Klein 2-Geometry VI

October 1, 2006

Posted by David Corfield

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It’s time to convene the sixth monthly session of the longest running Pro-Am math event in the blogosphere. The September session won’t go down in history as one of the most productive of the series. The Professional was justifiably distracted. To recap on the last steps taken. John asked us to

guess the precise description of the projective space associated to $k^{p, q}$ .

The answer to this was: ${kP}^{p - 1}$ worth of objects each with $k^{q}$ worth of automorphisms.

John then asked for a description of all categorified Grassmanians. I attempted an answer, raised a concern, and posed a question.

Posted at October 1, 2006 10:22 AM UTC

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Re: Klein 2-Geometry VI

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I’m afraid I’m still too distracted to push this Klein 2-geometry project forwards very quickly. But let me give it a nudge.

David wrote:

So the 2-space of $(n,0)$ sub-2-spaces of $k^{p, q}$ has the Grassmanian $G_{n, p}$ as objects and linear maps from $k^{n}$ to $k^{q}$ as automorphisms.

That sounds right to me. We’re going by the seat of our pants here, but I think it all works out quite simply.

In fact my main worry now is that it’s too simple to be very interesting! We’ll need to ask some tougher questions about these Grassmannian to see if they have hidden depths. For example, let’s study the “incidence relations” between the $(n,0)$ sub-2-spaces and $(m,0)$ sub-2-spaces of $k^{p, q}$ . Then we’ll finally be doing categorified incidence geometry.

Of course we’re using enough jargon and symbols now that nobody except us can possibly understand what we’re talking about. So, my claim that it’s “too simple” may strike them as unconvincing.

But maybe that’s how math advances: when you’re completely stuck and a problem seems incomprehensible, lots of people can understand what you’re whining about. But when you finally crack the problem and it seems completely lucid, nobody can understand you anymore!

To reach understanding it seems one must suffer confusion, even when there’s somebody standing there telling you anything you want to know. The neurons must be reconfigured.

Posted by: John Baez on October 2, 2006 7:28 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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If you had a moment to check on my worry. A subspace is surely a class of inclusions.

Thinking in concrete terms, we may as well look at the 2-spaces of (1,0) and (2,0) subspaces of $k^{3,1}$ . If we’re right, then in the second case we have $G_{2,3}$ worth of objects, with linear maps from $k^{2}$ to $k^{1}$ worth of automorphisms. That’s $k^{2}$ worth of automorphisms for each object. How should we picture chain maps?

Posted by: David Corfield on October 3, 2006 8:26 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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I think perhaps this worry you had is just part of the loosening of thinghood that goes with categorification. A “point” is no longer a thing, but instead a set or possibly a groupoid of ways of being that thing (up to isomorphism).

Does this make sense? Is this what you meant by a “class of inclusions”?

Posted by: Tim Silverman on October 6, 2006 12:17 AM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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I’ve been running hard to try and catch up with this discussion. Gosh, there’s a lot of material! But some of it is a record of confusion and clarification rather than finished new stuff, and it might confuse me to spend too long on it. Therefore, in an effort to avoid absorbing the whole lot, I want to try and take a shortcut, by taking some big intuitive leaps to get quickly to the point where I really need to start to actually think.

I think the trick with this sort of learning is to start with what one already understands and expand from there. With this in mind, and with your permission, I want to take a brief diversion, and momentarily go back to the last thing I think I thoroughly understand, namely the automorphism 2-group of the groupoid $S_{3} / / A_{3}$ (the pair-of-triangles). And I want to think about spaces and their figures in the way I’m most familiar with, namely as collections of points, and generalise from there. Since emphasising points as concrete fundamental entities is kind of contrary to the spirit of Klein, I’ll call this exercise “semi-Klein geometry”. Craving your lordships’ indulgence … hopefullly this won’t take too long … ulp! …. here goes …

So I see the pair-of-triangles as a space of two points, each of which is internally a triangle. Symmetries let us permute the points arbitrarily and map their internal structures isomorphically or (in the weak case) by any equivalence. 2-symmetries relate symmetries which permute the points the same way but differ by internal automorphisms or (in the weak case) internal auto-equivalences.

So generalising this as much as possible without actually thinking, we can add more points, we can cut down on the number of ways we’re allowed to permute them, we can give them more complicated internal structures (i.e. bigger/smaller/different internal automorphism/auto-equivalence groups/2-groups), and we can give different points within the space different internal structures from each other.

To express this more categorificationally:

With semi-Klein 1-geometry, we have a set of points and a group of symmetries on them. With semi-Klein 2-geometry, we get a category of points or (in the weak case) a 2-category of points, and a 2-group of symmetries and 2-symmetries on them. A symmetry of the space can map a given point to another given point only if the two points are isomorphic to each other or (in the weak case) equivalent; and the mapping must be an isomorphism or (in the weak case) an equivalence. I.e. we not only care which points go where but also how they get there (or at least what they look like when they arrive). We also have 2-symmetries which relate certain symmetries to each other, namely those symmetries that agree on which points get mapped to which, but differ by internal automorphisms/equivalences of the points.

I guess there are various uses we could put this concept of 2-symmetry to, but one way, perhaps, is to think of the 2-symmetries as gauge transformations. Then symmetries are only defined as fully distinct entities “up to gauge transformations”.

Does any of this make any sense so far? I’m kind of worried I don’t really understand how the strict/weak distinction affects things, or not as well as I should. Plus no doubt lots of other things.

But assuming we can still keep going down this path, we can think about cutting down the symmetries and 2-symmetries to a sub-2-group of the full 2-group of the space. In a spirit of wild reckless abandon, i.e. continuing to avoid actual thinking, I’ll try to make this as intuitively “injective” as possible by simply throwing away morphisms and 2-morphisms and hoping we can tidy it all up later.

So we can

a) Cut down on which points can go to which points, possibly foliating the space into subspaces of some kind (?)
b) Cut down on the internal symmetries of (some of) the points, thereby
   i) Reducing the number of ways to get from one point to another, and/or
   ii) Making the space of points of different kinds fall apart (or further apart) into more isomorphism/equivalence classes of different kinds of points, and of course
   iii) Adding extra structure to the points themselves

and also

c) by cutting down on the 2-symmetries, we can subdivide the internal symmetries into more kinds.
This will also affect whether and how we’re allowed to map between points.

I’m pretty sure I’m being somewhat “evil” here, at least in spirit, but I’m still hoping for “redemption” later, in the tidying-up phase. So to press on…

Now, since I’m thinking of the whole space as made of points, I want to think of figures in a pointy way too.

Figures in this semi-Klein 1-geometry I’ve been talking about are not the same as subgroups of the space’s symmetry group, but are made of points, namely the points in the orbit of a given point, under the appropriate subgroup. So in semi-Klein 2-geometry, we want to have figures which are “2-orbits” of points under sub-2-groups. In this case, a figure will be not a set of points, but a category of points or (in the weak case) a 2-category of points. This is because we care not only which points are in the figure, but also how we are allowed to get from one to another (via symmetries in the sub-2-group) and also which internal symmetries are isomorphic under (the remaining) 2-symmetries in the sub-2-group.

Then an incidence relation between figures becomes, depending on how virtuous we want to be, either a functor, or a set of functors, or a category of “functors together with natural transformations (or natural isomorphisms?) between them”.

And this makes sense because, in the full Klein geometry approach, the figures are sub-2-groups, and incidence relations are (closely related to?) inclusion relations among sub-2-groups, which we already know are functors, sets of functors or categories of functors. We are now in a position to analyse what we mean by “2-orbits” given the various kinds of “injectivity” and “surjectivity” conditions we want to impose on the functors, and given the ways we want functors to be naturally isomorphic or not. However, this requires me to think, so I will stop here.

Does any of this make any sense at all? (Or is it too mind-numbingly obvious? That would also be bad.)

The 2-Euclidean spaces that David was thinking about right at the beginning, where we make the base space $ℝ^{2}$ and the internal symmetries O(2) (or, more generally, the base space $ℝ^{n}$ and the internal symmetries O(n)), seem quite restrictive when viewed through this lens. The reason is that the internal space of a point contains all the information about the global geometry that isn’t coded in the location of the point. That is, to specify a symmetry of the space, we simply need one point (any point) and a morphism from that point. Is this true? Maybe the 2-symmetries allow a bit more wiggle-room. I guess it’s not surprising that a space specified as the quotient of two low-dimensional groups should be quite restrictive, and then mechanical categorification in the manner of going to the weak quotient doesn’t loosen things up much. Actually, that’s really mere oidisation rather than categorification in its full power and glory, so maybe there’s more to be done with these spaces (or spaceoids!)

Hmm, well, quickly looking at the posts on 2-vector spaces in the light of what I’ve just said, I think one way I may have been “evil” is by talking about individual points in the space as though they were well-defined entities, when maybe they should only be defined up to isomorphism or something. I suppose wondering if this is so might indicate progress of some sort in my understanding.

OK. I think I’ve now caught up and am ready to begin thinking.

We return you to your usual service.
Apologies for the interruption.

Posted by: Tim Silverman on October 5, 2006 10:50 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

Gosh, there’s a lot of material! But some of it is a record of confusion and clarification rather than finished new stuff, and it might confuse me to spend too long on it

It’s clearly not designed for pedagogy. There’s something I like, though, about leaving a trace of mathematical thinking as it happened rather than all tidied up.

John’s attention and mine seem to have moved elsewhere. It’s hard to predict when and in which direction enthusiasm will propel you. That’s not to say we didn’t find out some good things. There must be treasures to be had in forming weak quotients of 2-groups by their sub-2-groups. Perhaps when Derek Wise publishes his work on Cartan geometry, it’ll be an opportunity to take things up again.

Posted by: David Corfield on October 7, 2006 12:03 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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David said

John’s attention and mine seem to have moved elsewhere. It’s hard to predict when and in which direction enthusiasm will propel you.

It’s generally a safe bet that my enthusiasm will take me in the direction that everybody else has just abandoned. :-(

Since I’m there, however, I suppose I might as well put down my thoughts on 2-Grassmannians, which might be of some use to someone. Well, I think there is something of interest here, even though only half-digested.

In an effort to understand what we are doing here, I want to pick something even more concrete to work with. Since we are dealing with abstract chain complexes, why not draw pictures of some geometry giving rise to concrete chain complexes? For instance, consider the 1-dimensional cell complex below, let’s call it C:

$a \to b$
$↑ ↓$
$c \leftarrow d$

The 0-chains have an obvious basis of vertices $a$ , $b$ , $c$ and $d$ , and the 1-chains have a basis of oriented edges $(a \to b)$ , $(b \to c)$ , $(c \to d)$ and $(d \to a)$ . We’ll use the boundary operator to get from 1-chains to 0-chains. Let’s call the resulting chain complex C (with the bolding removed). Then both $C_{0}$ and $C_{1}$ are 4-dimensional. The kernel of the boundary operator is 3-dimensional. We’ll call the underlying field $k$ .

To get a 2-vector space V, we take $V_{0}$ to be $C_{0}$ , and $V_{1}$ to be $C_{0} + C_{1}$ , i.e. the space with a nice basis of ordered pairs $(x, y \to z)$ , $x$ being a vertex and $y \to z$ being an edge.

Then the identity map sends $x$ to $(x, 0)$
The source map sends $(x, y \to z)$ to $x$
The target map sends $(x, y \to z)$ to $z - y$

If we ask what the morphism $(x, y \to z)$ “does”, we find that it sends

$x \to x + \partial (y \to z) = x - y + z$ .

In particular, the target of $(x, x \to y)$ is $y$ , so the edge ‘acts’ as a morphism in the obvious way in this case.

What’s the dimension of this thing? I think that it’s (1, 1). That’s if I’ve understood this whole dimension business correctly.

Now we can consider chain maps, particularly inclusions.

First, consider the cell complex consisting of a single point; let’s call this cell complex P (and the single point p).

$. p$

The chain complex over this has dimension (1, 0). Let’s call it P.

Clearly maps from the chain complex P to the chain complex C comprise the vector space of linear maps whose basis is the four maps from the single vertex of P to each of the four vertices $a$ , $b$ , $c$ and $d$ of C.

This gives us a four-dimensional vector space of points.

However, this is before we mod out by the automorphisms of P, which reduces us to the projective space ${kP}^{3}$ , i.e. the Grassmannian $G_{1,4}$ .

However, we still haven’t reduced enough, because we haven’t thought about chain homotopies!

Note that internally in C, the points are all homotopic to one another, via the edges of C. This is because (or is another way of saying that) our cell complex has one component.

Basically, I think every map from $P_{0}$ to $C_{1}$ gives a chain homotopy. After modding out by automorphisms of $P_{0}$ , I think what we’re left with is that all projective transformations of k $P^{3}$ are chain homotopies, i.e. autoequivalences. I think this gives us a space of just 1 point, with a lot of automorphisms, corresponding to the fact that our original cell complex C has just one component.

This is looking increasingly like the example with the automorphisms of $S_{3} / / A_{3}$ , except with a more general 1-cell complex instead of a groupoid.

Anyway, onwards and upwards! What can we do to make this more complicated?

OK, so let’s take the cell complex Px2 consisting of two isolated points, called p and q, and no edges.

$. p$ $. q$

The dimension of its complex Px2 is (2, 0). If we want the map on 0-chains to be injective, then we have to map the two points of Px2 to separate points in C. So we get a 6-dimensional space of maps. Modding out by automorphisms of $Px 2_{0}$ , we get a 4-dimensional space, basically the Grassmannian $G_{2,4}$ in $C_{0}$ . Note that these automorphisms include ones related to exchanging p and q, since p and q are themselves isomorphic to each other.

Looking at the graph homotopies allowed on 2 points, and allowing only injective ones, we basically get a space of chain homotopies on the chain maps $Px 2 \to C$ which is whatever the equivalent of projective transformations is for more general Grassmannians. In particular, up to equivalence, I think there’s still only one point. That’s interesting and a bit disconcerting …

I wonder what happens if we let the chain homotopies (or the chain maps) condense the two points to 1. I guess in the first place, we’d allow degenerate maps that project planes to lines, and in the second, umm … moving swiftly on …

Let’s now take a cell complex consisting of a single edge with two ends:

$p . \to . q$

Let’s call this E, and its chain complex E. This has, I believe, dimension (1, 0).

Chain maps from the complex over this little fellow must surely derive from a map of the one edge of E to a single edge of C, also mapping the end vertices appropriately (including getting the orientation to match). Otherwise the boundary operator won’t carry over correctly. (Is this right?)

So we appear to get a four-dimensional space of maps, corresponding to the four edges of C.

This is a bit tricky. We have a 4-dimensional space of maps from $E_{0}$ to $C_{0}$ , and a 4-dimensional space of maps from $E_{1}$ to $C_{1}$ . But these are not at all independent of each other. Or are they?

OK, let’s think very, very hard …

Considering just the maps from $E_{0}$ to $C_{0}$

Once we’ve picked a pair of target points in C (ordered of course), we get a two-dimensional space of maps. It seems reasonable to count this space as a point in the Grassmannian $G_{2,4}$ of 2-dimensional subspaces of the 4-dimensional space $C_{0}$ . This Grassmannian is 4-dimensional.

However, the map is effectively determined (projectively, i.e. up to automorphisms of $E$ ) by the map of just one of the vertices. So AFAICS, we’re confined to a 3-dimensional subspace of $G_{2,4}$ , isomorphic to ${kP}^{3}$ .

Which point in this subspace of the Grassmannian we get is determined by the map from $E_{1}$ to $C_{1}$ , which is also ${kP}^{3}$ , even more obviously. I guess there is a bijection between these two copies of ${kP}^{3}$ , corresponding to the fact that the vertices are attached to the edge …

As for automorphism of this point, once again we can rotate the whole figure all the way around the four possible positions, so I’m guessing that all projective transformations of the ${kP}^{3}$ are automorphisms, and that up to automorphisms we have just one point.

I’m really not convinced I understand the relationship between the two copies of ${kP}^{3}$ here, but let’s just carry on assuming that the intuitive relationships are right …

Considering now the graph E2:

$p q r$
$. \to . \to .$

We appear to also have a space ${kP}^{3}$ here, for both $C_{0}$ (as a subspace of $G_{3,4}$ ) and for $C_{1}$ (as a subspace of $G_{2,4}$ ). Again, projective transformations are automorphisms. The two ways of mapping E to E2 give two projective transformations of ${kP}^{3}$ . That’s a rather interesting incidence relation. The two sets of maps are obviously homotopic to one another.

Not only that, but the graph Px3:

$p q r$
$. . .$

gives rise to the same pattern: since the three vertices can be exchanged with each other by automorphisms of Px3 and are hence projectively modded out by the map from the chain Px3, it seems we would get the same ${kP}^{3}$ pattern.

The graph Ex2 seems more interesting:

$p q r s$
$. \to . . \to .$

Because there’s so little room in C, if we want to be thoroughly injective then the images of the two components are compelled to be opposite each other, apparently making the corresponding space $k^{4}$ , and projectively ${kP}^{3}$ as before. But we can also swap the two components, since these are isomorphic to each other. This suggests the corresponding space is just $k^{2}$ , projectively k $P^{1}$ .

Finally, if we project a copy of C into C itself, the space will be the trivial $k^{0}$ , since projectively we mod out by all the automorphisms of C before starting.

An interesting phenomenon arises with images of the graph P+E

$p q r$
$. \to . .$

On the face of it, for $C_{0}$ , since we have two components which have some independence of movement, this might seem to give rise to a subspace of $G_{2, 4}$ . But on the other hand, the two components are not isomorphic, suggesting the projective maps lie in some larger space, maybe a space of 2-dimensional frames, which I don’t feel like thinking about right now.

Note that, if, instead of C, we’d used another graph with 2-chain of dimension (1, 1), but with many more vertices in the ring, then we’d have many more phenomena of this sort, with different non-isomorphic components being mapped into the target graph, e.g. E2+E3. Of course, with 3 or more components in our figure, we also come up against the phenomenon that different cyclic orderings of the images are not homotopic to one another, so giving rise to different homotopy components of the generalised Grassmannian.

Before I collapse in exhaustion, I just want to talk a little about spaces of dimension different from (1, 1). Consider the graph D of dimension (1, 2):

$. \to . \leftarrow .$
$↑ ↓ ↑$
$. \leftarrow . \to .$

Actually, I don’t think I have the energy to think about this right now, or about targets with multiple components, which obviously also need to be considered. Perhaps someone else could think about this for me. <Cheesy and ingratiating grin>

So I’ll wrap up and say what I think I’ve discovered. Just as finite-dimensional vector spaces and their subspaces are intimately related to finite sets and their subsets, so finite-dimensional 2-vector spaces and their 2-subspaces are intimately related to finite graphs and their subgraphs. These are obviously much more complicated and I’m not at all confident my analyses here are correct, particularly the identification of the various 2-Grassmannians associated to various graph maps. The basic idea seems sound though.

There are some quite odd new phenomena here, for instance the fact that, while there’s no injection that will send a 2-element set to a 1-element set, there are maps injective on vertices and edges that send a 2-component graph to a 1-component graph. Of course, if we’re being injective on vertices and edges, we might also want to see what happens if we impose injectivity on components as well.

We also might be interested in relaxing vertex injectivity so that vertex maps are only injective up to homotopy.

But the result is rather dull. In these conditions, every component of a graph is equivalent to one with only 1 vertex. The graph corresponding to the 2-vector space of dimension (p, q) is one with p components, each with one vertex and q edges. Taking maps from a space of dimension (m, n) to one of (p, q), mod automorphisms (i.e. projectively), is just a matter of picking one of the C(p, m) m-element subsets of the p components, and, for each chosen component, picking one of the C(q, n) n-element subsets of the q edges, corresponding to a point in the product of Grassmannians $G_{p, m} \times G_{q, n}$ . Automorphisms correspond to the homotopy operation of moving a vertex around a loop and back to itself, leaving everything unchanged. Up in the vector space over the vertex, I think you do at least get to multiply by an element of the base field, but projectively I think this is irrelevant anyway. So David C and JB were correct that this case is kind of boring.

Posted by: Tim Silverman on October 8, 2006 1:44 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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Well, I seem to have been terribly confused about how subspaces are related to subgraphs. Let me flounder a little more, possibly more usefully.

When comparing maps from P × 2 (2 disconnected vertices) with maps from E (2 vertices connected by an edge), the main real difference as far as $C_{0}$ is concerned now seems to me to be that the chain complex on the latter graph allows fewer automorphisms than that on the former, that is a 2-dimensional space of them instead of a 4-dimensional space. Hence, while the space corresponding to maps from P × 2 is 4-dimensional, basically the grassmanniann $G_{2,4}$ on $C_{0}$ , the space corresponding to maps from E is 6-dimensional.

We get this by calculating:

space of maps from 2-dimensional space $\to$ 4-dimensional space is $2 \times 4 = 8 -dimensional$ .

From 8, we subtract the dimension of the space of automorphisms.

For P×2 this is the space of invertible maps from $k^{2}$ to itself, of dimension $2 \times 2 = 4$ .

For E this is the direct sum of two copies of the space of invertable maps from $k$ to itself, of dimension $1 + 1 = 2$ .

On $C_{1}$ , on the other hand, we really do have the space of maps from $k \to k^{4}$ modulo maps from $k \to k$ , in other words ${kP}^{3}$ .

I guess the boundary map ties these together in an interesting way.

All this applies, mutatis mutandis, to the other examples too, of course.

I wonder if I’m really less confused now …

Posted by: Tim Silverman on October 8, 2006 4:31 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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Tim writes:

So I’ll wrap up and say what I think I’ve discovered. Just as finite-dimensional vector spaces and their subspaces are intimately related to finite sets and their subsets, so finite-dimensional 2-vector spaces and their 2-subspaces are intimately related to finite graphs and their subgraphs. These are obviously much more complicated and I’m not at all confident my analyses here are correct, particularly the identification of the various 2-Grassmannians associated to various graph maps. The basic idea seems sound though.

Yeah, that’s a great idea!

The way I see it, your fundamental idea is this. There’s a 2-functor from the 2-category of directed graphs to the 2-category of vector 2-spaces ( $≃$ 2-term chain complexes). And, it goes like this:

Any directed graph gives a 2-term chain complex.
Any map between directed graphs gives a chain map between 2-term chain complexes.
And, there’s also a kind of “homotopy between maps between directed graphs”, which gives a chain homotopy between chain maps between 2-term chain complexes.

(Do graph theorists ever think about those homotopies? They should!)

All this generalizes further, to an $(n + 1)$ -functor from $n$ -dimensional cell complexes to $(n + 1)$ -term chain complexes.

I’m not sure how to get some amazing new insights into projective $n$ -geometry from this way of thinking - but that’s not surprising, since I just read your post 2 minutes ago. We should mull on it.

Thanks!

Posted by: John Baez on October 8, 2006 7:25 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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John said, re this whole graphs $\to$ 2-vector spaces business:

Yeah, that’s a great idea!

Wahoo!

He also said:

(Do graph theorists ever think about those homotopies? They should!)

I’ve no idea. But I’ve thought about somewhat similar stuff before. It was a long time ago and it never came to anything, but I guess it was the first piece of “real” maths I did, so I have fond if dim memories of it, and look for opportunities to reanimate some of the ideas from time to time.

And also, even:

Thanks!

I owe you such an enormous debt for expository brilliance in twf and elsewhere, that if something interesting comes of this, it’s just a tiny return …

And elsewhere, when I threatened to “keep chuntering away”:

Yes, do! Though “chuntering” sounds sort of bad - I guess it’s one of those things only Brits know how to do, like “whinging” - but in fact you’re doing some cool stuff.

It’s actually not that bad. “Chuntering” isn’t in very common use, so I don’t know how well my definition matches up to other people’s, but I think of it as describing the activity of a small machine working away in a corner somewhere, possibly making a bit more noise and blowing out a bit more steam than strictly necessary, but basically doing something useful in a modest but reliable sort of way. So I was secretly being quite complimentary to myself! I think “slow but steady” kind of captures it.

Posted by: Tim Silverman on October 9, 2006 10:35 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

I couldn’t resist looking up the word “to chunter” - and here are some definitions I got:

Merriam-Webster. Chunter - Brit. To talk in a low inarticulate way: mutter.

Answers.com. Chunter - Brit. To mutter, murmur; to grumble, find fault, complain.

WordWeb Online. Chunter - Brit.. To make complaining remarks or noises under one’s breath - murmur, mutter, grumble, croak, gnarl, complain, kick, kvetch [N. Amer], plain [archaic], quetch, sound off.

You were doing something more like muttering than kvetching or sounding off, but I like your definition even better, as in “downstairs, the wood-fueled generator was chuntering away.”

Anyway, it’s great that you’ve joined the Klein 2-geometry team!

Posted by: John Baez on October 10, 2006 3:34 AM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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Keep up the effort. Treasure can’t be too far off!

One technical point: if your C has dimension (1,1), shouldn’t we be surprised to find sub-(2,0)-spaces, such as when you look for chain maps from Px2? To avoid ‘evil’, hadn’t these better look like injections into the 2-term chain complex $k \to k$ , the arrow being the zero map?

By the way, we haven’t been introduced. You can see my details from our front page. What’s your background?

Posted by: David Corfield on October 8, 2006 8:07 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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Tim Silverman writes:

And this makes sense because, in the full Klein geometry approach, the figures are sub-2-groups, and incidence relations are (closely related to?) inclusion relations among sub-2-groups, which we already know are functors, sets of functors or categories of functors. We are now in a position to analyse what we mean by “2-orbits” given the various kinds of “injectivity” and “surjectivity” conditions we want to impose on the functors, and given the ways we want functors to be naturally isomorphic or not. However, this requires me to think, so I will stop here.

Right, there are some decisions to be made in categorifying the Klein geometry program, and the only way I know to make these decisions is to try different choices and see which ones give more interesting math.

By the way, I’d say incidence relations are closely related to inclusion relations among subgroups (in ordinary Klein geometry), rather than being such inclusion relations. For example, in projective geometry one has stabilizer groups for points, lines, planes etc., each of which is a “maximal parabolic” subgroup of $SL (n)$ . None of these subgroups include each other, but when two such figures are incident the intersection of their subgroups is a nontrivial parabolic subgroup in its own right. I tried to explain this in week178. The whole story generalizes nicely when we replace $SL (n)$ by any other semisimple Lie group. One gets some very nice math this way, related to Dynkin diagrams.

So, it would be very tempting to categorify projective geometry and see how this incidence business generalizes. That’s (one reason) why David and I are currently at work categorifying projective geometry - in a rather leisurely manner.

Does any of this make any sense at all?

Yes, it makes perfect sense.

(Or is it too mind-numbingly obvious? That would also be bad.)

Well, I guess you could say David and I consider this stuff “obvious”, since it’s been our plan all along to categorify Klein geometry in precisely this way. But, it’s not “too mind-numbingly obvious”. In fact it’s great that you’re stating it so clearly: people reading what David and I have done so far are likely to be drowned in detail and not see the big picture.

As David notes, our project is moving slowly right now. I’ve been busy preparing my classes and my Long Now talk, while he has been emulating Aristotle. And, most of my spare time these days goes into learning what James Dolan has done over the summer.

So, now is actually a good time for you to catch up with us - and maybe shoot ahead!

Posted by: John Baez on October 8, 2006 6:56 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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Since no one’s reading this anyway, I might as well keep chuntering away.

Looking at chain maps from the edge graph E

$p \to q$

to the ring of four edges

$a \to b \to c \to d \to a$

a little thought about the nature of Grassmannians as homogeneous spaces makes it clear that the space of chain maps as far as vertex space is concerned is either $SO (4)$ or some quotient of $SO (4)$ by $Z_{2}$ , hence the 6 dimensions.

Trying to find a still simpler example to get a better picture of what’s going on, we consider maps from E to the graph

$a \begin{matrix} \to \\ \leftarrow \end{matrix} b$

which also has dimension (1, 1).

Clearly, as far as maps to the space of 0-chains are concerned, each vertex of E can be mapped to either point, which gives a 2-dimensional space of maps, and modding out by automorphisms of the space over that vertex, we get a copy of the projective line. So the space of projective maps is the product of two copies of the projective line over $k$ . E.g. for $k = ℝ$ , the space is topologically a torus.

The maps on 1-chains clearly also form a 2-dimensional space, projectively a projective line, and the boundary map sends a point $x \to (x - 1, 1 - x)$ , I think.

Compare this now with the slightly different graph

$a \begin{matrix} \to \\ \to \end{matrix} b$

we have drastically reduced the space of projective 0-chains to a single point, while the space of 1-chains remains the same.

OK, now we can go back up to more complicated graphs. If there are $p$ places we can map our edge graph E to, then the maps on the 0-chains will projectively form the space ${kP}^{p - 1} \times {kP}^{p - 1}$ while the maps on the 1-chains will projectively form the space ${kP}^{p - 1}$ . The boundary map will map it into each factor of the 0-chain map space in more or less the obvious way, but with one being in some way the mirror image of the other. Oh, wait … I need to distinguish how many target elements each source element can go to …

OK, more generally, suppose we have a source graph with some vertices and some edges. Suppose that the source graph can be mapped into the target graph in various ways such that a vertex $v$ can be mapped into $∣ v ∣$ different vertices of the target graph, and that an edge $e$ can be mapped into $∣ e ∣$ different edges of the target graph. Then the projective space of maps of 0-chains will be $\prod_{v} ({kP}^{∣ v ∣ - 1})$ and the projective space of maps of 1-chains will be $\prod_{e} ({kP}^{∣ e ∣ - 1})$ . The boundary map will send the latter into the former factor by factor according to the structure of the source graph.

No doubt there are many interesting homotopic things to be said about these maps, all well-known but not to me!

The above assumes there are no non-trivial automorphisms of the source graphs, which would give rise to extra automorphisms of the chains and would need to be modded out. Since these appear to be what give rise to more general Grassmannians than mere projective spaces, they are obviously also worth looking at.

Posted by: Tim Silverman on October 8, 2006 7:26 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

Tim Silverman writes:

Since no one’s reading this anyway…

Don’t be so sure of that! I read it as soon as you posted it!

… I might as well keep chuntering away.

Yes, do! Though “chuntering” sounds sort of bad - I guess it’s one of those things only Brits know how to do, like “whinging” - but in fact you’re doing some cool stuff.

Posted by: John Baez on October 8, 2006 7:37 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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Grr. )-(

I was getting so puzzled and bothered by the strange issue of edge orientation that I couldn’t sleep. I have been thinking once again about the boundary operation, and I think I am going to have to do a lot of backpedalling tomorrow. The maps on the vertices are tied to those on the edges by the boundary operation, but the source and target orientations don’t have to match (although it is still interesting to think about what happens if you force them to). That at least brings the graph back into line with the dimension of the chain complex.

Grumble. This always happens. As soon as it becomes clear, I realise I was doing it all wrong.

Posted by: Tim Silverman on October 9, 2006 12:48 AM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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So categorified incidence geometry has finally come up! I tried to think about this with little success a few months ago. The problem was how to define the dimension function - in the form of usual incidence geometry (in dimension $n$ ) this is a function from the set of figures to the ordinal $n$ (Better thought of as a functor from the lattice of figures to the lattice $n$ ). I toyed with Heyting algebras and got in a mess. :-(

It wasn’t until I saw Batanin’s talk at AustMS06 that a better idea came: use the `categorified natural numbers’, plane trees of height 2 (Section 2.1 in math.CT/0301221). The dimension should have target one of these. Then of course we get dimension labelled by two integers. So here’s a question - can a plane tree of height 2 (call it a 2-ordinal, say) be thought of as a 2-category, as an ordinal can be thought of as a category? Or the real question: can it be thought of as a 2-category in such a way that works with our concepts of categorified projective geometry?

Let’s go back a step - incidence geometry comes with a set $S$ and a function $\dim : S \to n$ (and some conditions, but they are properties, I think, so leave them for later). So if we have a category and a functor to a 2-ordinal, what can we say about the set up? A 2-ordinal is essentially a pair of ordinals and an order preserving map between them.

$T = k_{2} \to k_{1}$

Think of it as the category $2$ with some extra info (actually an ordinal internal to the category of ordinals). Then considering the object and morphism components of the functor to $T$ we get a pair of numbers which are candidates for the dimension.

The major problem with this idea is that it matters how the branches of the tree are organised - not a problem with 1-ordinals.

Tim: I personally think incidence is nicely thought of as spans in the suboject lattice of the whole space. Keep up the good work - I too tried to jump on the moving car, but I think I’m hanging off the spoiler instead ;)

Posted by: David Roberts on October 9, 2006 9:46 AM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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David Roberts wrote:

Tim: I personally think incidence is nicely thought of as spans in the suboject lattice of the whole space. Keep up the good work - I too tried to jump on the moving car, but I think I’m hanging off the spoiler instead ;)

Don’t be so sure. What’s a span? And what’s a subobject lattice?

OK, I can guess what a subobject lattice is basically, but I’m sure there are a lot of subtleties that I’m not aware of.

What’s a span? I’ve seen people talk about them, but I haven’t looked them up. Is there an easy explanation?

Posted by: Tim Silverman on October 10, 2006 7:23 PM | Permalink | Reply to this

Spans

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What’s a span?

By itself, a span is just a diagram of the form

(1)

\begin{matrix} s & \overset{p_{2}}{\to} & b \\ p_{1} ↓ \\ a . \end{matrix}

Spans internal to some category $C$ with pullbacks form a bicategory. Objects are the objects $a, b, c$ of $C$ , a 1-morphism

(2)

a \overset{(s, p_{1}, p_{2})}{\to} b

is a span, as above, composition of 1-morphisms $a \to b \to c$ is by pullback

(3)

\begin{matrix} s \times_{p} s' & \to & s' & \to & c \\ ↓ & ↓ \\ s & \to & b \\ ↓ \\ a . \end{matrix}

and a 2-morphism

(4)

\begin{matrix} a \overset{s}{\to} b \\ ⇓ \\ a \overset{s'}{\to} b \end{matrix}

is a diagonal arrow $s \to s'$ making

(5)

\begin{matrix} s & \to & b \\ ↓ & ↘ & ↑ \\ a & \leftarrow & s' \end{matrix}

commute.

For instance, a category internal to $C$ is nothing but a monad in spans in $C$ #.

Posted by: urs on October 10, 2006 8:02 PM | Permalink | Reply to this

Intersections of Subobjects

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I personally think incidence is nicely thought of as spans in the suboject lattice of the whole space. #

I can’t read David Robert’s mind here, but I guess what he has in mind is this:

Let $V$ be some object (a vector space, say), and let $a, b, c$ be subobjects of $V$ .

If $V$ is a vector space then its subobjects are nothing but sub vector spaces. Moreover, one subobject $a$ of $V$ may also be subobject of another subobject $b$ of $V$ , i.e. we may have monomorphisms

(1)

a \to b \to V .

In terms of these, consider a span

(2)

\begin{matrix} c & \to & b \\ ↓ \\ a \end{matrix} .

The existence of this span says that $c$ is a subobject of $a$ as well as of $b$ .

But both $a$ and $b$ are also subobjects of $V$

(3)

\begin{matrix} c & \to & b \\ ↓ & ↓ \\ a & \to & V \end{matrix} .

So $c$ is in fact a subobject of $V$ that sits inside both $a$ and $b$ .

In terms of vector spaces, $c$ sits inside the intersection of $a$ with $b$ .

I guess we are interested here in the “largest” $c$ with this property (the full intersection of $a$ with $b$ ).

This should be nothing but the terminal object in

(4)

{Hom}_{spans} (a, b),

namely the object $a \cap b$

(5)

\begin{matrix} a \cap b & \to & b \\ ↓ \\ a \end{matrix}

such that for any other $c$ with

(6)

\begin{matrix} c & \to & b \\ ↓ \\ a \end{matrix}

there is

(7)

\begin{matrix} c \\ ↘ \\ a \cap b \end{matrix}

such that

(8)

\begin{matrix} c & \to & b \\ ↓ & ↘ & ↑ \\ a & \leftarrow & a \cap b \end{matrix}

commutes.

I assume David Roberts is telling us that we should look at the same diagrams, but internal to 2-vector spaces.

Posted by: urs on October 10, 2006 8:59 PM | Permalink | Reply to this

Re: Intersections of Subobjects

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Urs: spot on!

Tim: As you have seen in the discussion, when doing category theory, one doesn’t think of things as being inside other things, but as being mapped to them 1-1 (monically). For any object $x$ in a category with an initial object (a topological space, say) consider all the monic maps to that object, and all the monic maps to their sources etc. This forms a new category with at most one map between any two objects. What’s more, we have a final object, $x$ , and an initial object $\emptyset$ . This gives us a bounded lattice. A nicer example is the lattice of open subsets of a topological space.

I used `subobject’, but to connect with the Klein geometry, `subfigure’ sounds better (well, more logical to me), especially once it all gets categorified.

And, we are (eventually) not only interested in, say, vector spaces, but things invariant under a group action (e.g. spheres).

So, back to urs’ post:

Spans internal to some category C with pullbacks form a bicategory.

I would say $a$ and $b$ are incident if $a \cap b$ , the terminal object in the spans between $a$ and $b$ was not the initial object. This clearly gives a reflexive, symmetric relation.

In traditional incidence geometry, the incidence relation is not transitive. This corresponds with not being able to compose morphisms (=spans) with a source-target match. So we don’t really need spans in a category (the subfigure lattice) with pullbacks, just one where we can form the binary product $a \cap b$ , and with an initial object (hence all finite products)

When we talk about the dimension of a figure, we don’t want something of a higher dimension sitting inside something of smaller dimension. Thus, if the notion of dimension is defined for all objects in the `subfigure lattice’ $Sub$ , we have a map of lattices $Sub \to n = {0 < 1 < \dots < n}$ , where $n$ is the dimension of our space.

I know I’ve sidetracked a lot here from Klein geometry, but I just wanted to turn the problem upside down.

Posted by: David Roberts on October 11, 2006 8:21 AM | Permalink | Reply to this

Re: Intersections of Subobjects

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In traditional incidence geometry, the incidence relation is not transitive. This corresponds with not being able to compose morphisms (=spans) with a source-target match. So we don’t really need spans in a category (the subfigure lattice) with pullbacks, just one where we can form the binary product a∩b, and with an initial object (hence all finite products)

I guess that depends on what one is trying to achieve. I could turn that argument around and say that one should not care so much about the incidence relation, but just about the category of spans.

Composing $a \cap b$ with $b \cap c$ yields $a \cap b \cap c$ . It’s well defined. As you say, it may be empty (= the terminal object) even if $a \cap b$ and $b \cap c$ are not.

[…] subfigure […]

In order to apply the concept of spans in the subobject lattice to 2-geometry, we need to finally figure out what notion of sub-2-object we really need.

I am a little surprised that apparently there is no literature on notions of sub-2-objects. Or is there?

Posted by: urs on October 11, 2006 11:37 AM | Permalink | Reply to this

Re: Intersections of Subobjects

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urs wrote:

In order to apply the concept of spans in the subobject lattice to 2-geometry, we need to finally figure out what notion of sub-2-object we really need.

I am a little surprised that apparently there is no literature on notions of sub-2-objects. Or is there?

I’m not an authority, but none comes immediately to mind. I was going to say something about Mark Weber’s recent paper before, but nothing constructive would come.

The gist of it is: the forgetful functor ${Set}_{*} \to Set$ plays the role, inside $Cat$ that $1 \to {0,1}$ does in $Set$ . A subcategory of $C$ is given by (weak) pullback of this subobject classifier along a functor $C \to Set$ . One would think that just as the subobject lattice of a set $S$ arises from $Hom (S, {0,1})$ , the sub-2-objects would be given by $[C, Set]$ . A stupid question: can this have much to do with doctrines?

Presumably Benabou has something to say about sub-bicategories.

Posted by: David Roberts on October 12, 2006 3:38 AM | Permalink | Reply to this

Re: Intersections of Subobjects

So to think of the category 1 as a subcategory of the category with 2 objects, each with 2 automorphisms, you map one of the objects into the empty set, and make sure that the non-identity automorpism of the other object goes to a map with no fixed point.

Posted by: David Corfield on October 12, 2006 10:10 AM | Permalink | Reply to this

subobject classifier

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pullback of this subobject classifier

I need to learn this.

How do we deal with subobjects of objects with extra structure this way?

For $G$ a group, not every morphism in ${Hom}_{Set} (G, {0,1})$ describes a subgroup of $G$ .

For the particular case of groups, I could of course regard the group as a category and look at ${Hom}_{Cat} (Σ (G), Set)$ . (Does that really give me subgroups? I need to think about that. But have to run now.)

But how does it work in general?

Posted by: urs on October 12, 2006 11:56 AM | Permalink | Reply to this

Re: subobject classifier

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Is this right? For a subgroup $H$ of $G$ , you need a $G$ -set $X$ , so that the action of $H$ fixes an element of $X$ , while the elements of $G$ \ $H$ don’t fix it.

Hmm. Something makes me think that this sort of process wouldn’t produce a unique pullback - the stabilizer of other points would feature. Perhaps all the points of $X$ have to be fixed by $H$ , like if $X$ is the set of cosets.

Posted by: David Corfield on October 12, 2006 12:23 PM | Permalink | Reply to this

Re: Intersections of Subobjects

I guess that depends on what one is trying to achieve. I could turn that argument around and say that one should not care so much about the incidence relation, but just about the category of spans.

Precisely. Having thought some more about the existence of pullbacks, I would now say we do need them. I was getting confused with the definition of the incidence relation, which is `the existence of a non-initial (=nonempty) span’. The mere existence of spans of any sort is transitive, since any two figures are at least `incident in the empty set’. I’m just matching the new definition to the old when talking incidence relations.

Posted by: David Roberts on October 12, 2006 4:40 AM | Permalink | Reply to this

Re: Intersections of Subobjects

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[…] no literature on notions of sub-2-objects. Or is there?

Maybe this one

Marco Grandis Weak subobjects and weak limits in categories and homotopy categories Cah. topol. géom. differ. 1997, vol. 38, no4, pp. 301-326 (25 ref.) (inist)

Posted by: urs on October 12, 2006 12:56 PM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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Each vector 2-space is equivalent to a skeletal vector 2-space (did we ever decide that that was better than ‘2-vector space’?), and one of these we thought could be characterised by two integers corresponding to the 2-term chain complex, $k^{p} \to k^{q}$ , the map sending all of $k^{p}$ to 0. A cell complex which would generate this chain complex is a set of $q$ points, with a set of $p$ loops, each based at one of the points.

You now introduce categorified ordinals in the shape of Batanin’s 2-trees, but note that

The major problem with this idea is that it matters how the branches of the tree are organised

This would seem to correspond to there being a difference between the (2,2) vector 2-space generated as the chain complex of a pair of points each having a loop based there, and the one generated by a pair of points, one of which has two loops based there, the other none. Intuitively, you’d think they were different.

Finally, the issue of ordinal or cardinal. In the last case I mentioned of a cell complex, you wouldn’t think that it would matter which order the points came in, i.e., whether the point with 2 loops was point 1 or point 2.

Posted by: David Corfield on October 11, 2006 8:22 AM | Permalink | Reply to this

Re: Klein 2-Geometry VI

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David C wrote:

Finally, the issue of ordinal or cardinal. In the last case I mentioned of a cell complex, you wouldn’t think that it would matter which order the points came in, i.e., whether the point with 2 loops was point 1 or point 2.

The ordinal is to keep track of the lower dimensional things, not the things of the same dimension. For regular vector spaces (length 1 chain complex of a collection of points), the we’d use an ordinal, and subspaces would be mapped to their dimension. Having drawn some diagrams for $k^{2,2}$ it looks like only one tree is not enough per 2-space. There is more structure present, though, so maybe it will work.

Posted by: David Roberts on October 12, 2006 8:42 AM | Permalink | Reply to this

Re: Klein 2-Geometry VI

So let’s get this straight. We seem to be saying that John was wrong when he said:

Up to equivalence, 2-vector spaces are classified by two natural numbers - the “first and second Betti numbers”. (7 August)

And that this, of course, carries over to constructions like the projective 2-space associated with a vector 2-space, in that there are non-equivalent categories of injective maps of, say, (1,0) into the two different forms of (2,2). In other words the (p,q) notation had best be dropped.

Posted by: David Corfield on October 12, 2006 4:41 PM | Permalink | Reply to this

skeletal 2-vector spaces and cohomology

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Up to equivalence, 2-vector spaces are classified by two natural numbers - the “first and second Betti numbers”.

This is correct.

Proof.

Let

(1)

0 \to V_{1} \overset{δ}{\to} V_{0} \to 0

be the chain complex representing an arbitrary BC 2-vector space $V$ .

Let

(2)

0 \to \ker (δ) \overset{0}{\to} V_{0} / im (δ) \to 0

be the cohomology of the above complex, representing a skeletal 2-vector space $H (V)$ .

In order to demonstrate the equivalence

(3)

V ≃ H (V)

I construct weakly inverse functors going back and forth as follows:

first, I choose on both $V_{1}$ and on $V_{0}$ a scalar product. I use this to decompose

(4)

V_{1} = \ker (δ) \oplus (\ker (δ))^{⊥}

and

(5)

V_{0} = im (δ) \oplus (im (δ))^{⊥} .

Given this choice, there is an obvious functor

(6)

f : V \to H (V)

represented by the chain map

(7)

\begin{matrix} 0 & \to & V_{1} & \overset{δ}{\to} & V_{0} & \to & 0 \\ ↓ & ↓ \\ 0 & \to & \ker (δ) & \overset{0}{\to} & V_{0} / im (δ) & \to & 0 \end{matrix},

where the vertical arro

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October 1, 2006