nLab
causet

Contents

Idea

A causet or causal set is essentially the same thing as a poset, more precisely a locally finite poset.

The difference in terminology indicates the restriction to certain applications: regarding a poset as a causal set means to regard it as a model for something like a space with time evolution: a morphism aba \to b in the poset is read as encoding the information: “there is a time evolution process which takes aa to bb”. Equipped with this interpretation as causal sets, posets find applications 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 physics, where every globally hyperbolic Lorentian metric on a manifold equips that 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).

Definition

A causal set, or causet is a poset XX such that for all objects x,yx,y the interval [y,x][y,x] is finite.

References

Revised on November 21, 2010 17:48:26 by Tim Porter (95.147.237.23)