Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJun 8th 2025
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexDeleted the article. Here is its former content: * tic {:toc} This is an experiment in collaboration. I want to write an article about discrete causal spaces. Please help. Coauthors welcome! - [[Eric Forgy|Eric]] # Goals # 1. Construct a generalization of [[ericforgy:Discrete differential geometry on causal graphs|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). 1. The continuum limit of a discrete causal space should be a [[directed space]], or more specifically, a [[smooth Lorentzian space]]. 1. Etc # Abstract # ... # 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. # References # * Marco Grandis, _Directed homotopy theory, I. The fundamental category_ ([arXiv](http://arxiv.org/abs/math.AT/0111048)) * 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](http://drops.dagstuhl.de/opus/volltexte/2007/898/pdf/06341.PorterTimothy.Paper.898.pdf)) * Fajstrup, Rosicky, _A convenient category for directed homotopy_ ([arXiv](http://arxiv.org/abs/0708.3937)) * Dan Christensen, and Louis Crane, _Causal sites as quantum geometry_ ([arXiv](http://arxiv.org/abs/gr-qc/0410104)) +--{.query} [[Eric]]: They use the word "diamond". I hope that catches on. =-- * Louis Crane, _Model categories and quantum gravity_ ([arXiv](http://arxiv.org/abs/0810.4492)) * Bombelli, Henson, Sorkin, _Discreteness without symmetry breaking: a theorem_ ([arXiv](http://arxiv.org/abs/gr-qc/0605006)) * Dagstuhl Seminar Proceedings 04351, _Spatial Representation: Discrete vs. Continuous Computational Models_, ([web](http://drops.dagstuhl.de/portals/index.php?semnr=04351)) See also: * [[directed homotopy theory]]. *** # 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 &#8212; that I sometimes think of as differential geometry over GF(2) &#8212; for quite a while now. You might take a gander at my [Magnum Opiate](http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0) and we could see if it fits in here somewhere. *** [[Eric]]: Motivated by some emails from [[Tim Porter|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: <a href="http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c021347">Authorship</a>) _[[Urs Schreiber|Urs]] says_: I think Eric wants a [[partial order|poset]] of sorts. It seems that a good formalization of "[[smooth Lorentzian space|smooth Lorentzian space(time)]]" is something like: a [[partial order|poset]] [[internal category|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 [[causet|causal subset]] in a poset $X$ is what John in his latest <a href="http://golem.ph.utexas.edu/category/2009/01/hopf_algebras_from_posets.html">entry</a> calls an "interval", namely for two objects $x,y$ in the poset the [[under category|under-over category]] $$x\darr X\darr y$$ of all objects in the past of $x$ and in the future of $y$. _[[Tim Porter|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 caf&#233;) 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 [[!redirects Discrete Causal Spaces]] <a href="https://ncatlab.org/nlab/revision/diff/Discrete+causal+spaces+%3E+history/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/Discrete+causal+spaces+%3E+history/17">v17</a>, <a href="https://ncatlab.org/nlab/show/Discrete+causal+spaces+%3E+history">current</a>

   Deleted the article. Here is its former content:

   • tic {:toc}

   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 $n$-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

   Abstract

   …

   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.

   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) +–{.query} 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)

   See also:

   • directed homotopy theory.

   ---

   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

   !redirects Discrete Causal Spaces

   diff, v17, current