they spot pointers to $n$Lab entries where they would like to check out the recent edits.

I think I can guarantee, though, that in all cases I did edits that should be entirely uncontroversial, their main defect being that in many cases they leave one wish for more exhaustive discussion.

• I've been meaning to write this for a while. Now I need to look at Bourbaki this weekend to explain their approach.

• Hi guys,

The situation with my habilitation has been resolved.
I decided to postone it to more favourable times.

You can refer to my book and link it.

Best,

Frédéric
• I have created a stub quantum affine algebra as a means to collect some references, alluded to here.

If there is any expert on the matter around, he or she should please feel invited to add an illuminating Idea-section to the entry.

• I created types and calculus and seven trees in one. Both entries as yet contain just references.

It would be nice to have more articles expanding on the reltion of calculus and (higher) category theory /type theory.

• Maybe I am not searching correctly, but it seems to me that until 2 minutes ago the rather remarkable diagram of LCTVS properties was linked to from exactly none non-meta $n$Lab page. It was effectively invisble unless one explicitly searched for “SVG”.

Let me know if there is a reason for it remaining invisible. Assuming that there isn’t, I have now added it to locally convex space and to functional analysis - contents (which I restructured slightly, moving the two such overview diagrams prominently to the top, where they can be recognized as what they are).