# Goals

1. Construct a generalization of Discrete differential geometry on causal graphs to $n$-dimensional discrete causal spaces that are locally like $n$-diamonds (see the paper for a description of $n$-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

# Introduction

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

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

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

Example: $\mathbb{Z}^n$ with the its obvious $r$-diamond faces, $0\le r\le n$ is an $n$-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)

# 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 $n$-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 $X$ is what John in his latest entry calls an “interval”, namely for two objects $x,y$ in the poset the under-over category

$x\darr X\darr y$

of all objects in the past of $x$ and in the future of $y$.

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

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