Coherent space

In proof theory, a coherent space is a concept introduced in the semantic study of linear logic.

Let a set C be given. Two subsets S,T ⊆ C are said to be orthogonal, written S ⊥ T, if S ∩ T is ∅ or a singleton. For a family of C-sets (i.e., F ⊆ ℘(C)), the dual of F, written F ^⊥, is defined as the set of all C-sets S such that for every T ∈ F, S ⊥ T. A coherent space F over C is a family C-sets for which F = (F ^⊥) ^⊥.

In topology, a coherent space is another name for spectral space. A continuous map between coherent spaces is called coherent if it is spectral.

In Proofs and Types coherent spaces are called coherence spaces. A footnote explains that although in the French original they were espaces cohérents, the coherence space translation was used because spectral spaces are sometimes called coherent spaces.

References

• Girard, J.-Y.; Lafont, Y.; Taylor, P. (1989), Proofs and types, Cambridge University Press.
• Girard, J.-Y. (2004), "Between logic and quantic: a tract", in Ehrhard; Girard; Ruet; et al. (eds.), Linear logic in computer science, Cambridge University Press.
• Johnstone, Peter (1982), "II.3 Coherent locales", Stone Spaces, Cambridge University Press, pp. 62–69, ISBN 978-0-521-33779-3.