Quiver geometry is an experiment in defining a kind of discrete geometry purely in terms of directed graphs with a particular structure of edge labels on them.
I began thinking about this project in late 2020 when thinking about the abstract geometry induced by computational rewriting systems, having found nothing in the existing literature that seemed like a good “port” of the methods and theories of classical differential geometry to the setting of discrete geometry.
I started to create a kind of casual introduction and illustration of the ideas at quivergeometry.netquivergeometry.net, which gave me an opportunity to experiment with different pedagogical approaches, and in particular an software toolchaingithub.com to create beautiful visualizations and illustrations.
I’ve since discovered that much of what I was trying to define is already part of the canon of category theory, or can be set up fairly easily within it. Roughly speaking:
|functors between categories freely generated by a directed graph
|path groupoid of a quiverquivergeometry.net
|morphism composition groupoid in the above
|discrete fiber bundlesquivergeometry.net
|monoidal structure on the category of quivers
Anyway, it was still useful to work on this project, and I learned a lot, leading me to algebraic geometry, sheaf theory, and cohomology, and obviously category theory.
I do still need to understand how cardinal structure can be understood categorically – it feels like some incarnation of a homotopy lifting property. I need to understand more about fibered categories!
The website may also be useful to people who don’t want to engage with category theory but do want a template for how to think about discrete geometry – I may continue to work on it for that reason at some point in the future. There are also sections of general mathematical exposition that I am proud of, such as the exploration of multisetsquivergeometry.net and fiber bundlesquivergeometry.net.
I’ve also presented quiver geometry to the Wolfram Physics Projectwolframphysics.org, where it has formed part of the basis for several student projects at the annual Wolfram summer school such as “Topological invariants in discrete lattice via graph rewriting”community.wolfram.com, “Gauge field theories on discrete principal fibre bundles”community.wolfram.com, “Full Discretization of Local Gauge Invariance”community.wolfram.com. Who knows were that will go?