The Internal Description of a Causal Set:¶What the Universe Looks Like from the Inside

Abstract: We describe an algebraic way to code the causal information of a discrete spacetime. The causal set ? is transformed to a description in terms of the causal pasts of the events in ?. This is done by an evolving set, a functor which to each event of ? assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set. A causal quantum theory can be based on the quantum version of evolving sets, which we briefly discuss.