On reflective subcategories of varieties (Q1886829)

A regular generator in a category \(\mathcal K\) is a small collection \(\mathcal F\) of objects such that, for every object \(K\) of \(\mathcal K,\) the canonical morphism \(e_K:\coprod_{A\in\mathcal F}\mathcal K (A,K)\circ A\to K\) is well defined (i.e., the coproduct in the domain exists) and is a regular epimorphism (\(M\circ A\) denotes the copower of \(A\) indexed by \(M\)). In this paper the full reflective subcategories of varieties are characterized as the cocomplete categories with a regular generator, or as classes of algebras presented by ``pre-equations'', i.e. formulas of the following form \[ \forall (x_{u})_{u\in U}[\wedge_{i\in I}\alpha _{i}(x_{u})\to\;\exists !(y_{v})_{v\in V} \wedge_{j\in J} \beta _{j}(x_{u},y_{v})]. \] As a byproduct, the authors present a solution to the problem of describing \(\omega\)-orthogonality classes of locally finitely presentable categories in terms of closure properties.