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A family of definable arrows covers a formula when the theory proves that their images jointly exhaust that formula. Such covers encode disjunction and existential quantification. The resulting site presents the classifying topos of the geometric theory. Adding geometric axioms adds covering sieves, yielding the duality between geometric quotients and subtoposes.

Geometric theories are Morita-equivalent when their classifying toposes are equivalent. This gives equivalent internal model categories pseudonaturally in every Grothendieck topos. A common classifying topos transports intrinsic invariants between different presentations; agreement of set-valued model categories alone is insufficient. The duality between geometric quotients and subtoposes transfers geometric theory extensions along such an equivalence.

Morita equivalence of geometric theories means equivalence of their classifying toposes. Equivalently, their categories of models in every Grothendieck topos are equivalent pseudonaturally with respect to inverse image. Agreement only of their set-based model categories, without this natural internal-model structure, is not the definition.

Suppose $\mathbf{Set}[\mathbb{T}]\simeq \mathbf{Set}[\mathbb{S}]=\mathcal{E}$. A property expressed intrinsically in terms of the topos is the same under either presentation. One can therefore translate a site or logical characterization of that property from $\mathbb{T}$ into one for $\mathbb{S}$. Examples include Booleanity, connectedness, atomicity and the structure of the subtopos lattice. The bridge is the common invariant $\mathcal{E}$, rather than an assumed literal identification of the two signatures.

The duality between geometric quotients and subtoposes makes this precise. A geometric quotient theory ${\mathbb{T}}^{\prime }$ of $\mathbb{T}$ adds geometric axioms over the same signature. Quotients are identified when they prove the same geometric sequents. The theorem givesThe subtopos corresponding to ${\mathbb{T}}^{\prime }$ is its classifying topos. Stronger axioms correspond to smaller subtoposes under inclusion.

The site mechanism explains the correspondence. On ${\mathcal{C}}_{\mathbb{T}}$, an additional sequent requires the associated family of definable images to cover its antecedent. Adding these covering sieves produces a topology $J^{\prime }\supseteq J_{\mathbb{T}}$ and a geometric embeddingConversely, subtoposes correspond to such larger topologies; requiring their extra definable covers gives the corresponding deductively closed quotient theory. Pulling the universal model into the subtopos supplies the universal model of the quotient.

Given a quotient ${\mathbb{T}}^{\prime }$, transport its subtopos along the chosen equivalence of classifying toposes. Apply the duality again to obtain a quotient ${\mathbb{S}}^{\prime }$ of $\mathbb{S}$. Their classifying toposes are equivalent, soThis is an explicit transfer principle for whole families of theory extensions and their order relations. Intrinsic constructions such as open, closed or Boolean subtoposes can likewise be described in each syntax. Different descriptions may look unrelated, but their equality is explained by the common subtopos. No universal translation of individual formulas is asserted without choosing the relevant equivalence and interpretations.

A subtopos is a Grothendieck topos embedded by a geometric embedding, considered up to equivalence over the ambient topos. On a site $(\mathcal{C},J)$, subtoposes correspond to Grothendieck topologies $J^{\prime }\supseteq J$. On a classifying topos this is the duality between geometric quotients and subtoposes.