Free-algebra functors from a coalgebraic perspective

Abstract

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} ${\Sigma }^{\prime }$ 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 $\Sigma ⊢{\Sigma }^{\prime }$, i.e. $\Sigma$ derives its derivative ${\Sigma }^{\prime }$. 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.