This article fully characterizes categorically universal varieties of (0, 1)-lattices (that is, lattices with a least element 0 and a greatest element 1 regarded as nullary operations), thereby concluding a series of partial results [3, 5, 8, 10, also 14] which originated with the proof of categorical universality for the variety of all (0, 1)-lattices by Grätzer and Sichler [6].
A category C of algebras of a given type is universal if every category of algebras (and equivalently, according to Hedrlín and Pultr [7 or 14], also the category of all graphs) is isomorphic to a full subcategory of C. The universality of C is thus equivalent to the existence of a full embedding Φ : G→C of the category G of all graphs and their compatible mappings into C. When Φ assigns a finite algebra to every finite graph, we say that C is finite-to-finite universal.
