nLab causet




The concept of causal set – or causet, for short – is a concept with an attitude: In itself it is just a partially ordered set (or poset, for short), but meant to be understood as a set of spacetime events subject to the relation of causality.

As such, causal sets play a role in:

  • computer science, where causal sets model concurrent computations: there exists a morphism aba \to b in the poset if aa and bb are states of some machine and if operating the machine can take it from state aa to state bb. A simple example would be type of computation which can be done in a single step inoutin \to out, but which needs to be done on two inputs of the same kind. The causal set modelling this situation is

    {(in 1,in 2) (out 1,in 2) (in 1,out 2) (out 1,out 2)} \left\lbrace \array{ (in_1, in_2) &\to& (out_1, in_2) \\ \downarrow && \downarrow \\ (in_1, out_2) &\to& (out_1,out_2) } \right\rbrace
  • relativistic field theory, where every globally hyperbolic Lorentzian structure equips the underlying set of points in a manifold naturally with the structure of a causal set: there is a morphism from the point xx to the point yy in the manifold precisely if yy is in the future of xx, i.e. precisely if there exists a smooth path from xx to yy whose tangent vector is everywhere non-spacelike with respect to the Lorentzian metric. Moreover, the Lorentzian metric on a manifold can essentially (need to dig out the details here, see discussion at smooth Lorentzian space) be reconstructed from this poset structure and from a measure. This has led to some attempts to use posets as a foundational concept for relativistic physics.

    For more see smooth Lorentzian space

The only technical distinction between the notion of posets and that of causal sets is that for a causal set the under-over category xXyx\darr X\darr y for all objects xx and yy in the poset (the category of “two-step time evolution paths” from yy to xx) is required to be finite. This means these are required to have a finite set of objects (and hence necessarily, being a poset, a finite set of morphisms).


A causal set, or causet is a poset XX such that for all objects x,yx,y the interval [x,y]={zxzy}[x,y] = \{z \mid x \le z \le y\} is finite.


Discussion in relativistic field theory:

See also:

Last revised on June 9, 2023 at 08:56:16. See the history of this page for a list of all contributions to it.