Duality of Equations and Coequations via Contravariant Adjunctions

Abstract

Abstract. In this paper we show duality results between categories of equations and categories of coequations. These dualities are obtained as restrictions of dualities between categories of algebras and coalgebras, which arise by lifting contravariant adjunctions on the base categories. By extending this approach to (co)algebras for (co)monads, we retrieve the duality between equations and coequations for automata proved by Ballester-Bolinches, Cosme-Llópez and Rutten, and generalize it to dynamical systems.

Guarded Kleene Algebra with Tests: Coequations, Coinduction, and Completeness

“…Coequations have appeared in the coalgebra literature in a variety of contexts, e.g. [3,1,5,29,30], and notably in the proof of generalized Eilenberg theorems [35,2]. The use of coequations in completeness proofs is, as far as we are aware, new.…”

How to write a coequation

“…It describes an unusual approach to dualising Birkhoff's HSP theorem which reduces the problem to the ordinary version of the theorem by turning every coalgebra X Ñ F X into an algebra 2 F X Ñ 2 X , and by introducing an infinitary equational logic extending that of complete atomic boolean algebras to define varieties of such algebras. The idea of establishing a bridge between coequations and equations was explored again in [80] and [88] where conditions for a full duality between equations and coequations are given.…”

“…The naturality of the inverse in not strictly necessary, but makes the presentation easier, see[68] 12. The existence of a natural transformation δ ´1 : GL Ñ T G is also key to the duality between varieties and covarieties and between equations and coequations developed in[88]. It is also crucial to strong completeness proofs in coalgebraic modal logic[81,86].…”

Minimisation in Logical Form

“…in [17] to relate coalgebraic and algebraic specifications in terms of observations and constructors. In this context most notable is the use of Stone-type dualities between automata and varieties of formal languages [33,34,59] which recently culminated into a general algebraic and coalgebraic understanding of equations, coequations, Birkhoff's and Eilenbergtype correspondences [4,5,14,[62][63][64].…”