E.g., in the case of category theory, one will want to talk about universal properties. In the case of product, we know this is right adjoint to duplication.

]]>At natural deduction it says

natural deduction with computation rules gives a formulation of computation. See computational trinitarianism for discussion of this unification of concepts.

But I don’t see anything at these links to help explain the role of the C in FIEC (formation-introduction-elimination-computation). Philosophers call the satisfaction of computation rules ’harmony’, and there’s some reflection on why this is desirable, and of course arguments against.

There should presumably be perspectives on this from the three corners of the trinity of trinitarianism. What can we say from the perspectives of computing and of category theory?

]]>Re: #9. Yes, I completely agree. As mentioned in another thread, I’m working on something which should make this easier.

Re: #10. Yes, from the time the backups began being made. It would need some work to extract data like the author and date of the change, but it could be done.

]]>Is the nlab history included in github backup or only the present version ?

]]>If I were writing to a generic audience the word “trick” would serve a purpose of establishing communication without irritating the audience by a perspective which, even if superior, could cause confusion with the uninitiated.

If I were writing to an audience that I expect to appreciate the foundational role of HoTT, I would try to explain that, far from being a trick, this is a phenomenon fundamental to the meaning of the whole field.

In order to unify these two perspectives, it may be worth recalling that, historically, the word “group” is a shorthand for “symmetry group”, witnessing the original idea that a group necessarily is a group of transformations of something. This original idea, which may seem naïve from the point of view of modern mathematics, finds its re-incarnation, at a higher level of insight, in the fact that $\infty$-groups are equivalently the loop space objects of pointed connected homotopy types.

]]>Right, but what do you think of calling it a “trick”?

If you recall we were hereabouts before when discussing equivalence of physical theories:

]]>Urs: This may seem like a cheap trick, but I actually think this is a useful perspective.

David: That’s the nub of it. What’s the perspective that will continue to hold that this is a cheap trick, whatever you go on to say, because of some principles which, say, would require the expression of that colimit? And conversely, can we understand your perspective to be more than just ’useful’, but getting things ’right’?

Urs: …maybe we learn from it to stick, where they exist, to elementary concepts equivalent to concepts that would need simplicial constructions.

Mike: An interesting point, that perhaps one of the things HoTT (and our current inability to deal with ∞-coherences therein) teaches us is to avoid higher homotopy coherences whenever possible. Of course, now I can hear my advisor saying “we knew that decades ago!”…

That “trick” has been driving much of what we discussed here over the years. In our arXiv:1207.0248 this is highlighted as theorem 2.19, citing Lurie’s lemma 7.2.2.1 in “Higher Topos Theory” and theorem 5.1.3.6 of “Higher Algebra”.

In the base $\infty$-topos $\infty Grpd$ this is a classical theorem, in its simplicial incarnation this is due to Kan, Milnor; it is also a special case of the May recognition principle.

]]>What do people here make of

Note that we have crucially used a trick to study higher groups in HoTT, namely that these can be represented by pointed, connected types?

I see the point that this feature is not available for higher monoids (at least until directed HoTT appears), but it’s surely not merely a piece of luck that there happens to be a convenient way to represent higher groups.

]]>And we still want to know whether something here emerges from the brane bouquet associated to the superpoint $R^{0|3}$.

In case some enthusiastic young person is tuning in, this idea was to do for other superpoints what Urs and John Huerta did for $R^{0|2}$ in M-theory from the Superpoint.

Urs had pointed out that this might be of interest: J. Ambjorn, Y. Watabiki, Creating 3, 4, 6 and 10-dimensional spacetime from W3 symmetry.

]]>That might be interesting.

But, just to highlight, what I am after in #5 above here is crucially not a variant of the splitting principle where we ask whether it generalizes tori to higher tori.

Instead, I am trying to see if the role of approximation of ordinary tori, hence approximation by homotopy 1-types, is a way to understand conceptually what the DF-algebra is doing.

The logic in the supergravity literature going back to D’Auria-Fré 82, section 6 is as follows:

First they show that 11d SuGra is governed by the supergravity Lie 3-algebra, only that that’s not what they really say, since they have no concept of higher (super-)Lie algebra. Accordingly, next they insist that they must force it to become an ordinary super Lie 1-algebra.

If we write

$\mathfrak{m}2\mathfrak{brane}$ for the super Lie 3-algebra, which is the higher central extension of $\mathbb{R}^{10,1\vert\mathbf{32}}$ by the M2-brane 4-cocyle $\mu_{M2} = \tfrac{i}{2} \overline{\psi}\Gamma_{a_1 a_2} \psi \wedge e^{a_1} \wedge e^{a_2}$

$T_{exc,s}\mathbb{R}^{10,1\vert \mathbf{32}}$ for the super Lie 1-algebra whose CE-algebra is the DF-algebra at parameter $s$ (Bandos-Azcarraga-Izquierdo-Picon-Varela 04) (a fermionic extension of the “M-theory super Lie algebra”, but introduced long before the latter got a name)

then what they show is that there is a homomorphism

$\array{ T_{exc,s}\mathbb{R}^{10,1\vert\mathbf{32}} && \overset{comp}{\longrightarrow} && \mathfrak{m}2\mathfrak{brane} \\ & \searrow && \swarrow \\ && \mathbb{R}^{10,1\vert\mathbf{32}} }$such that pullback $comp^\ast$ along it *injects* the degree-3 generator $c \in CE(\mathfrak{m}2\mathfrak{brane})$ which witnesses the higher central extension, in that $d c = \mu_{M2}$.

Moreover, from the details of the construction it seems clear that at $s = -6$ the left hand $T_{exc,s}\mathbb{R}^{10,1\vert\mathbf{32}}$ is the *smallest* super Lie 1-algebra that has this property, though I don’t have a rigorous proof for this.

So, you see, the key point here is that a super Lie 1-algebra, hence from the point of view of rational super homotopy theory a super torus, “approximates” a higher super homotopy type, where the nature of “approximation” might remind one of the splitting principle.

Concretely, there is a 7-cocycle $\tilde \mu_{M5}$ on $\mathfrak{m}2\mathfrak{brane}$, which is also injected into the cohomology of $T_{exc,s}\mathbb{R}^{10,1\vert\mathbf{32}}$ now, under $comp^\ast$, and in the given applications one would really like to have that under $comp^\ast$ the $\tilde \mu_{M5}$-twisted rational cohomology of $\mathfrak{m}2\mathfrak{brane}$ injects into the $comp^\ast(\tilde \mu_{M5})$-twisted rational cohomology of $T_{exc,s}\mathbb{R}^{10,1\vert \mathbf{32}}$.

This really makes the analogy to the standard splitting principle clear, I think. Still, it’s all a bit different, due to the twists, but mostly due to an overall shift of degree as compared to the standard story.

]]>I have added a remark on the relation of the exceptional Jordan algebra to $\mathbb{R}^{10,1\vert \mathbf{16}}$, here

]]>We had a paragraph on split ocotnions buried in the entry *composition algebra*.

In order to be able to link to it, I have given that paragraph its own entry, now *split octonions*. But this deserves to be expanded of course.

I think the category:people label is moderately useful, but other than that I agree that ToCs are probably more useful.

]]>I do wonder whether Ganter’s categorical tori, which sit inside eg the String 2-groups, exhibit a form of the splitting principle

]]>There is a simple argument in Severa 05, p.1 (have added the reference) for the H-cohomology of graded symplectic forms. This should generalize to the case that I need by the double complex spectral sequence. But not tonight…

]]>Thanks for the pointer, I had actually missed both of these passages in the article.

But I am happy to have invited Samuel Monnier to a “Durham Syposium” that we are organizing later this year, “Higher Structures in M-Theory” (not much online at the moment, just the brief item 109 in the table here)

]]>Well it’s all a mystery to me. I dare say the younger ones, like Tachikawa (#17) and Monnier (worked with Gregory Moore at Rutgers, but now seems to be in the maths department at Geneva), will turn first.

Hamiltonian anomalies from extended field theories acknowledges the nLab as

a very useful reference for many of the higher categorical concepts appearing in the present paper,

and the author knows of the larger picture:

]]>Our construction generalizes the construction of the classical Dijkgraaf-Witten theory by Freed… and is strongly inspired by this work. Note that such theories have been constructed using elaborate technology under the name of $\infty$-Chern-Simons theories.

David, just search for “Postnikov” in arXiv:0906.0795, arXiv:hep-th/0701244 and so forth.

]]>You could try a bounty on MO. That seems to motivate some people.

]]>Sorry that the quote issue has not been fixed yet. I am currently working on something more major which I hope will lead to a significant speed up of the nLab, and make it significantly easier to respond to issues/requests. Maybe I’ll have something to show sometime this week or next.

If people think that the quote stuff is very urgent, I can take a look.

]]>It’s clearly due to the same issue that killed our blockquote environments.

But I suggest to everyone *not* to pursue this category:-labelling of entries, but instead adopt my convention, that entries get a floating-TOC to put them into their topic cluster. For two reasons:

1) It is much easier for the reader to *spot* in the first place. I doubt any actual user ever noticed this category-label business.

2) It is much more useful for the reader once spotted, since in the floating TOC we have control over how to organize the information about the related entries, instead of just producing a blind string of keywords

]]>Yes, but John Baez had just been just been explaining about Postnikov data.

Still maybe he’s playing down what he knows.

]]>Why isn’t

category:∞-groupoid

working at the bottom of the page?

]]>It’s getting all the more interesting, in that just three weeks back a new, alternative “splitting principle” of the M5-7-cocyle-twisted cohomology on the M2-brane extension of $\mathbb{R}^{10,1\vert\mathbf{32}}$ was found (not presented in this perspective, of course) in Ravera 18a, with the most curious property that now the super Lie 1-algebra is non-abelian, in fact a super-extension of $Lie(Spin(10,1))$.

This is exactly what I want to see appear in section 2.4 of *From higher to exceptional geometry (schreiber)*