June 12, 2008

Klein 2-Geometry XI

Posted by David Corfield

Everything’s gone very quiet at the Café. For my part it’s due to having fried my brains over the past three days as Chief Examiner, having to keep a thousand details in my head. The one tiny thought I managed to have in the past few days though came via a question to myself:

Why when trying to categorify Klein geometry did we never think to look at internal categories in some category of geometric spaces?

Next question then is

What is a category of geometric spaces?

Presumably much has been done on this. From a brief foray it seems that there are choices to be made. In version 1 of this article, Wolfgang Bertram says

The question is simply: what shall be the morphisms in the “category of projective spaces (over a given field or ring $K$)”? - shall we admit only maps induced by injective linear maps (so we get globally defined maps of projective spaces) or admit maps induced by arbitrary non-zero linear maps (so our maps are not everywhere defined)? The answer is of course that both definitions make sense and thus we get two different categories.

From a review of Faure and Frölicher’s Modern Projective Geometry it appears that forming such categories allows novel insights:

Opening a book on projective geometry, we expect an investigation of objects occurring in projective space. We expect to meet subspaces, quadrics, algebraic subvarieties, differential submanifolds, and many other objects. The book under review is not of this type, and this explains perhaps, why it carries the title Modern Projective Geometry. The main aim of the book is to introduce the category of projective geometries. This means that the authors’ goal is to look at projective geometries not only from inside, but also from outside.

Anything stopping us looking for categories internal to such categories?

Posted at June 12, 2008 12:27 PM UTC

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Re: Klein 2-Geometry XI

I invariably find your musings interesting.
I’ve read two ideas recently which from my
sea level view seem related to your post. Perhaps I didn’t grasp some disqualifying difference. Also, from my pov, if you have ‘maps that are not everywhere defined’ that category is less suitable for Machine Learning, one would need a more concrete method.

http://www.tac.mta.ca/tac/
Every Grothendieck Topos Has a One-Way Site
Abstract. “Lawvere has urged a project of
characterizing petit toposes which have
the character of generalized spaces and
gros toposes which have the character of
categories of spaces. Etendues and locally
decidable toposes are seemingly petit and
have a natural common generalization in
sites with all idempotents identities.
This note shows every Grothendieck topos
has such a site.” By Colin Mclarty

Posted by John Baez May 17, 2008
Convenient Categories of Smooth Spaces
“The category of all sheaves on a site
is extremely nice: it is a topos. Here,
following ideas of Dubuc [11], we show
that the category of concrete sheaves
on a concrete site is also nice, but
slightly less so: it is a ‘quasitopos’
[38]. This yields many of the good
properties listed above.”

Posted by: Stephen Harris on June 12, 2008 5:37 PM | Permalink | Reply to this

Re: Klein 2-Geometry XI

I seem to be able to imagine a Euclidean space of morphisms with source and target maps to a Euclidean space of objects. A straight line segment in the former space joining two points projects as a pair of line segments in the latter space where the source moves along the line at the same rate proportionally as the target moves along its line. Not everything is picked up by this projection.

Posted by: David Corfield on June 12, 2008 6:14 PM | Permalink | Reply to this

Re: Klein 2-Geometry XI

I have a Baez-Crans 2-vector space, i.e., a category internal to vector spaces, with vector spaces $V$ and $W$ and linear maps $s$ and $t$ and $i$ satisfying the relevant equations, as described on p. 9 of HDA VI. Can I projectivise everything in sight so that I now have a category internal to the category of projective spaces with morphisms partial maps arising from arbitrary non-zero linear maps, as mentioned by Bertram above? If so, would such an entity be good for anything?

Posted by: David Corfield on June 16, 2008 10:16 AM | Permalink | Reply to this

Re: Klein 2-Geometry XI

Sorry to take a while to join in here! I’m in Barcelona; I’ve given my talk; my hotel has wireless internet access. So, let me start.

David wrote:

I have a Baez-Crans 2-vector space, i.e., a category internal to vector spaces…

Why when trying to categorify Klein geometry did we never think to look at internal categories in some category of geometric spaces?

Last Thursday I was going to ask you if $Vect$ counts as a ‘category of geometric spaces’. (I have little idea what that phrase means).

And if you said “yes,” I was going to remind you that we’d spent quite a bit of time in our Klein 2-Geometry discussions talking about internal categories in $Vect$, and their associated ‘projective 2-spaces’.

But, at that point my internet connection died, so I never posted that comment.

Can I projectivise everything in sight so that I now have a category internal to the category of projective spaces with morphisms partial maps arising from arbitrary non-zero linear maps, as mentioned by Bertram above?

Well, we certainly have a good idea what the ‘projective 2-space’ associated to a 2-vector space should be. But, we didn’t obtain it in the way you describe! So, it’s an interesting puzzle whether this way works.

For it to work, you need a category of projective spaces, let’s call it $Proj$, and a ‘projectivization’ functor

$P : Vect \to Proj$

that preserves pullbacks. (A functor $F: X \to Y$ sends categories internal to $X$ to categories internal to $Y$ if it preserves pullbacks.)

People love to talk about the projective space $P V$ associated to a vector space, but I’ve never seen much about making this $P$ into a functor. Surely people have thought about this! Wolfgang Bertram touches on the first key questions:

The question is simply: what shall be the morphisms in the “category of projective spaces (over a given field or ring $K$)”? - shall we admit only maps induced by injective linear maps (so we get globally defined maps of projective spaces) or admit maps induced by arbitrary non-zero linear maps (so our maps are not everywhere defined)?

To be able to projectivize a rich supply of 2-vector spaces, we really want our functor $P$ to be defined on all linear maps between vector spaces.

But, as Bertram notes, this forces us into a rather weird concept of the category $Proj$. For example, the map $f: V \to W$ with $f(v) = 0$ for all $v \in V$ will give a nowhere defined map from $P V$ to $P W$. Or something like that…

Maybe some algebraic geometers will jump in and tell us something helpful.

Posted by: John Baez on June 17, 2008 6:59 AM | Permalink | Reply to this

Re: Klein 2-Geometry XI

So the question is whether we get the same thing when we projectivise the 2-term complex picture as when we projectivise the internal category picture?

Back with the 2-term picture, when we had skeletal 2-vector spaces with 0-th betti number zero, you said the projective 2-space was empty. But if we were working with categories internal to projective spaces with partial maps would it be possible to have space of objects empty while space of morphism non-empty with source and target everywhere nondefined?

Posted by: David Corfield on June 17, 2008 9:54 AM | Permalink | Reply to this

Re: Klein 2-Geometry XI

Maybe so.

But I’m really reluctant to think about categories internal to some category of projective spaces until some algebraic geometer answers my question: is there some reasonable category of projective spaces which makes ‘projectivization’ a functor from $Vect$ to this category?

It may just be the fear of jumping into an empty pool and making a fool of myself… but I have a suspicion that the answer is no, you idiot!

Posted by: John Baez on June 17, 2008 9:43 PM | Permalink | Reply to this

Re: Klein 2-Geometry XI

I tend to think ‘no’ too. You’d certainly need totally undefined maps.

If there were such a functor, there would be a $q = 1$ deformation to a functor from pointed sets to sets with partial maps which dropped the point. Now, is there anything wrong there?

But even if it weren’t a functor, there’s nothing to stop someone looking for internal categories, is there?

But anyway, was our original way of forming projective 2-spaces to say all skeletal 2-vector spaces with 0th betti number 0 get sent to the empty category?

Posted by: David Corfield on June 17, 2008 10:18 PM | Permalink | Reply to this

Re: Klein 2-Geometry XI

The obvious choice of projectivization is surely a functor, but not a product preserving one. $\mathbb{R}^4$ is isomorphic to $\mathbb{R}^2 \times \mathbb{R}^2$, but $RP(3)$ is not $RP(1) \times RP(1)$.

So categories internal to Vect, won’t get sent to categories internal to Proj.

So is it that categories with underlying, right adjoints to Set have internal categories which are also ordinary categories, while internal categories in categories like sets and partial maps are not?

Is there any interest in something like the latter? A set of morphisms with partially defined source and target maps, composition defined where source and target match in a defined place.

What do the symmetries of an internal category form?

Posted by: David Corfield on June 20, 2008 9:34 AM | Permalink | Reply to this

Re: Klein 2-Geometry XI

Perhaps we’d better start learning about restriction categories. Example 15 on page 8 describes the category of real projective spaces and homogeneous polynomial morphisms, rather than our limitation to linear morphisms.

It seems,

As the notion of restriction category is not self-dual, we should not expect colimits and limits in restriction categories to behave in the same manner. The notion of colimit in the restriction context is quite straightforward, but limits are more delicate. The suitable notion of limit turns out to be a kind of lax limit, satisfying certain extra properties.

Sounds like pullbacks for internal categories might be tricky.

Posted by: David Corfield on June 19, 2008 9:28 AM | Permalink | Reply to this

Re: Klein 2-Geometry XI

I think I see what I’ve been struggling with here. There’s something unhomogeneous about a vector space, i.e., there’s a privileged 0. So before you go on to develop geometries – Euclidean, affine, etc. – you rip out the 0 and factor out by the field. In other words you form the projective space.

Now if you have a Baez-Crans 2-vector space $d = 0: F^n \to F^m$ and rip out the component of the zero object, you have $F^{n + m} - F^n$ worth of arrows, which when quotiented gives you $P(F^m) \times F^n$.

So it’s inhomogeneous. But this is obvious because the identity arrows in the space of arrows are privileged, unless they’re all that there are, i.e, $n = 0$.

In the case of vector spaces over the field with one element, or $F_{un}$ as the Francophones call it, these are pointed sets, and they project to sets with partial maps. These categories happen to be equivalent, but not isomorphic. You can have internal categories within pointed sets, i.e., pointed categories, so there should equivalently be a kind of category internal to sets and partial maps. I was hoping to do something similar for other fields.

Posted by: David Corfield on June 26, 2008 12:28 PM | Permalink | Reply to this

Re: Klein 2-Geometry XI

Another attempt to capture projective geometry category theoretically. Anders Kock has just put Abstract Projective Lines on the arXiv:

We describe a notion of (abstract) projective line over a field as a set equipped with a certain first order structure, and a projectivity between projective lines as a bijection preserving this structure. The structure in question is that of a groupoid, with certain properties. This leads to a natural notion of bundle of projective lines, forming a stack.

The symmetries of this groupoid will form a 2-group. Might we have seen it already?

Posted by: David Corfield on December 15, 2009 9:35 AM | Permalink | Reply to this

Re: Klein 2-Geometry XI

I’m not sure we ever wrote in down formally, but I seem to recall we thought about something similar to Pronk and Scull’s orbit 2-category, $\mathcal{O}_G$, p. 30 here. For a Lie group $G$, this has objects $G/H$, for closed subgroups $H$; $G$-maps as arrows $G/H \to G/K$; and homotopy classes of paths as 2-cells.

Presumably one could map between such orbit 2-categories $\mathcal{O}_G \to \mathcal{O}_{G'}$, given a Lie group mapping $G \to G'$.

Posted by: David Corfield on July 19, 2010 3:56 PM | Permalink | Reply to this

Re: Klein 2-Geometry XI

In two papers back some, Ronnie, Marek Golasinski, Andy Tonks and myself used a simplicially enriched category version of this. We were generalising results on classifying spaces for crossed complexes to the equivariant case. If anyone is interested I can give references and summarise the theory.

Posted by: Tim Porter on July 25, 2010 12:04 PM | Permalink | Reply to this
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