The n-Category Café

Jamie wrote:

I always had the impression that the free bialgebra on a vector space does always exist! After all, the free monoid on a vector space certainly exists, …

That’s free as an algebra, which already hints that it’s not free as a bialgebra, since a bialgebra has extra structure not determined by the algebra structure.

… and that has a canonical bialgebra structure, so there’s an obvious functor from a category of vector spaces to its category of internal bialgebras and bialgebra homomorphisms.

Yup, but being ‘obvious’ doth not make the functor ‘free’.

Is the idea that there’s no adjoint to the forgetful functor from the category of internal bialgebras to the underlying category of vector spaces?

More precisely, no left adjoint — being a left adjoint is what makes a functor count as ‘free’.

Check out the material starting around page 3 of the paper by Mélliès and Tabareau — it’s pretty darn readable.

Thanks for those comments, Todd and John. From Todd’s insight, it follows that for any algebraic structure containing at least a unit and a counit, and where morphisms require the unit and counit to be preserved, there can be no free functor unless the tensor unit in the underlying category is the zero object. An important class of monoidal categories where this is impossible are the nontrivial rigid monoidal categories. This is good to know!

John said that being a left adjoint is what makes a functor count as ‘free’. Surely this is too strong a statement — doesn’t it only apply when our structure is ‘monoid-like’? For example, the free functor to the category of comonoids in Vect${}_k$ is right-adjoint to the forgetful functor, not left-adjoint. Using this logic, when our structure is both monoid- and comonoid-like, I suppose we would expect the free and forgetful functors to be ambidextrous adjoints.

I’ve added a bit more on how Mélliès and Tabareau think about constructing free algebras of T-algebraic theories. For those unwilling to reread week276 and look for the new stuff, here it is:

Suppose we have any map of algebraic theories

Q: B → C

that is, a finite-product-preserving functor that sends the special object b in B to the special object c in C. Then composition with Q gives a functor

Q*: Mod(C,X) → Mod(B,X)

For example, if B is the theory of groups and C is the theory of rings, C is “bigger”, so we get an inclusion

Q: B → C

and then Q* is the functor that takes a ring object in X and forgets some of its structure, leaving us a group object in X.

It’s interesting to ask when Q* has a left adjoint. And the answer is: it always does!

The proof uses a left Kan extension followed by what Mélliès and Tabaraeu call a “miracle” - see page 5 of their paper. And, it’s this miracle they want to understand and generalize.

Here’s the basic idea. If we write Hom(C,X) for the category with

• arbitrary functors F: C → X as its objects;
• natural transformations between these as its morphisms

then composition with Q gives a functor

Hom(C,X) → Hom(B,X)

and this has a left adjoint

Hom(B,X) → Hom(C,X)

thanks to a well-known trick called “Kan extension”, or more precisely “left Kan extension”. Since Mod(B,X) is included in Hom(B,X), we can compose this inclusion with the functor above:

Mod(B,X) → Hom(B,X) → Hom(C,X)

And now comes the miracle: this composite functor actually lands us in Mod(C,X), which sits inside Hom(C,X). This gives us a functor

Mod(B,X) → Mod(C,X)

which turns out to be what we wanted: the left adjoint of

Q*: Mod(C,X) → Mod(B,X)

Kan extensions are a very general concept, so the hard part is understanding and generalizing this miracle.

They generalize algebraic theories to “T-algebraic theories” for any pseudomonad T on Cat. And then, they generalize the “miracle” to any situation where we have a little T-algebraic theory sitting inside a bigger one

Q: B → C

and the bigger one only has extra operations, not co-operations.

Of course I said “the major contributors include” because I didn’t claim to give a complete list.

Yes, Zoran Škoda has been contributing plenty of entries and material, and is trying to popularize the idea of contributing among the visitors at the Max-Planck institute in Bonn; Ronnie Brown is serving as a constant source of references and further pointers in the background and Eric Forgy has put effort into connecting abstract entries to earth.

We are happy about every new contributor.

Last week I was talking with Bruce Bartlett about why, even though things are going well, we don’t have even more contributions yet. Bruce said: “Don’t forget, it’s lot of work to contribute to the nLab”.

I would like to dispel this idea that the $n$Lab is extra work. Instead, it is a means to reduce work.

Don’t add anything that would distract you from your daily work. Instead, use it to do your daily work.

Every day, I think each of us comes along some fact or insight or reference that looks worth remembering. Sometimes one makes a note about this in some notebook, only to lose that later. Or one just tells a colleague, who forgets it later. Or one starts typing a short note, which remains incomplete.

In all these cases, just open the $n$Lab and feed the material in there. It may well be just a short note first. It will be fun to come back to it later and see how it has grown.

Or, if you give a lecture or some other class and think about typing up your notes on this. Think about using the $n$Lab as your notepad.

Andrew Stacey suggested that more people might feel like contributing to the $n$Lab if it were more transparent who is “in charge of it”, so that it be clearer what may happen to the material one contributes in the long run. Together we decided to take some action to improve on this siuation. I’ll post a link soon.

There’s an upper bound on the luminosity of a star: if they get too bright, their own radiation pressure blows them apart.

But this maximum luminosity, called the ‘Eddington luminosity’, seems to scale linearly with the mass. I don’t know why there’s an upper bound on the mass. There seems to be one, of roughly 120 solar masses.

A great example of a star close to the brink is Eta Carinae. It’s mass is about 100 times that of our Sun, and its luminosity is more than a million times that of our Sun.

I’m pretty sure that when people talk about the size of the star, they refer to the volume in which radiation is in thermal equilibrium with the matter. That is, the variation of temperature over the mean free path of a photon is small. This is not the case for nebulae, as far as I understand.

Let me try a couple of calculations. I don’t know whether Betelgeuse is scheduled to become a type I or type II supernova, but never mind… maybe you can check me on this number crunching.

Very roughly, a supernova is 10 billion times more luminous than the Sun. Say Betelgeuse is 600 light years away. That’s about 38 million AU. If Betelgeuse is 38 million times farther than the Sun, it’s dimmed by a factor of 38 million squared, or about $1.4 \cdot 10^{15}$. So, its apparent luminosity as a supernova will be about

$$10^{10} / 10^{15} = 10^{-5}$$

times that of the Sun. Okay: the Sun will look 100,000 times brighter.

At this point the silly astronomical ‘magnitude’ system may come in handy. It’s logarithmic: a factor of 10 luminosity corresponds to roughly -2.5 ‘magnitude’. So, the apparent magnitude of Betelgeuse gone supernova would be

$$5 \times 2.5 = 12.5$$

more than that of the Sun. The advantage of knowing this is that we can consult a chart. The Sun has magnitude -26.7, the full Moon has magnitude -12.6; supernova Betelgeuse would have magnitude roughly -14.2. Hmm, that’s just a bit brighter than the Moon, not ‘brighter than a million moons’. Did I screw up?

Let me try again, avoiding this ‘magnitude’ stuff. From that chart I see the Sun is 450,000 times brighter than the full Moon. If the Sun is 100,000 times brighter than supernova Betelgeuse, that means this supernova will only be 4.5 times brighter than the Moon.

Of course it will be essentially a point source, so it might still damage your eyes to stare at it. Dunno.

I was surprised that you omitted this. Not that I dislike the Clarke story. Larry Niven attended my alma mater Caltech for some time. As fact-checked at wikipedia:

“Inconstant Moon” is a science fiction short story by American author Larry Niven that was published in 1971. Inconstant Moon is 1973 anthology of Larry Niven’s short stories that includes the title piece. The title is a quote from the balcony scene in William Shakespeare’s Romeo and Juliet. The anthology was assembled from the US collections The Shape of Space and All the Myriad Ways. The short story won the 1972 Hugo Award for best short story.”

“First appearance: 1971 short story collection All the Myriad Ways. Stan, the narrator, notices that the moon is glowing much brighter than ever before. The people he meets as the story begins all praise the moon’s increased beauty but lack the scientific background to understand its cause. However the narrator surmises that the Sun has gone nova, the day side of the Earth is already destroyed, and this is the last night of his life. He then calls and visits his girlfriend Leslie, presuming her ignorant of the situation, but she realizes it independently when Jupiter brightens with appropriate delay; they then enjoy their last night on the town, before rain and winds start.”

“Later, he realizes one other possibility. In case he is right, they find appropriate supplies and seek refuge from the coming natural disasters in Leslie’s high-rise apartment. The second possibility turns out to be correct: the Earth has merely been struck by an enormous solar flare. The vaporized seawater leads to torrential rains, hurricanes and floods. Most (if not all) people on the Eastern Hemisphere are presumed dead. The story ends at the break of an overcast, gray morning, with the narrator “wonder[ing] if our children would colonize Europe, or Asia, or Africa”.

“In 1996, the story was made into an episode of the Outer Limits television series with Niven himself writing the script - see Inconstant Moon (The Outer Limits). Jo Walton in 1997 wrote a short poem, ‘The End of the World in Duxford’ as ‘an unauthorised version of Inconstant Moon, a British equivalent….’”

Let me try doing the calculation a different way, to check my work. According to Michael Redmond’s page, a type Ia supernova would be as bright as the Sun if it were .3 parsecs away, while a type II would be as bright as the Sun if it were .25 parsecs away.

I don’t know which type of supernova Betelgeuse is likely to become. (If anyone does, let me know, and say why!) So, let’s split the difference and say .25 parsecs.

Betelgeuse is about 600 light years away. A parsec is about 3.26 light years, so that’s about 180 parsecs.

So, Betelgeuse is about $180/.25 = 720$ times the distance it would need to be as bright as the Sun when it goes supernova. So, supernova Betelgeuse will be about

$$\frac{1}{720^2} \sim \frac{1}{500,000}$$

as bright as the Sun.

As mentioned, a full moon is $1/450,000$ as bright as the Sun. So, it seems that supernova Betelgeuse will be just a little brighter than a full moon.

Given how rough my calculations are, this counts as consistent with my previous estimate that supernova Betelgeuse will be 4 times brighter than a full moon. Same order of magnitude. Nothing like ‘brighter than a million full moons’.

This slide is commonly used to make star size comparisons on You Tube. But it got measurements wrong, for example, Betelgeuse should be half the diameter of the biggest star, but it is much smaller. The stars of the last slide, with the biggest stars, should show all of them with aproximately the same size.
And they put the homunculus nebula, that evolves Eta Carinae as it were the star itself…

Here is a list of the largest known stars.

Daniel wrote:

This slide is commonly used to make star size comparisons on You Tube. But it got measurements wrong, for example, Betelgeuse should be half the diameter of the biggest star, but it is much smaller.

Hmm, it looks like you’re right! I’ll remove that slide from week276.

Here’s a picture which shows the enormous plume coming out of Betelgeuse:

It was taken at the Very Large Telescope or ‘VLT’ in Chile. The VLT is very cool.