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Jonathan Beardsley (May 05 2025 at 15:56):I recently posted a couple of questions on MathOverflow that I think people here probably would have some insight into (https://mathoverflow.net/questions/491964/are-there-categories-of-incidence-structures-and-projective-geometries, https://mathoverflow.net/questions/492095/dense-generators-for-models-of-universal-horn-theories). The basic idea is that I'm trying to figure out a way of carefully writing down the foundations of discrete projective geometry as a category, probably using universal algebra. It's pretty clear that you can set up incidence systems as a 2-sorted relational structure, a la Adamek-Rosicky. Then, I believe, you can write down formulas giving a universal Horn theory describing projective geometries themselves (e.g. every pairs of points has a unique line through it, satisfies the "projective law," etc.). This shows straight off that projective geometries are an epireflective subcategory of this simple category of models of "incidence systems." Adamek and Rosicky tell you that this category has a set of "regular projective generators," which are the free incidence systems reflected into geometries (see also 3.1 here: https://ncatlab.org/nlab/files/Barr-ModelsOfHornTheories.pdf). Morevoer, Rosicky showed that this category is locally presentable (the main theorem here https://www.sciencedirect.com/science/article/pii/002240499490023X). But what I'm really after is which objects are generating this category under colimits.
Intuitively, one would expect these to be things like the trivial geometry with two points and a line through those points. You could potentially glue this together to get any geometry you wanted (although you'd have to localize again to get an actual geometry and not just some wacky incidence system). These seem like they're probably "free geometries?"
Does anyone know if it's possible to find a dense set of generators for the category of models of a universal Horn theory? Maybe if this isn't the case in general this situation is still simple enough though...
Jonathan Beardsley (May 05 2025 at 16:08):I mean, I guess I should say that there's also the obvious approach of just "write down the category, and start proving things," without using any universal algebra at all. But it would be nice if there was an off-the-shelf method for proving the things that seem intuitive.