nLab
Discrete causal spaces

This is an experiment in collaboration. I want to write an article about discrete causal spaces. Please help. Coauthors welcome! - Eric

Goals

  1. Construct a generalization of Discrete differential geometry on causal graphs to nn-dimensional discrete causal spaces that are locally like nn-diamonds (see the paper for a description of nn-diamonds until a page is created).

  2. The continuum limit of a discrete causal space should be a directed space, or more specifically, a smooth Lorentzian space.

  3. Etc

Abstract

Introduction

We need a good category-friendly definition of an nn-diamond. Here’s a first stab that is incomplete, but hopefully gets the ball rolling.

(Tentative/Incomplete) Definition: An nn-diamond is a minimal causet.

(Tentative/Incomplete) Definition: An nn-diamond complex is a directed n-graph in which each node has exactly nn edges directed into it and exactly nn nodes directed away from it and each ??? has fillers.

Example: n\mathbb{Z}^n with the its obvious rr-diamond faces, 0rn0\le r\le n is an nn-diamond complex.

References

  • 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)
  • 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)

See also:


Discussion

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.


Eric: Motivated by some emails from Tim, I think we could use pages on causal site? and dg-quiver. I am particularly interested in seeing a definition of dg-quiver.


Eric: I wonder if nn-diamonds should be more closely related to n-fold categories rather than posets?


(From nCafe: Authorship)

Urs says: I think Eric wants a poset of sorts.

It seems that a good formalization of “smooth Lorentzian space(time)” is something like: a poset internal to a category of measure spaces.

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.

A causal subset in a poset XX is what John in his latest entry calls an “interval”, namely for two objects x,yx,y in the poset the under-over category

xXyx\darr X\darr y

of all objects in the past of xx and in the future of yy.

Tim: In a poset one can define a diamond as being an order convex subset possibly with extra properties: U is order convex if xx, zz are in UU with x<zx \lt z, and yy lies between xx and zz (x<y<zx\lt y \lt z).

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.

category: drafts

Revised on October 19, 2010 13:06:03 by Eric Forgy (219.78.208.88)