Substitution systems revisited

Matthes and Uustalu (TCS 327(1–2):155–174, 2004) presented a categorical description of substitution systems capable of capturing syntax involving binding which is independent of whether the syntax is made up from least or greatest fixed points. We extend this work in two directions: we continue the analysis by creating more categorical structure, in particular by organizing substitution systems into a category and studying its properties, and we develop the proofs of the results of the cited paper and our new ones in UniMath, a recent library of univalent mathematics formalized in the Coq theorem prover. In previous work, Matthes and Uustalu [6] define a notion of “heterogeneous substitution system”, the purpose of which is to axiomatize substitution and its desired properties. Such a substitution system is given by an algebra of a signature functor, equipped with an operation— which is to be thought of as substitution—that is compatible with the algebra structure map in a suitable sense. The term “heterogeneous” refers to the fact that the underlying notion of signature encompasses variable binding constructions and also explicit substitution a. k. a. flattening. More precisely, the signature is based on a rank-2 functor H (an endofunctor on a category of endofunctors) for the respective domain-specific signature, to which a monadic unit is explicitly added. The latter corresponds to the inclusion of variables into the elements that are considered as terms (in a quite general sense) over their variable supplies. The name “rank-2 functor” stems from the rank of the type operator that transforms type transformations into type transformations—hence has kind (Set → Set) → (Set → Set)—which may be seen as backbone of H in case the base category is Set. In this rank-2 setting, the carrier of the algebra is an endofunctor, and since a monadic unit is already present, a natural question is if one obtains a monad. In that paper, it is then shown that for any heterogeneous substitution system this is indeed the case; multiplication of the monad is derived from the “substitution” operation which is parameterized by a morphism f of pointed endofunctors and consists in asking for a unique solution that makes a certain diagram commute. Monad multiplication and one of the monad laws is obtained from the existence of a solution in the case that f is the identity, while the other monad laws are derived from uniqueness for two other choices of f . Furthermore, it is shown there that “substitution is for free” for both initial algebras as well as—maybe more surprisingly—for (the inverse of) final coalgebras: if the initial algebra, resp. terminal coalgebra, of a given signature functor exists, then it, resp. its inverse, can be augmented to a substitution system (for the former case, and in order to easily use generalized iteration [4], it is assumed that the functor − · Z has a right adjoint for every endofunctor Z). Indeed, it was one of the design goals of the axiomatic framework of heterogeneous substitution systems to be applicable to non-wellfounded syntax as well as to wellfounded syntax, whereas related work (e.g., [5, 2]) frequently only applies to wellfounded syntax. Examples of substitution systems are thus given by the lambda calculus, with and without explicit flattening, but also by languages involving typing and infinite terms. ∗The work of Benedikt Ahrens was partially supported by the CIMI (Centre International de Mathématiques et d’Informatique) Excellence program ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02 during a postdoctoral fellowship.