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.

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