It is a classical result of categorical algebra, due to Lawvere and Linton, that finitary varieties of algebras (in the sense of Birkhoff) are dually equivalent to finitary monads on $Set$. Recent work of Ad\'amek, Dost\'al, and Velebil has established that analogous results also hold in certain enriched contexts. Specifically, taking $V$ to be one of the cartesian closed categories $\mathsf{Pos}$, $\mathsf{UltMet}$, $\omega$-$\mathsf{CPO}$, or $\mathsf{DCPO}$ of respectively posets, (extended) ultrametric spaces, $\omega$-cpos, or dcpos, Ad\'amek, Dost\'al, and Velebil have shown that a suitable category of $V$-enriched varieties of algebras is dually equivalent to the category of strongly finitary $V$-monads on $V$. In this paper, we extend and generalize these results in two ways: by allowing $V$ to be an arbitrary complete and cocomplete cartesian closed category that is concrete over $Set$, and by also considering the multi-sorted case. Given a set $S$ of sorts, we define a suitable notion of (finitary) $V$-enriched $S$-sorted variety, and we say that a $V$-monad on the product $V$-category $V^S$ is strongly finitary if its underlying $V$-endofunctor is the left Kan extension of its restriction to a suitable full sub-$V$-category of $V^S$. Our main result is that the category of $V$-enriched $S$-sorted varieties is dually equivalent to the category of strongly finitary $V$-monads on $V^S$. By taking $S$ to be a singleton and $V$ to be $\mathsf{Pos}$, $\mathsf{UltMet}$, $\omega$-$\mathsf{CPO}$, or $\mathsf{DCPO}$, we thus recover the aforementioned results of Ad\'amek, Dost\'al, and Velebil. We provide several classes of examples of $V$-enriched $S$-sorted varieties, many of which admit very concrete, syntactic formulations.

翻译：范畴代数的经典结果（归功于Lawvere和Linton）表明：Birkhoff意义下的有限泛代数簇与集合范畴上的有限单子呈对偶等价。Adámek、Dostál和Velebil近期的工作证实，在特定丰富化语境下也存在类似结论。具体而言，取$V$为Cartesian闭范畴$\mathsf{Pos}$（偏序集）、$\mathsf{UltMet}$（扩展超度量空间）、$\omega$-$\mathsf{CPO}$（$\omega$-有向完全偏序集）或$\mathsf{DCPO}$（有向完全偏序集）之一时，他们证明了：合适的$V$-丰富化代数簇范畴与$V$上的强有限单子范畴呈对偶等价。本文从两方面推广这些结果：其一，允许$V$为任意完备且余完备、且关于集合范畴具体的Cartesian闭范畴；其二，同时考虑多类情形。对于给定的类集$S$，我们定义了（有限）$V$-丰富化$S$-类簇的恰当概念，并称$V^S$上的$V$-单子为强有限的，若其底$V$-函子是对限制到$V^S的某满子$V$-范畴的函子的左Kan扩展。主要结论是：$V$-丰富化$S$-类簇范畴与$V^S$上的强有限$V$-单子范畴呈对偶等价。取$S$为单点集且$V$为$\mathsf{Pos}$、$\mathsf{UltMet}$、$\omega$-$\mathsf{CPO}$或$\mathsf{DCPO}$时，我们即可恢复Adámek、Dostál和Velebil的上述结果。本文还给出了若干$V$-丰富化$S$-类簇的例子类，其中许多具有非常具体的句法形式。