Classifying topos ($\mathrm{Geom}(\mathcal{F},{\mathcal{E}}_{\mathbb{T}})\simeq \mathbb{T}\text{-}\mathrm{Mod}(\mathcal{F})$)

A topos classifies a geometric theory when geometric morphisms into it from any Grothendieck topos correspond naturally to internal models of that theory. Pulling back one generic model gives the corresponding model. For a finitary algebraic theory, the classifying topos is the covariant functor category on finitely presented models.

On the opposite of finitely presented commutative rings, cover the zero ring by the empty family and cover $A$ by $A/(a),A/(b)$ whenever $ab=0$. The generated topology classifies nontrivial integral domains, with generic model the sheafified tautological ring. It is not subcanonical: at $\mathbb{Z}/4$, the covering quotient to $\mathbb{Z}/2$ identifies the distinct sections $0$ and $2$ of the polynomial-ring representable.

In the generic domain, negation of simultaneous invertibility of finitely many elements implies that one is zero. At a ring stage, localize at their product $t$. All become units, so a tuple satisfying the negation makes the localized stage empty. The empty-cover criterion for the domain-classifying site then says $A[1/t]=0$, hence $t$ is nilpotent. Repeated zero-product covers force one factor to vanish locally. The two-variable property conversely forces the integral-domain axiom in any nontrivial internal ring.

For a finitely presented ring $A$, its sheafified representable is initial exactly when $A$ is the zero ring. The zero ring has a generating empty cover. Every nonzero ring maps to a field by quotienting by a maximal ideal, and that set-based domain gives a point at which the representable has a section, precluding initiality.

Adding coherent or geometric axioms to a theory determines a coverage on a site for its classifying topos. The antecedent presentation is covered by presentations where its consequent disjuncts and witnesses hold. Sheafification imposes those axioms on the generic model. An inconsistent antecedent gets an empty cover.

In $[{\mathbb{T}}_{fp},\mathbf{Set}]$, the generic model assigns to each finitely presented algebra its underlying set at each sort, with the algebraic operations pointwise. It is covariant on algebra homomorphisms. Pullback along a geometric morphism yields its classified internal model.