Coherent space

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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 a sober space with a basis of compact sets in which finite intersections preserve the property of being compact open. A continuous map between coherent spaces is called coherent if its associated preimage map takes compact open sets to compact open sets.

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.

The compact open sets of a coherent space form a distributive lattice under union and intersection operations, and Stone's representation theorem for distributive lattices states that any distributive lattice may be represented as set unions and intersections in this way. This correspondence between coherent spaces and distributive lattices extends to a category-theoretic duality (Johnstone 1982).

[edit] 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., Linear logic in computer science, Cambridge University Press, http://iml.univ-mrs.fr/~girard/LLcup.pdf.gz .

• Johnstone, Peter (1982), "II.3 Coherent locales", Stone Spaces, Cambridge University Press, pp. 62–69, ISBN 9780521337793 .