余代数视角下的自由代数函子
Free-algebra functors from a coalgebraic perspective
摘要：
给定一组方程$Σ$，自由代数函子$F_Σ$将每个变量集合$X$与$X$上的自由代数$F_Σ(X)$相关联。扩展的概念，\emph{derivative} $Σ'$的任意一组$Σ$方程，最初定义的登特，Kearnes和Szendrei，我们表明，$F_Σ$保留原像当且仅当$Σ\vdash Σ'$，即$Σ$衍生$Σ'$。如果$F_Σ$弱保持核对，则每个方程$p(x,x,y)=q(x,y,y)$产生项$s(x,y,z,u)$，使得$p(x,y,z)=s(x,y,z,z)$和$q(x,y,z)=s(x,x,y,z)$。在这种情况下，n-可置换变种必须已经是可置换的，即Mal'cev。相反，如果$Σ$定义了一个Mal'cev簇，那么$F_Σ$弱保留核对。作为一种工具，我们证明了任意$Set-$内函子$F$弱保持核对的充要条件是它们弱保持epis的回调。
英文摘要：
Given a set $Σ$ of equations, the free-algebra functor $F_Σ$ associates to each set $X$ of variables the free algebra $F_Σ(X)$ over $X$. Extending the notion of \emph{derivative} $Σ'$ for an arbitrary set $Σ$ of equations, originally defined by Dent, Kearnes, and Szendrei, we show that $F_Σ$ preserves preimages if and only if $Σ\vdash Σ'$, i.e. $Σ$ derives its derivative $Σ'$. If $F_Σ$ 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 $Σ$ defines a Mal'cev variety, then $F_Σ$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
