Free-algebra functors from a coalgebraic perspective

Given a set $\Sigma$ of equations, the free-algebra functor $F_{\Sigma}$ associates to each set $X$ of variables the free algebra $F_{\Sigma}(X)$ over $X$. Extending the notion of \emph{derivative} $\Sigma'$ for an arbitrary set $\Sigma$ of equations, originally defined by Dent, Kearnes, and Szendrei, we show that $F_\Sigma$ preserves preimages if and only if $\Sigma \vdash \Sigma'$, i.e. $\Sigma$ derives its derivative $\Sigma'$. If $F_\Sigma$ weakly preserves kernel pairs, then every equation $p(x,x,y)=q(x,y,y)$ gives rise to a term $s(x,y,z,u)$ such that $p(x,y,z)=s(x,y,z,z)$ and $q(x,y,z)=s(x,x,y,z)$. In this case n-permutable varieties must already be permutable, i.e. Mal'cev. Conversely, if $\Sigma$ defines a Mal'cev variety, then $F_\Sigma$ weakly preserves kernel pairs. As a tool, we prove that arbitrary $Set-$endofunctors $F$ weakly preserve kernel pairs if and only if they weakly preserve pullbacks of epis

Duality of Equations and Coequations via Contravariant Adjunctions

International audienceIn 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