The n-Category Café

This looks great, but I wonder if there’s any chance of making the video files available for download rather than just streaming? My effective bandwidth seems to be a factor of 4 too small to receive the stream – and trying to watch a lecture that plays for 5 seconds then halts for 15 is pretty painful – but if I could download the files to play smoothly I wouldn’t care how long that took.

I can stream this fine (from Nebraska, not very far), but I’d still like to download if possible! My Internet connections go in and out, and I have plenty of disk space, so I like to download anything that I’m liable to want to watch again (in case there’s no Internet when I decide to review it).

On a related note, does anybody know how to get Firefox (or Shockwave Flash) to tell me where it’s storing the temporary file behind any given display? I could download all of the Catsters’ videos from YouTube, since I found them in my operating system’s temp folder; other times, I find thing in the browser’s cache folder. But this video I can’t find anywhere!

Congratulations on the video! However I too have been unable to watch very much because of the streaming problem.

Thank you for your efforts on this project. These videos, the catsters, and several others are at the leading edge of these exciting times for undergrad through post-grad level communication.

(“several others” is vague, and I can’t actually think of any others that are this good…)

I have a remark/question/observation on the groupoidification program.

One of the big messages of this program is, I gather, that in order to understand representations well we ought to be looking at the corresponding action groupoids.

So, if a group $G$ acts on a set or space $V$

$$\rho (g):V\to V$$

we’d form the groupoid

$$V//G$$

whose objects are the elements of $V$ and which has all morphisms of the form

$$v\stackrel{\rho (g)}{\to }\rho (g)(v)$$

for all $v\in V$ and $g\in G$.

Now, what is an action groupoid, abstractly speaking? One striking property of $V//G$ is that it is still equipped with an action of $G$:

$$\tilde{\rho }(g):V//G\to V//G$$

These $\tilde{\rho }(g)$ now are functors. This means they can have natural transformations between them.

Once could say that the action groupoid $V//G$ has precisely the right morphisms in order to make all group element actions homotopic.

Namely the action $\tilde{\rho }$ on $V//G$ has the property that

$$\begin{array}{cc}& \nearrow {\searrow }^{\tilde{\rho }(g)}\\ V//G& {\Downarrow }^{\simeq }& V//G\\ & \searrow {\nearrow }_{\tilde{\rho }(g\prime )}\end{array}$$

any two group element actions $\tilde{\rho }(g)$ and $\tilde{\rho }(g\prime )$ are related by a unique natural transformation.

This is of course just another aspect of the statement that the weak quotient $V//G$ is equivalent to the strict quotient $V/G$ (regarded as a discrete category).

But how can we describe the existence of these unique 2-morphisms abstractly?

I believe that one way to do it is this:

let me write

$$\Sigma \mathrm{Aut}(V)$$

for the category which contains the single object $V$ and all its automorphisms. This way our representation is a morphism

$$\rho :\Sigma G\to \Sigma \mathrm{Aut}(V)$$

I am thinking that the action groupoid $V//G$ is the strict pushout (in $2\mathrm{Cat}$) of

$$\begin{array}{ccc}\Sigma G& \hookrightarrow & \Sigma (G//G)\\ {\downarrow }^{\rho }\\ \Sigma \mathrm{Aut}(V)\end{array}$$

in that

$$\begin{array}{ccc}\Sigma G& \hookrightarrow & \Sigma (G//G)\\ {\downarrow }^{\rho }& & \downarrow \\ \Sigma \mathrm{Aut}(V)& \stackrel{\widetilde{(\cdot )}}{\to }& \Sigma \mathrm{Aut}(V//G)\end{array}$$

is the universal strict completion of this cone.

Here’s a comment by Apoorva Khare on the homework exercise in Lecture 1:

Dear Prof. Baez,

Hi, while doing the homework, in order to get the ($q$ or usual) multinomial coefficient — as the answer for the number of $D$-flags on $F_q^n$ (for $q$ a prime power or $q=1$) — I realised:

When you write a $D$-flag on a set — in the form of an UNCOMBED Young diagram, do you want to specify if the integers/subsets in each row are written in INCREASING (or DECREASING) order? Because the subsets

$$X_0\subseteq X_1\subseteq \dots$$

are the only data given in a $D$-flag, i.e. $X_{i+1}-X_i$ is NOT given in a specific order, but just as a set.

The reason this was left out was because you only drew $D$-flags on sets with $n=1+1+\cdots +1$, so this case never arose.

Thanks,
Apoorva.

Here’s my reply:

You just answered your question, but I’ll do it more slowly.

A D-flag on an $n$-element set $X$ is a bunch of nested subsets

$$\varnothing =X_0\subseteq X_1\subseteq \dots \subseteq X_k=X$$

where the cardinality of $X_{i+1}-X_i$ is the number of boxes in the $i$th row of the uncombed Young diagram $D$.

So, for example, if $D$ looks like this:

XX
X

there are 3 $D$-flags on the set $\{\mathrm{1,2,3}\}$, namely:

$$X_1=\{\mathrm{1,2}\}{X}_2=\{\mathrm{1,2,3}\}$$ $$X_1=\{\mathrm{2,3}\}{X}_2=\{\mathrm{1,2,3}\}$$ $$X_3=\{\mathrm{1,3}\}{X}_2=\{\mathrm{1,2,3}\}$$

You’re suggesting that we cleverly keep track of these by putting numbers in the boxes of our Young diagram. We can do that:

12
3

23
1

13
2

Now to the point: the order of the numbers within each row doesn’t matter, since it’s just a notation for a set $X_i-X_{i+1}$. So, we can without loss of generality write them in increasing order, as I’ve done.

Best,
jb

Here’s a comment by Jagannatha Prasad Senesi on Lecture 1:

Hi,

One thing you mentioned on Thursday when you wrote down the vector space ${ℂ}^X$ is that we should not confuse this with the set of functions from $X$ to $ℂ$. But that’s exactly what it is, isn’t it?

In fact if we think of the freely generated vector space ${ℂ}^X$ as ‘Vector space valued functions on $X$’, where the vector space is just $ℂ$, then this begins to sound very similar to the construction of an induced representation (from a subgroup H to a group G), one description of which goes something like…

‘Vector valued functions on $G$ which are $H$-equivariant’.

These induced representations are also freely generated.

-Prasad

Here’s my reply:

There are two different vector spaces, which one should not mix up:

1. The complex vector space of all complex functions on the set $X$.
2. The complex vector space having the set $X$ as basis.

These are canonically isomorphic when $X$ is finite — that’s why it’s tempting to mix them up. But, they’re NOT ISOMORPHIC AT ALL when $X$ is infinite. And, even when $X$ is finite, they’re not naturally isomorphic.

(You can ask Jim about the difference between ‘naturally’ and ‘canonically’.)

The notation $A^B$ usually means ‘all functions from $B$ to $A$’. I warned the class that I’m using ${ℂ}^X$ to mean ‘the complex vector space with $X$ as basis’, not ‘the set of all functions from $X$ to $ℂ$’. But I added that since $X$ for us will often be finite, we can often ignore the difference between these, as long as we keep our wits about us.

Similarly, your description of induced representations is fine when $H$ and $G$ are finite, but potentially problematic otherwise.

I’ll cc this to some other people in the class, since I bet you’re not the only one who was puzzled by my remark.

Best,
jb

Here’s a comment by Chris Rogers on Lecture 1:

Hi Dr. Baez,

I have a quick question: In the quantum case we are talking about Vect, which intuitively seems to have a lot of structure built into it that we care about i.e. structure that is important to quantum mechanics. Aren’t we “cheating” classical mechanics by just treating it as Set, when classical mechanics really lives in the category of symplectic manifolds, which has a lot more relevant structure going on than Set? And if we are talking about group theory in this context, aren’t we eventually going to want to say something about the relationship between canonical transformations (not just functions between sets) and representations of unitary operators? I guess I’m a little confused.

Thanks,
Chris

Here’s my reply:

Aren’t we “cheating” classical mechanics by just treating it as Set, when classical mechanics really lives in the category of symplectic manifolds, which has a lot more relevant structure going on than Set?

Right, definitely. When I wrote “classical” on the board, what I really meant is not so much “classical mechanics” as “classical logic” — i.e., the way we treat Set as the foundations of mathematics. I actually hinted at this, but I didn’t want to make a big deal about it. There’s really too much to say about this…

The category of symplectic manifolds, or even better (maybe) Poisson manifolds, is actually much more like the category of vector spaces than people tend to realize. They’re both non-cartesian - see

Quantum quandaries: a category-theoretic perspective.

for details on what ‘cartesian’ means. The category Set is cartesian. For more on cartesian versus noncartesian categories, try:

Spans in quantum theory.

In any event, what matters most in this seminar is how group actions on sets are related to group representations on vector spaces, and the extent to which we can find a ‘purely combinatorial’ description of portions of quantum mechanics. We’re not really going to talk much about classical mechanics.

Best,
jb

Here’s an article on Vermeer’s camera.

duu ~ nope ~ sorry ~ can’t get lectures here ~ vid or otherwise :( just the pdf’s would be nice ~ for starters at least ~ vids for download would also be best ~ in the interim ~ just drop them on utube ~ that works for some :)

The lecture streams beautifully from Sheffield! And that’s just on bog-standard home broadband. I like the software - click and play, simple and effective. Also, it seems one can instantaneously jump to various times in the video.

Whose satchel was that obscuring the right hand board?

I think we are all sorely missing Derek Wise’s notes!