Modality via Iterated Enrichment

Abstract

This paper investigates modal type theories by using a new categorical semantics called change-of-base semantics. Change-of-base semantics is novel in that it is based on (possibly infinitely) iterated enrichment and interpretation of modality as hom objects. In our semantics, the relationship between meta and object levels in multi-staged computation exactly corresponds to the relationship between enriching and enriched categories. As a result, we obtain a categorical explanation of situations where meta and object logics may be completely different. Our categorical models include conventional models of modal type theory (e.g., cartesian closed categories with a monoidal endofunctor) as special cases and hence can be seen as a natural refinement of former results. On the type theoretical side, it is shown that Fitch-style modal type theory can be directly interpreted in iterated enrichment of categories. Interestingly, this interpretation suggests the fact that Fitch-style modal type theory is the right adjoint of dualcontext calculus. In addition, we present how linear temporal, S4, and linear exponential modalities are described in terms of change-of-base semantics. Finally, we show that the change-of-base semantics can be naturally extended to multi-staged effectful computation and generalized contextual modality a la Nanevski et al. We emphasize that this paper answers the question raised in the survey paper by de Paiva and Ritter in 2011, what a categorical model for Fitch-style type theory is like.

Logic of computational semi-effects and categorical gluing for equivariant functors

“…(A similar construction is also found in the semantics of multi-staged computation [31].) Because a lax monoidal functor models Applicative in Haskell, this serves as a model of the example presented at the end of Section 3.…”

Dual-context Modal Logic as Left Adjoint of Fitch-style Modal Logic

“…So, we do not refer any more in this paper. Whereas the Gentzen-style calculus naively represents a cartesian closed category with a lax monoidal endofunctor, the following Fitch-style calculus represents an infinitely enriched category [19].…”

“…A survey paper [9] by de Paiva and Ritter might be helpful for understanding Fitch-style. A proof relevant translation is discussed in our work [19] via the semantics. A ccc with a normal monoidal endofunctor G is an instance of our semantics of the Fitch-style calculus.…”

“…In this paper, we show an adjunction by embedding both the Fitch-style and dual-context calculi into the adjoint calculus [3], [4], which was proposed in order to represent monoidal adjunctions. To achieve our aim, we refine the Fitch-style and the dual-context with levels along the line of our previous work [19]. Due to the assignment of levels, it can be shown that the union of the 2-level calculi exactly corresponds to the adjoint calculus.…”