Clones, coclones, and the duality between relations and operations
Eli Hazel – 29 May 2024, 5 June 2024
In general (universal) algebra, a concrete clone describes algebraic structure in terms of the operations available on a fixed base set. Further,
concrete clones may be regarded as representations of abstract clones, known to be equivalent to Lawvere's algebraic theories in category
theory. Relational clones arise as fixpoints of a Galois connection (adjunction) between operations and relations on a base set; concrete clones
are the fixpoints on the other side. When the base set is finite, clones and coclones form dually isomorphic lattices --- a duality between clones of
operations and clones of relations.
We will present our work on a category theoretic treatment of this situation in talks over two consecutive weeks:Part I. Syntactic categories and functorial semantics.
We will see that a relational clone corresponds to a certain category, and these `coclone theories' fit in with the formalism of a crowning
achievement of the categorical approach to logic. We proceed to describe functorial semantics, to characterise the categories of models, and to
consider a number of examples along the way.Part II. Abstract coclones, duality.
A key insight into the structure of the syntactic category of a relational clone provides a basis from which we introduce the notion of `abstract
coclone'. This leads us to an account of the duality between clones and coclones as a dual equivalence between categories whose objects on each
side are certain classes of finite limit theories; the algebraic (Lawvere) theories of the clones are recoverable as certain subcategories thereof.This work provides an answer to an invitation extended by editors John Power and Cai Wingfield in the Proceedings of the 2013 Workshop on
Algebra, Coalgebra and Topology to obtain a category theoretic treatment of the connection between clones and coclones in terms of Lawvere
theories.
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