Silva A, Jacobs B.
2014.

Initial Algebras of Terms With Binding and Algebraic Structure. Categories and Types in Logic, Language, and Physics - Lecture Notes in Computer Science. 8222:211-234.

AbstractOne of the many results which makes Joachim Lambek famous is: an initial algebra of an endofunctor is an isomorphism. This fixed point result is often referred to as ``Lambek's Lemma''. In this paper, we illustrate the power of initiality by exploiting it in categories of algebra-valued presheaves $\EM(T)^{\N}$, for a monad $T$ on $\Sets$.

The use of presheaves to obtain certain calculi of expressions (with variable binding) was introduced by Fiore, Plotkin, and Turi. They used set-valued presheaves, whereas here the presheaves take values in a category $\EM(T)$ of Eilenberg-Moore algebras. This generalisation allows us to develop a theory where more structured calculi can be obtained. The use of algebras means also that we work in a linear context and need a separate operation $\bang$ for replication, for instance to describe strength for an endofunctor on $\EM(T)$. We apply the resulting theory to give systematic descriptions of non-trivial calculi: we introduce non-deterministic and weighted lambda terms and expressions for automata as initial algebras, and we formalise relevant equations diagrammatically.