Monads on Categories of Relational Structures

Abstract: We introduce a framework for universal algebra in categories of relational structures given by finitary relational signatures and finitary or infinitary Horn theories, with the arity $\lambda$ of a Horn theory understood as a strict upper bound on the number of premisses in its axioms; key examples include partial orders ($\lambda=\omega$) or metric spaces ($\lambda=\omega_1$). We establish a bijective correspondence between $\lambda$-accessible enriched monads on the given category of relational structures and a notion of $\lambda$-ary algebraic theories (i.e. with operations of arity $<\lambda$), with the syntax of algebraic theories induced by the relational signature (e.g. inequations or equations-up-to-$\epsilon$). We provide a generic sound and complete derivation system for such relational algebraic theories, thus in particular recovering (extensions of) recent systems of this type for monads on partial orders and metric spaces by instantiation. In particular, we present an $\omega_1$-ary algebraic theory of metric completion. The theory-to-monad direction of our correspondence remains true for the case of $\kappa$-ary algebraic theories and $\kappa$-accessible monads for $\kappa<\lambda$, e.g. for finitary theories over metric spaces.