This is an experiment in collaboration. I want to write an article about discrete causal spaces. Please help. Coauthors welcome! - Eric
Construct a generalization of Discrete differential geometry on causal graphs to -dimensional discrete causal spaces that are locally like -diamonds (see the paper for a description of -diamonds until a page is created).
We need a good category-friendly definition of an -diamond. Here’s a first stab that is incomplete, but hopefully gets the ball rolling.
(Tentative/Incomplete) Definition: An -diamond is a minimal causet.
(Tentative/Incomplete) Definition: An -diamond complex is a directed n-graph in which each node has exactly edges directed into it and exactly nodes directed away from it and each ??? has fillers.
Example: with the its obvious -diamond faces, is an -diamond complex.
Marco Grandis, Directed homotopy theory, I. The fundamental category (arXiv)
Tim Porter: Enriched categories and models for spaces of evolving states, Theoretical Computer Science, 405, (2008), pp. 88–100.
Tim Porter, Enriched categories and models for spaces of dipaths. A discussion document and overview of some techniques (pdf)
Fajstrup, Rosicky, A convenient category for directed homotopy (arXiv)
Dan Christensen, and Louis Crane, Causal sites as quantum geometry (arXiv)
Eric: They use the word “diamond”. I hope that catches on.
Louis Crane, Model categories and quantum gravity (arXiv)
Bombelli, Henson, Sorkin, Discreteness without symmetry breaking: a theorem (arXiv)
Dagstuhl Seminar Proceedings 04351, Spatial Representation: Discrete vs. Continuous Computational Models, (web)
Note. Topics will be separated by lines and each topic is presented in reverse chronological order.
JA: Just starting reading this and don’t know much about diamonds yet, but I have been working on logic-based approaches to discrete dynamics — that I sometimes think of as differential geometry over GF(2) — for quite a while now. You might take a gander at my Magnum Opiate and we could see if it fits in here somewhere.
(From nCafe: Authorship)
There are some immediate possibilities about graph versions of this statement that come to mind. For one, a discrete “Lorentzian spacetime” should be a poset such that all causal subsets are finite set.
x\darr X\darr y
of all objects in the past of and in the future of .
Tim: In a poset one can define a diamond as being an order convex subset possibly with extra properties: U is order convex if , are in with , and lies between and ().
One possible interpretation of what Eric wants is that one specifies a collection of order convex subsets with inclusion on them making a directed set or possibly a lattice. (sort of needing to capture that diamonds should intersect in diamonds (or not at all). Something like this is used by John Roberts in some of his work I seem to remember.
Reminder (‘cos I thought this had been mentioned in the café) it may also pay to look at gr-qc/0410104 for some ideas but I never convinced myself that that was really what was needed.