A coherence theorem for canonical morphisms in cartesian closed categories

Abstract

In this paper there is proved a coherence theorem in proof-theoretic formulation: all derivations of a balanced sequent are equivalent. (A sequent is called balanced if each variable appears in it no more than twice.) Canonical morphisms in a Cartesian closed category are morphisms which can be obtained from those explicitly mentioned in the definition of a Cartesian closed category (i.e., the left and right projectionsl: A x B → A and r:A x B → B, ε:A x hom (A, B) → B, etc.) with the help of composition of functors x, hom and the operation +. Let the objects A and B be constructed from the objects C₁,..., C_n with the help of the functors x and hom. Then, generally speaking, not all canonical morphisms from A to B will be equal. For example, if A is C₁ x C₁, and B is C₁, then the left and right projections are different morphisms. The coherence theorem asserts that if one does not make superfluous identifications of objects, then all canonical morphisms from A to B will be equal, i.e., all diagrams of canonical morphisms beginning in A and ending in B will commute. There is a familiar translation of certain concepts of the theory of categories into the language of proof theory, under which to objects correspond formulas, and the functors x and hom are interpreted as the connectives & and ⊃. Under this translation, to canonical morphisms from A to B correspond derivations in the (&, ⊃)-fragment of the intuitionistic prepositional calculus of the sequent A → B. Morphisms are equal if and only if the derivations corresponding to them are equivalent, i.e., certain of their normal forms coincide, or, what is the same thing, their deductive terms are equivalent. The theorem proved in this paper is equivalent with the coherence theorem in the algebraic formulation. There are given two proofs of this theorem, obtained independently by the authors, in one of which there are considered natural derivations and the apparatus of deductive terms is used, and the other is based on reduction of the depth of formulas preserving equivalence of derivations, specialization of forms of inference in Gentzen L-systems, and analysis of links in sequences.