Clones and the characterization of varieties as categories In Grätzer's Universal Algebra, he mentions the clone of an algebra at some point, but very quickly and not diving into its properties etc. He mentions that the study of clones has been applied to the characterization of varieties as categories, and this remark sparked my curiosity.
My question is: how precise is that characterization of varieties among categories ? Is it elementary, or very technical ? Has there been progress on this topic since Grätzer wrote his book ? 
 A: There are a variety of categorical perspectives on clones.
For your purpose, the notion of an algebraic category and notions derived from it, namely bounded monadic categories seem to be what you are looking for, though cartesian multicategories may also be an interesting place to look.
From the nLab page for algebraic category and the monadicity theorem: a concrete category, i.e. a category equipped with a faithful functor, $U$, into $\mathbf{Set}$ is bounded monadic iff $U$ is monadic and for every $\kappa$-directed colimit for some cardinal $\kappa$, the universal cocone is surjective. For $\mathbf{Set}$-valued functors, $U$ being monadic can be checked via checking that it has a left adjoint and that $U$ creates kernel pairs and equalizers. Every monadic category is the category of algebras for some algebraic variety. A bounded monadic category is exactly a category of algebras for an algebraic variety with at most a set of operations.
Somewhat in the vein of Kevin Carlson's comment, the notion of an algebraically exact category as a category which has limits and sifted colimits which distribute over those limits. A category of algebras for a variety is algebraically exact. You may find the referenced paper How Algebraic is Algebra interesting.
