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A coverage specifies covering families on a small category, stable under refinement along arrows: for a covering family into $C$ and $f:B\to C$, there is a covering family into $B$ whose composites with $f$ factor through the original family. Passing to generated sieves and imposing maximality and local transitivity gives its associated Grothendieck topology, with the same sheaves on a site. One can equivalently specify the saturated covering sieves directly. Rigidity refers to these covering sieves, so it is unchanged by replacing a family presentation by its generated topology.

The dense sieves form the double-negation Grothendieck topology. Its sheaf topos is a Boolean topos: the union of a subobject with its pseudocomplement is dense and thus becomes the whole object after sheafification. This need not be the atomic topology, which requires every nonempty sieve to be dense.

A Grothendieck topos is a category equivalent to the sheaves on a site for a small Grothendieck topology. It is an elementary topos with all small colimits and a small generating family. Small disjoint coproducts, effective quotients and the set of subobjects of any fixed object allow many constructions by unions.

A sieve is J-closed when membership is local with respect to the Grothendieck topology $J$. Pullback preserves J-closedness. Their presheaf is the subobject classifier of the sheaf topos: a section of a sheaf is sent to the sieve of arrows on which it belongs to the chosen subsheaf.

A dense sieve, also called a persistently nonempty sieve, is one whose every pullback sieve on a category is nonempty: a member factoring through $f$ has the form $fh$, corresponding to a member $h$ of $f^{\ast }R$. The maximal sieve on a category clearly has this property. If $R$ is persistently nonempty, then $u^{\ast }R$ is too: for any $v$ into the domain of $u$, apply density to $uv$ to find an extension $uvh\in R$, giving $vh\in u^{\ast }R$.

For transitivity, suppose $R$ is dense on $n$ and $S$ is a sieve on a category such that $r^{\ast }S$ is dense for each $r\in R$. Given $f:m\to n$, choose $h$ with $fh\in R$. Density of $(fh{)}^{\ast }S$, tested on its identity arrow, supplies $k$ with $fhk\in S$. Thus $S$ is dense. These are the maximality, pullback and transitivity axioms of a Grothendieck topology. Therefore the specified sieves on a category form the coverage $J$, namely the double-negation topology. The empty sieve on a category is never covering, because its identity pullback is empty.

A geometric formula is built from atomic formulas, including equality, using finite conjunctions, arbitrary set-indexed disjunctions and existential quantification in a finite variable context. Truth is the empty conjunction and falsity the empty disjunction. A geometric theory is a set of sequents $\varphi {⊢}_{\vec{x}}\psi$ between such formulas over a many-sorted signature. The context supplies the universal force of an axiom; unrestricted universal quantification, implication and negation are not formula constructors in this fragment.

These choices are exactly suited to inverse image functors of geometric morphisms. Finite-limit preservation handles equality and conjunction; colimit and image preservation handle disjunction and existential quantification. Therefore an inverse image carries an internal model of a geometric theory to another model.

Construct the geometric syntactic category ${\mathcal{C}}_{\mathbb{T}}$ as follows. Objects are formulas in context $\{\vec{x}.\varphi \}$, modulo provable renaming and equivalence. A morphism to $\{\vec{y}.\psi \}$ is an equivalence class of formulas $\theta (\vec{x},\vec{y})$ that are provably total and single-valued:The last notation abbreviates componentwise equality. Identity is equality of the context variables; composition is existential conjunction over the intermediate tuple. The category has finite limits, formed through conjunction and equality.

Give it the geometric syntactic topology $J_{\mathbb{T}}$. A family of arrows represented by ${\theta }_i({\vec{y}}_i,\vec{x})$ into $\{\vec{x}.\varphi \}$ covers whenPullback stability is substitution, and transitivity comes from distributing existential conjunction through the covering disjunctions. Thus the generated sieves define a Grothendieck topology. Empty covers impose the interpretation of falsity as the initial object.

The classifying topos isIts universal model $U_{\mathbb{T}}$ assigns a sort $A$ the sheafified representable of $\{x^A.\top \}$, functions their definable graph morphisms and relations their definable subobjects. The syntactic covering conditions make the axioms valid in this model.

For the universal property, interpreting formulas in a model $M$ in a Grothendieck topos $\mathcal{F}$ gives a finite-limit-preserving, $J_{\mathbb{T}}$-continuous functor ${\mathcal{C}}_{\mathbb{T}}\to \mathcal{F}$: covers go to jointly epimorphic families precisely because the relevant sequents hold. The Diaconescu equivalence for geometric morphisms associates to this functor a geometric morphism $\mathcal{F}\to \mathbf{Set}[\mathbb{T}]$. Conversely, pulling $U_{\mathbb{T}}$ back along a geometric morphism gives a model. The constructions on models and their homomorphisms are inverse up to natural isomorphism, yieldingnaturally in $\mathcal{F}$. This proves the required classifying property, not merely classification of set-valued models.

In a small regular category $\mathcal{C}$, the regular coverage has one-arrow covers: each regular epimorphism $\alpha :B\twoheadrightarrow A$ is a covering family by itself. Identity maps, composites and pullbacks of such arrows are again covers, so this is a basis for a Grothendieck topology. A presheaf is a sheaf precisely when every compatible section over a covering arrow descends uniquely along that arrow.

It is subcanonical. A section of the representable presheaf $yX=\mathcal{C}(-,X)$ over $B$ is a map $g:B\to X$. Its matching condition is $g{\pi }_1=g{\pi }_2$ on the kernel pair $B{\times }_{A}B$. Since a regular epimorphism is the coequalizer of its kernel pair, there is a unique $\bar{g}:A\to X$ with $\bar{g}\alpha =g$. This is exactly the sheaf condition for $yX$.

For a subfunctor $F^{\prime }\subseteq F$ of a regular-coverage sheaf, define local membership byThis is a subfunctor. If $t:C\to A$ and $\alpha$ witnesses membership, pull back $\alpha$ along $t$; the resulting cover of $C$ witnesses membership of $F(t)x$, because $F^{\prime }$ is stable under restriction.

It is a sheaf as well. Let $\beta :B\twoheadrightarrow A$ cover and let $y\in F^{\prime \prime }(B)$ satisfy the kernel-pair matching condition. The sheaf $F$ gives a unique $x\in F(A)$ with $F(\beta )x=y$. Choose a cover $\alpha :C\twoheadrightarrow B$ with $F(\alpha )y\in F^{\prime }(C)$. The composite $\beta \alpha$ is a cover of $A$ and witnesses $x\in F^{\prime \prime }(A)$. Uniqueness comes from $F$. Thus the inclusion $F^{\prime \prime }\hookrightarrow F$ is a mono between sheaves and is closed for the associated local operator.

Moreover, any closed subobject $S\subseteq F$ containing $F^{\prime }$ is itself a sheaf. If $x\in F^{\prime \prime }(A)$, its restriction along a witnessing cover lies in $S(B)$, and the sheaf condition for $S$ descends it to a section in $S(A)$. Its image in $F(A)$ is $x$ by uniqueness. Hence $F^{\prime \prime }$ is the least closed subobject containing $F^{\prime }$, andThis proves the local membership closure for the regular coverage, with a single regular epimorphism as witness.

Suppose $yA$ is the union, in the sheaf topos, of a family of subsheaves $F_i$. The union in presheaves is the pointwise union $F^{\prime }={\bigcup }_i{F}_i$; its closure is the union in sheaves. Thus $1_A\in F^{\prime \prime }(A)$. A witnessing cover $\alpha :B\twoheadrightarrow A$ belongs to $F_i(B)$ for one index $i$. The two restrictions of $\alpha$ to its kernel pair agree. Since $F_i$ is a subsheaf, this matching section descends in $F_i$ to $1_A\in F_i(A)$. Every map $t:C\to A$ is the restriction of $1_A$ along $t$, so $F_i=yA$. Therefore every representable sheaf is irreducible, even with respect to arbitrary unions. There are no empty covering families in this coverage, and $1_A$ also shows that the representable cannot be initial.

Finally, in the regular syntactic category of a regular theory, the context formula $\varphi$ defines an object $A$. Each formula $\varphi \wedge {\psi }_i$ defines a mono $A_i\hookrightarrow A$, and the associated representable subsheaf is its interpretation inside $yA$ in the generic model. A derivation of the coherent disjunction says that these finitely many subsheaves have union $yA$. Irreducibility gives $y{A}_i=yA$ for some $i$. The Yoneda embedding is full and faithful, since the coverage is subcanonical, and therefore reflects this isomorphism. Hence $A_i\hookrightarrow A$ is invertible in the syntactic category: $\varphi$ entails ${\psi }_i$. ThusThe disjunction is an outer coherent conclusion; the theory and all constituent formulas are regular. This disjunction property of regular theories follows from single-arrow descent, rather than from ordinary first-order compactness.

The required condition is the common-refinement condition for nonempty-sieve coverage: for every pair $f:V\to U$, $g:W\to U$, there are arrows $h:T\to V$, $k:T\to W$ withNecessity follows by pulling back the nonempty sieve generated by $f$ along $g$: a member $k$ of the pullback sieve supplies such an $h$. Conversely, this condition makes the pullback of every nonempty sieve on a category nonempty. The maximal sieve is nonempty, and the transitivity axiom holds: if a sieve $S$ is locally covering along every member of a nonempty covering sieve $R$, choose $f\in R$ and then $g\in f^{\ast }S$; their composite is in $S$. Hence the nonempty sieves form a Grothendieck topology, called the atomic topology.

For all functions between nonempty finite sets, the two maps from a singleton to different points of a two-point set have no common refinement. Every potential domain remains nonempty, so the two constant composites cannot agree. The condition fails.

For surjections it holds: $V{\times }_{U}W$ is nonempty and both projections are surjective. Work from now on in a small skeleton $\mathcal{D}$ of nonempty finite sets and surjections. Every morphism $f:P\twoheadrightarrow U$ is a regular epimorphism, with kernel pair $P{\times }_{U}P\rightrightarrows P$, and is the coequalizer of that pair in $\mathcal{D}$.

A matching family in a representable $yV$ on the sieve generated by $f$ is determined by a surjection $t:P\to V$ equalizing that kernel pair. It factors uniquely through a function $U\to V$, which is surjective because $t$ is. This gives the unique amalgamation. A general nonempty covering sieve contains such an $f$; after amalgamating there, common refinements with any other member force agreement on the entire sieve. Therefore every representable is a sheaf, so this atomic site is subcanonical.

For any sheaf $F$, every restriction $F(f)$ is injective: equality after a covering arrow forces equality by the separated part of the sheaf condition. We shall also use descent along any surjection $q:n\twoheadrightarrow k$:is an equalizer of sets. These are the descent identities for the atomic finite-surjection site.

Consider primitive $x\in F(m)$, $y\in F(n)$ with a common restriction along $\alpha :P\twoheadrightarrow m$, $\beta :P\twoheadrightarrow n$. Suppose $a,b\in P$ have $\alpha (a)=\alpha (b)$ but $\beta (a)\ne \beta (b)$. Let $q:n\twoheadrightarrow n-1$ identify just the two points $\beta (a),\beta (b)$. Define the finite nonempty setBoth projections $t_1,t_2:T\twoheadrightarrow P$ are surjective, since $T$ contains every diagonal pair. There is also a surjectionIndeed the target consists of diagonal pairs, which are reached because $\beta$ is surjective, and the two off-diagonal pairs corresponding to $\beta (a),\beta (b)$, reached by $(a,b)$ and $(b,a)$.

Since $\alpha t_1=\alpha t_2$, the common-restriction equality gives $F(\beta t_1)y=F(\beta t_2)y$. If ${\pi }_1,{\pi }_2$ are the target kernel-pair projections, this isInjectivity of $F(h)$ gives the kernel-pair matching condition on $y$. Descent along $q$ then writes $y=F(q)y^{\prime }$, contradicting primitivity. Thus $\ker \alpha \subseteq \ker \beta$; interchange the roles to obtain equality. This is the primitive-element kernel rigidity lemma.

Equal kernels produce a unique bijection $\gamma :m\to n$ with $\beta =\gamma \alpha$. Now $F(\alpha )x=F(\alpha )F(\gamma )y$, and injectivity impliesIn particular equivalent primitive elements have the same cardinality and differ only by transport along a bijection.

Every element $z\in F(p)$ descends to a primitive one: whenever it is not primitive, descend along a surjection reducing the cardinality by one; this process terminates at or before cardinality one. Kernel rigidity shows that all primitive ancestors of $z$ lie in one equivalence class. Let $F_C(P)$ be the elements with primitive-ancestor class $C$. Restriction along a surjection preserves this class, so each $F_C$ is a subfunctor andpointwise. Each $F_C$ is a sheaf. A matching family glues in $F$, and one member along a nonempty covering arrow already determines the primitive class of the glued element; it must be $C$.

Choose a representative primitive $x\in F(m)$ of $C$. The Yoneda map $ym\to F$ named by $x$ has image exactly $F_C$. It reaches all descendants of $x$, and every equivalent primitive ancestor is its transport along a bijection. It is therefore pointwise surjective onto $F_C$ and is epic as a map of sheaves. We obtain the primitive decomposition of an atomic finite-surjection sheafThis includes the empty coproduct for an empty sheaf.

Each nonempty $F_C$ is an atom in a topos. If a sheaf subobject $S\hookrightarrow F_C$ has an element $F(\alpha )x$ at some $P$, membership descends along the covering surjection $\alpha :P\to m$, so $x$ belongs to $S$. All its restrictions then belong to $S$, giving $S=F_C$. Thus every subobject of any $F$ selects entire components of this coproduct, and its complementary selection is again a sheaf subobject. Its characteristic map sends selected components to $\top$ and all others to $\perp$.

The constant two-element presheaf is a sheaf: a matching family on a nonempty sieve has the same value on all its arrows, since any two have a common refinement. The value extends uniquely. It therefore supplies these characteristic maps, with truth the inclusion of the $\top$ value. Equivalently, a J-closed sieve here is either empty or maximal, because every nonempty sieve covers. Hence

A site is a category equipped with a Grothendieck topology, specifying which sieves are covering. The sheaves on a site satisfy unique amalgamation of matching families on those sieves. A small site presents a Grothendieck topos.

A Zariski cover of a scheme is a family of open immersions whose images cover its underlying topological space. Such families define the Zariski topology as a Grothendieck topology on the category of schemes; a cover need not be finite.