  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 dual-context 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.
