Logical systems II: Free semantics

Abstract

This paper is a sequel to [1]. It provides a general 2-categorical setting for extensional calculi and shows how intensional and extensional calculi can be related in logical systems. We define Yoneda triangles as relativisations of internal adjunctions, and use them to characterise universes that admit a notion of convolution. We show that such universes induce semantics for lambda calculi. We prove that a construction analogical to enriched Day convolution works for categories internal to a locally cartesian closed category with finite colimits. Categorical 2-powersTo better understand our definition of "2-powers", let us first recall how one may define ordinary powers. With every regular category 4 C there is associated 1 In other words -if the coding is effective then it has to be ambiguous. 2 The result generalises to any higher-order type theory [14]. 3 Otherwise, by the axiom of union we could form A0 = k U k , and A = P (A0) would not be classified by any U k . 4 A category is called regular (Chapter 2, Volume 2 of [11], Chapter A1.3 of [4], Chapter 4, Section 4 of [12], (Chapter 1.5 of [15]) if it has finite limits, regular epimorphisms are stable under pullbacks, and every morphism factors as a regular epimorphism followed by a monomorphism.