Regular theory ($\mathbb{T}$)

A regular theory has axioms that are sequents between regular formulas. It has a regular syntactic category and a classifying topos of sheaves for the regular coverage. Its regular formulas enjoy the disjunction property of regular theories when a finite disjunction is allowed as an outer coherent conclusion.

If a regular theory proves an outer coherent sequent $\varphi ⊢{\bigvee }_{i=1}^n{\psi }_i$ with all formulas regular, it proves $\varphi ⊢{\psi }_i$ for some $i$. In the classifying topos, the corresponding subobjects cover the representable associated with $\varphi$. Irreducibility of that representable and full faithfulness of the syntactic Yoneda embedding produce the desired derivation.

The regular syntactic category has regular formulas-in-context as objects and provably functional regular relations as arrows. It has finite limits and stable image factorizations, with existential quantification supplying images. The regular coverage on this category presents the classifying topos of the theory.