Duality between geometric quotients and subtoposes ($\{{\mathbb{T}}^{\prime }\text{ geometric quotient of }\mathbb{T}\}\leftrightarrow \mathrm{Subtop}(\mathbf{Set}[\mathbb{T}])$)

Each geometric quotient theory is classified by the corresponding subtopos of the original classifying topos. Extra axioms impose extra covers on the geometric syntactic topology. Conversely additional definable covers give the corresponding deductively closed quotient. Under Morita equivalence of geometric theories, transporting a subtopos produces a corresponding quotient of the other theory, with equivalent classifiers. Stronger axioms correspond to smaller subtoposes under inclusion.