Coherent category

A coherent category has finite limits, pullback-stable regular-epi/mono image factorizations, and finite unions of subobjects stable under pullback. Each subobject lattice is distributive. These structures interpret all constructors of coherent logic.

A functor between coherent categories is coherent when it preserves finite limits, image factorizations and finite unions of subobjects, including the empty union. It consequently preserves the interpretation of every coherent formula. A coherent functor from a coherent syntactic category to a topos is equivalent to a model of the corresponding coherent theory; natural transformations correspond to ordinary model homomorphisms.

The coherent topology on a small coherent category is generated by finite families of arrows whose image subobjects have union equal to the whole target. Pullback stability of images and unions and closure under composition make these a coverage. On a coherent syntactic category, this is the site topology used to construct the classifying topos of its coherent theory.

In a coherent category, maps $A\to Y$ and $B\to Y$ agreeing on the intersection define a relation from $A\cup B$ to $Y$ by taking the union of their graphs. It is total because the two subobjects cover their union, and single-valued because the maps agree on overlap. Total functional relations define morphisms, giving a unique glued map. This is the pushout universal property without assuming a general supply of pushouts.