### 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?

## 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.”