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

• 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).

• Danny Stevenson was so kind and completed spelling out the proof of the pasting law for $\infty$-pullbacks here at (infinity,1)-pullback.

• I created a stub for Kirchhoff’s laws to go with the $n$Café-discussion here. Maybe somebody feels like expanding it, I don’t really have the time for this right now.

• I wrote Hamiltonian action.

I tried to say precisely what the action is by. In the literature (but also in a previous version of our moment map entry) there is often (for instance on Wikipedia, but also in many other sources) an imprecise (not to say: wrong) statement, where an action by Hamiltonian vector fields is not distinguished from one by Hamiltonians.

• I have decided to splitt off a stand-alone entry symplectic reduction from BRST-BV formalism (which used to be the redirect). Still just a stub. Lots of material and references still needs to be copied or moved from the latter to the former.

• I have started a table geometric quantization - contents and added it as a floating TOC to the relevant entries.

Parts of this remain a bit unfinished. The $n$Lab is pretty much unusable in the last hours. I’ll give up now, have wasted too much time with this already. Maybe later it has recovered.

• I have created a table geometric quantization extensions - table.

Mostly I have been editing aspects of the entries listed in this table here and there. Also included the table in the Properties-section of various of these entries.

But I ran out of steam before being entirely satisfied with the result.

• I realized that infinite-dimensional manifold and all entries related to it are still very stubby.

I am not attempting to change this now, but I thought a first step to make progress is to list what stubs we actually have. I came up with this list

and added it to manifolds and cobordisms - contents and made sure that all these entries point to each other.

Now somebody go and add more content to these entries! :-)

• In case you see the activity in the logs and are wondering, I should say that I have been working on a new entry higher geometric quantization (that used to redirect to n-plectic infinity-groupoid).

I have started adding some survey-tables. But not done yet with the entry as a whole.

• I felt it was time for another table: homotopy-homology-cohomology

The structure is just a first attempt, begun in a brief moment of leisure. I’ll try to think about how to improve on it. Let me know what you think.

I have started to include this into relevant entries.

• started an entry associated infinity-bundle

in order to summarize the thesis by Matthias Wendt on associated $\infty$-bundles in arbitrary $(\infty,1)$-toposes, generalizing the classical old results by Stasheff and May from $\infty Grpd$.

Also added some remarks on the relation to the discussion at principal infinity-bundle. Hopefully to be continued tomorrow.

• I am now going through the section Structures in a cohesive oo-topos and polish and expand the discussions there.

First thing I went through is the subsection Geometric homotopy and Galois theory. It gives the definition of the fundamental $\infty$-groupoid functor, a proposition on its consistency (which we had mentioned elsewhere), the definition of locally constant $\infty$-stacks in the sense of $Disc Aut(F)$-principal $\infty$-bundles, and then the central theorem of Galois theory, proven by applying the $\infty$-Yoneda lemma iteratively.

(This is material appearing in one form or other in other entries and at this point does not invoke the $\infty$-locality, but I want to have here all in one place a nice comprehensive discussion of the whole situation in a cohesive $\infty$-topos.)

• Hi guys,

I suppressed the reference to my course on global analytic geometry. These notes were not well written enough and i put them into the basket. Please, don't pull back the reference.

Cheers,

Fred
• I created minimal fibration which could be merged with minimal Kan fibration. The idea-section says that this notion is needed to give a well defined notion of n-category. However there are other applications which I didn’t mention.

• I made redirects to Online Resources, namely the math blogs, online resources. Before we were complaining to Online Resources for many reasons including that it is not of all resources but only of blogs and wikis in relevant areas. No list of main institutes and archives like arXiv, numdam, jstor etc. there. As the list is long, and hard to scroll, I suggest not to add those to the current page. I think we should rename the current page to math blogs eventually and keep Online Resources (especially because of John's reference in his AMS Notices paper) as a redirect and create new pages for other stuff as well as organize the whole system around a top page math resources which will link to math blogs, math archives, math institutions (and maybe more) as well as very comprehensive central AMS-kept list of math resources.

I know it is not only about math here, but math is a short abbreviation for page name.

Up to now I have realized a large part of an above program, see math archives, math institutions and the supposed top resource page math resources, except that I was cautious not to rename the page Online Resources as people may disagree even with keeping the old redirect and because it may be tricky with the cache bug, while the page is of central importance. I think it would be useful if the pages like math institutions and the top page math resources stay not much longer than they are now, to have quick links and nice readability/visibility. This is the most effective organization, I think. For smaller institutions societies and alternative small lists of resources, it is better to go via links at AMS, EMS and IMU which are already efefctively linked. We can not do better there than those societies do, apart from listing few extra main resources of our main interest. We can have a separate page just for categories or some other things. But the list of blogs is of different character, unlike going to AMS page or jstor, one does not need to go that quickly through list of less-organized stuff like blogs. So the blog list math blogs should grow indefinitely...I have chosen plural as before in these pages, without singular redirect at the moment.

Maria Emilia Maietti, Modular correspondence between dependent type theories and categories including pretopoi and topoi, Math. Struct. in Comp. Science (2005), vol. 15, pp. 1089–1149

• I have created energy ex nihilo. Take that, Hermann von Helmholtz!

• I may have written something at Kervaire invariant, but it is at best a stub for the moment

• Some reorganization and added material at type theory. In particular, I added some of the basic syntax of type theories, and also some comments about extensional vs. intensional type theories.

• We already have the entry predicate calculus (or first-order logic); I have created separate stub first-order theory, almost empty now, and which could just have been redirect to first-order logic, though I think eventually it would be good to have them separate, as under first-order theories version one could list lots of standard examples of first-order theories, what does not really fit into predicate logic entry, where one should really deal more with predicate calculus. PLus one should do other views of first-order theories.

• Stub constructible set, not yet precise (e.g. the universe is not a Boolean algebra as it is a proper class), but gives idea what we could work on in the entry.

• New entry structure but the $n$Lab is down so I save here the final version of editing, which is probably lost in $n$Lab:

The concept of a structure is formulated as the basic object of mathematics in the work of Bourbaki.

In model theory, a structure of a language $L$ is the same as model of $L$ with empty set of extra axioms. Given a first-order language $L$, which consists of symbols (variable symbols, constant symbols, function symbols and relation symbols including $\epsilon$) and quantifiers; a structure for $L$, or $L$-structure is a set $M$ with an interpretation for symbols:

• if $R\in L$ is an $n$-ary relation symbol, then its interpretation $R^M\subset M^n$

• if $f\in L$ is an $n$-ary function symbol, then $f^M:M^n\to M$ is a function

• if $c\in L$ is a constant symbol, then $c^M\in M$

Interpretation for an $L$-structure inductively defines an interpretation for well-formed formulas in $L$. We say that a sentence $\phi\in L$ is true in $M$ if $\phi^M$ is true. Given a theory $(L,T)$, which is a language $L$ together with a given set $T$ of sentences in $L$, the interpretation in a structure $M$ makes those sentences true or false; if all the sentences in $T$ are true in $M$ we say that $M$ is a model of $(L,T)$.

Some special cases include algebraic structures, which is usually defined as a structure for a first order language with equality and $\epsilon$-relation both with the standard interpretation, no other relation symbols and whose function symbols are interpreted as operations of various arity. This is a bit more general than an algebraic theory as in the latter, one needs to have free algebras so for example fields do not form an algebraic theory but are the algebraic structures for the theory of fields.

In category theory we may talk about functor forgetting structure (formalizing an intuitive, related and in a way more general sense), see

!redirects structures

• I added in the definition of algebraic group the requirement ”field” into ”algebraically closed field”. Alternatively one could omit ”field” in the definition at all since this is implicit in ”variety”.