Meaning of distributivity (a + (b × c) ≅ (a + b) × (a + c)) for general Lawvere theories

Ascertain the conceptual meaning and significance, for an arbitrary Lawvere algebraic theory, of requiring that the opposite category Op(C) be distributive—equivalently, determine the interpretation of the isomorphism a + (b × c) ≅ (a + b) × (a + c) in general Lawvere-theoretic terms.

Background

The paper’s two-dimensional stable semantics relies on categories C with both products and coproducts for which Op(C) is distributive, yielding the unusual isomorphism a + (b × c) ≅ (a + b) × (a + c) in C. This distributivity condition ensures that the sifted colimit completion Func(C^op, Set)[×] is cartesian closed and that the Yoneda embedding preserves exponentials.

Although this property is known to hold in certain algebraic categories (e.g., distributive lattices, commutative rings), its general conceptual interpretation within the framework of Lawvere algebraic theories is not clarified in the paper, motivating a precise articulation of its meaning and role.