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

• I added the following remark to classifying topos of a localic groupoid.
It would be nice if somebody more competent in this area expanded it.

The above equivalence of categories can in fact be lifted to an equivalence between the bicategory of localic groupoids, complete flat bispaces, and their morphisms and the bicategory of Grothendieck toposes, geometric morphisms, and natural transformations. The equivalence is implemented by the classifying topos functor, as explained in

Ieke Moerdijk, The classifying topos of a continuous groupoid II,
Cahiers de topologie et géométrie différentielle catégoriques 31, no. 2 (1990), 137–168.
• I have cross-linked the two entries homotopical algebra and higher algebra.

At homotopical algebra I moved the text that had existed there into a subsection “History”, because that’s what it is about, right? I added a section “Idea” but so far only included a link to higher algebra there. We could maybe merge the two entries.

• I have been adding to AdS/CFT in the section AdS7 / CFT6 a (of course incomplete) list of available evidence for what is going on.

This is triggered by the fact that we have a proposal for a precise formalization of the effective 7d theory.

• one possible bit of information at flux

• I have added an explanatory paragraph to n-poset in reply to this MO question.

Also, at poset itself I have added a word (“hence”) to indicate that if something is a category with at most one morphism between any ordered pair of objects, then it is already implied that if there are two morphisms back and forth between two objects, then these are equal.

• at n-connected/n-truncated factorization system I have created an Examples-section with a brief indication of what this factorization “means” for low values of $n$ (from $n= -2$ to $n = 0$).

• I plan to redo measurable space, and the outline of the plan is now at the bottom.

For the nonce, I’ve moved some material to a new article sigma-algebra, and some of that thence to the previous stub Borel subset.

• I wrote about the boolean algebra of idempotents in a commutative ring. There’s also stuff in there about projection operators (that page doesn’t exist).

• Tall-Wraith monoid

Updated the reference to "The Hunting of the Hopf Ring" since it's now appeared in print.

• I added a comment to the end of the discussion at predicative mathematics to the effect that free small-colimit completions of toposes are examples of locally cartesian closed pretoposes that are generally not toposes.

• I added the notion of a regular curve to curve. In differential geometry, for most purposes only regular curves are useful: the parametrized smooth curves with never vanishing velocity. Smooth curves as smooth maps from the interval are not of much use without the regularity condition: their image may be far from smooth, with e.g. cusps and clustered sequences of self/intersections.

• Do you have some ideas on how to define a general/higher notion of local Kan extension in an n-category, that gives back the usual notion in a 2-category? I am talking of local kan extension Lan_F G, with F and G two morphisms, that is given by a 2-cell with F and G on the boundary that is universal among such 2-cells.

I would define it using as in the nlab page, the corepresentation of the functor Hom(G,F^*_) but this does not make sense in a weak n-category. I don't want of a Lurie type kan extension given by adjoint to F^*. Want something weaker.

One could also use simply truncation to a 2-category, but is there something finer than that?

The applications i have in mind are related to higher doctrines and theories, derived algebras and their universal properties.

Is there in the litterature something finer than that and useful?
• added stuff to Lie 2-group: more in the Idea-section, more examples, some constructions, plenty of references.

• I am working on further bringing the entry

infinity-Chern-Weil theory introduction

into shape. Now I have spent a bit of time on the (new) subsection that exposes just the standard notion of principal bundles, but in the kind of language (Lie groupoids, anafunctors, etc) that eventually leads over to the description of smooth principal oo-bundles.

I want to ask beta-testers to check this out, and let me know just how dreadful this still is ! ;-) The section I mean is at

Principal n-bundles in low dimension

• I found an interesting question on MO (here) and merrily set out to answer it. The answer got a bit long, so I thought I’d put it here instead. Since I wrote it in LaTeX with the intention of converting it to a suitable format for MO, it was simplicity itself to convert it instead to something suitable for the nLab.

The style is perhaps not quite right for the nLab, but I can polish that. As I said, the original intention was to post it there so I started writing it with that in mind. I’ll polish it up and add in more links in due course.

The page is at: on the manifold structure of singular loops, though I’m not sure that that’s an appropriate title! At the very least, it ought to have a subtitle: “or the lack of it”.

• I’ve written path and loop, in order to record the misconception in the last paragraph of the definition of the latter. Along the way, I noticed that the graph-theoretic concepts are special cases of the topological ones, so enjoy.

• I created Frobenius map, since I had linked to it in several places.

• On well-order and elsewhere, I’ve implied that a well-order (a well-founded, extensional, transitive relation) must be connected (and thus a linear order). But this is not correct; or at least I can’t prove it, and I’ve read a few places claiming that well-orders need not be linear. So I fixed well-order, although the claim may still be on the Lab somewhere else.

Of course, all of this is in the context of constructive mathematics; with excluded middle, the claim is actually true. I also rewrote the discussion of classical alternatives at well-order to show more popular equivalents.

• Created factorization structure for sinks, and added remarks to Grothendieck fibration, M-complete category, and topological concrete category about constructing them. Largely I wanted to record the proof of the theorem that in an factorization structure for sinks $(E,M)$, the class $M$ necessarily consists of monomorphisms. It’s a nice generalization of Freyd’s theorem about complete small categories, which has more of the feel of a useful theorem than of a no-go theorem like Freyd’s.

At first I thought that the lifting of factorization structures described at topological concrete category would work for solid functors too, but then I couldn’t see how to do it. Does anyone?

• [Jim Stasheff means to post the following message here to the $n$Forum, but accidentally (I think) posted it here instead (I guess because that is the forum that comes up when one googles for “nForum”)]:

Solutions of the KZ equations are usually given in terms of assymptotic behavior in certain regions. Since the region on which the eqns are defined has a nice compactication, what is the obstruction to extending the solutions to the compactification?

• I started a stub

symmetric space

and added a bit of overlapping material to quandle. I would like to talk about Lie and Jordan triple systems, but I need this introductory material first.

• created

(to go with the discussion at ∞-gerbe).

There are other notions of “center” of $\infty$-groups. But this is one of them.

• I have created lax morphism, with general definitions and a list of examples. It would be great to have more examples.

• I made an initial foray into explaining the coalgebraic aspects of recursion schemes (following Taylor) by editing well-founded relation, by including a new section “Coalgebraic formulation”. (The title is slightly awkward when it appears just after the section “Alternative formulations”; that section was on alternative formulations which are possible in classical logic, whereas this section is on a different language for presenting the intuitionistic case. Therefore I didn’t want to make it a subsection of “Alternative formulations” as currently written.)

Also some words there on the coalgebraic formulation of simulations.

Edit: I decided to rename “Alternative formulations” by “Formulations in classical logic”; I hope no one minds.