The n-Category Café

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.”

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.

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?

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.

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?

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 $G \to G'$.