  We give an account of the basic combinatorial structure underlying the notion
of type dependency. We do so by considering the category of all dependent
sequent calculi, and exhibiting it as the category of algebras for a monad on a
presheaf category. The objects of the presheaf category encode the basic
judgements of a dependent sequent calculus, while the action of the monad
encodes the deduction rules; so by giving an explicit description of the monad,
we obtain an explicit account of the combinatorics of type dependency. We find
that this combinatorics is controlled by a particular kind of decorated ordered
tree, familiar from computer science and from innocent game semantics.
Furthermore, we find that the monad at issue is of a particularly well-behaved
kind: it is local right adjoint in the sense of Street--Weber. In future work,
we will use this fact to describe nerves for dependent type theories, and to
study the coherence problem for dependent type theory using the tools of
two-dimensional monad theory.
