Abstract Clones for Abstract Syntax

We give a formal treatment of simple type theories, such as the simply-typed ?-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional algebraic structure, one can further axiomatize second-order, variable-binding operators. This provides a syntax-independent representation of simple type theories. We describe multisorted second-order presentations, such as the presentation of the simply-typed ?-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as ?- and ?-equality. We give a construction of free algebras and derive a corresponding induction principle, which facilitates syntax-independent proofs of properties such as adequacy and normalization for simple type theories. Working only with clones avoids some of the complexities inherent in presheaf-based frameworks for abstract syntax

Adjoint functor theorems for lax-idempotent pseudomonads

For each pair of lax-idempotent pseudomonads $R$ and $I$, for which $I$ is locally fully faithful and $R$ distributes over $I$, we establish an adjoint functor theorem, relating $R$-cocontinuity to adjointess relative to $I$. As special cases, we recover variants of the adjoint functor theorem of Freyd, the multiadjoint functor theorem of Diers, and the pluriadjoint functor theorem of Solian--Viswanathan. More generally, we recover enriched $\Phi$-adjoint functor theorems for weakly sound classes of weight $\Phi$.Comment: 12 page

Monadic and Higher-Order Structure

Simple type theories, ubiquitous in the study of programming language theory, augment algebraic theories with higher-order, variable-binding structure. This motivates the definition of higher-order algebraic theories to capture this structure, permitting the study of simple type theories in a categorical setting analogous to that of algebraic theories. The theory of higher-order algebraic theories is in one sense much richer than that of algebraic theories, as we may stratify the former according to their order: for instance, the first-order algebraic theories are precisely the classical algebraic theories, the second-order algebraic theories permit operators to abstract over operators, the third-order algebraic theories permit operators to abstract over operators that themselves abstract over operators, and so on. We study the structure of the category of (n + 1)th-order algebraic theories, demonstrating that it may be viewed as a construction on the category of nth-order algebraic theories, facilitating an inductive construction of the category of higher-order algebraic theories. In turn, this description leads naturally to a monad–theory correspondence for higher-order algebraic theories, subsuming the classical monad–theory correspondence, and providing a new, monadic understanding of higher-order structure. In proving the monad–theory correspondence for higher-order algebraic theories, we are led to reconsider the traditional perspective on the classical monad–theory correspondence. In doing so, we reveal a new understanding of the relationship between algebraic theories and monads that clarifies the nature of the correspondence. The crucial insight follows from the consideration of relative monads, which are shown to act as an intermediary in the correspondence. To support our proposal that this be viewed as the correct perspective of the monad–theory correspondence, we show how the same proof may be carried out in a formal 2-categorical setting. The classical monad–theory correspondence, as well as those in the literature for enriched and internal categories, then follow as corollaries of a general theory.Sansom Premium Scholarshi

The formal theory of relative monads

We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad qua trivial relative monad are stronger than the classical notions of algebra object and opalgebra object for a monad qua monad. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the $j$-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment $\mathbb{V}\text{-}\mathbf{Cat}$ of categories enriched in a monoidal category $\mathbb{V}$, though many of our results are new even for $\mathbb{V} = \mathbf{Set}$.Comment: 85 pages; v2: improved exposition, modified notation, new references and example

Algebraic models of simple type theories: a polynomial approach

We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed $\lambda$-calculi, the computational $\lambda$-calculus, and predicate logic. Simple type theories are given models in presheaf categories, with structure specified by algebras of polynomial endofunctors that correspond to natural deduction rules. Initial models, which we construct, abstractly describe the syntax of simple type theories. Taking substitution structure into consideration, we further provide sound and complete semantics in structured cartesian multicategories. This development generalises Lambek's correspondence between the simply-typed $\lambda$-calculus and cartesian-closed categories, to arbitrary simple type theories