Strongly Finitary Monads for Varieties of Quantitative Algebras

Quantitative algebras are algebras enriched in the category Met of metric spaces or UMet of ultrametric spaces so that all operations are nonexpanding. Mardare, Plotkin and Panangaden introduced varieties (aka 1-basic varieties) as classes of quantitative algebras presented by quantitative equations. We prove that, when restricted to ultrametrics, varieties bijectively correspond to strongly finitary monads T on UMet. This means that T is the left Kan extension of its restriction to finite discrete spaces. An analogous result holds in the category CUMet of complete ultrametric spaces

Limits of small functors

For a small category K enriched over a suitable monoidal category V, the free completion of K under colimits is the presheaf category [K*,V]. If K is large, its free completion under colimits is the V-category PK of small presheaves on K, where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on PK.Comment: 17 page

Monads on Categories of Relational Structures

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 ? of a Horn theory understood as a strict upper bound on the number of premisses in its axioms; key examples include partial orders (? = ?) or metric spaces (? = ??). We establish a bijective correspondence between ?-accessible enriched monads on the given category of relational structures and a notion of ?-ary algebraic theories (i.e. with operations of arity < ?), with the syntax of algebraic theories induced by the relational signature (e.g. inequations or equations-up-to-?). 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 ??-ary algebraic theory of metric completion. The theory-to-monad direction of our correspondence remains true for the case of ?-ary algebraic theories and ?-accessible monads for ? < ?, e.g. for finitary theories over metric spaces

Quasivarieties and Varieties of Ordered Algebras: Regularity and Exactness

We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets. We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, in the enriched sense)

Notions of Lawvere theory

Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how this equivalence, and the basic results of universal algebra, can be generalized in three ways: replacing Set by another category, working in an enriched setting, and by working with another class of limits than finite products. An important special case involves working with sifted-colimit-preserving monads rather than filtered-colimit-preserving ones.Comment: 27 pages. v2 minor changes, final version, to appear in Applied Categorical Structure

Two-dimensional regularity and exactness

We define notions of regularity and (Barr-)exactness for 2-categories. In fact, we define three notions of regularity and exactness, each based on one of the three canonical ways of factorising a functor in Cat: as (surjective on objects, injective on objects and fully faithful), as (bijective on objects, fully faithful), and as (bijective on objects and full, faithful). The correctness of our notions is justified using the theory of lex colimits introduced by Lack and the second author. Along the way, we develop an abstract theory of regularity and exactness relative to a kernel--quotient factorisation, extending earlier work of Street and others.Comment: 37 page