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For a categorical presheaf $P$ on the open subsets, define the functors on objects byTo prove $L⊣\Delta$, a natural transformation $t:P\to \Delta A$ must satisfyfor every open $U$, since the restrictions of $\Delta A$ are identities. Thus it is determined by the function $P(\varnothing )\to A$. Conversely such a function defines all $t_U$ by that equation, and the categorical presheaf restriction laws make the components compatible. HenceFor the other adjunction, a transformation $s:\Delta A\to P$ is determined by its component $s_S:A\to P(S)$, sinceEvery function $A\to P(S)$ supplies compatible components by this formula. Therefore $\mathrm{Nat}(\Delta A,P)\cong \mathrm{Set}(A,P(S))$, giving the adjoints to constant presheaves on open sets:These are adjoints of the stated constant-presheaf functor, with its value at every open set including the empty one; no sheafification is involved.

A poset becomes a category whose objects are its elements and whose hom-set $X(x,y)$ is a singleton if $x\le y$ and empty otherwise. Reflexivity supplies identity arrows and transitivity supplies composition; all possible composites are unique, so the category axioms hold. Antisymmetry says that isomorphic objects are equal. A functor between such categories is exactly an order-preserving function: an inequality must be sent to an inequality, and there is no further choice of arrow map.

The Freyd general adjoint functor theorem is as follows. Let $G:\mathcal{D}\to \mathcal{C}$, with $\mathcal{D}$ a complete category which is locally small. Then $G$ has a left adjoint if and only if it preserves all small categorical limits and satisfies the solution-set condition: for each $c\in \mathcal{C}$, there is a set-indexed familysuch that every $u:c\to Gd$ can be written $u=G(h)u_i$ for some $i$ and some $h:d_i\to d$. The target category need not be complete. We prove all the existence steps.

First establish the initial-object lemma for complete categories with a weakly initial set. Let $\mathcal{E}$ be locally small and complete, with a weakly initial set $\{w_i\}$, meaning every object receives a map from at least one $w_i$. Its categorical product $w={\prod }_i{w}_i$ is weakly initial: compose a categorical product projection with the map from the appropriate $w_i$. Local smallness makes $\mathcal{E}(w,w)$ a set. Let $e:v\to w$ be the simultaneous equaliser of every endomorphism of $w$ and the identity. Explicitly it is the equaliser of the two arrowswith components $k$ and $1_w$. Thus $ke=e$ for every endomorphism $k$, and $e$ is monic.

Every object $x$ receives a morphism from $v$, by composing $e$ with a map $w\to x$. To prove uniqueness, take parallel maps $a,b:v\to x$ and their equaliser $m:y\to v$. Weak initiality of $w$ gives $t:w\to y$. The composite $emt$ is an endomorphism of $w$, so $(emt)e=e$. Cancelling the monomorphism $e$ gives $mte=1_v$. Hence $m$ is both monic and a split epimorphism, and is therefore invertible. Since $am=bm$, this forces $a=b$. We have proved $v$ is initial, without using a class-sized categorical product or equaliser.

For sufficiency in the theorem, fix $c$ and consider the comma category $(c\downarrow G)$. Its objects are pairs $(d,u:c\to Gd)$; a morphism $h:(d,u)\to (d^{\prime },u^{\prime })$ satisfies $G(h)u=u^{\prime }$. This category is locally small because each hom-set is a subset of $\mathcal{D}(d,d^{\prime })$. It is complete: for a small diagram of such pairs, take the underlying categorical limit $d$ in $\mathcal{D}$. The arrows from $c$ form a cone to its image under $G$, and preservation of the categorical limit gives a unique $u:c\to Gd$ inducing them. The same universal property gives all mediators in the comma category. The solution-set family is a weakly initial set in this comma category, so the lemma supplies an initial pairFor $f:c\to c^{\prime }$, initiality defines a unique $Ff:Fc\to F{c}^{\prime }$ satisfying $G(Ff){\eta }_c={\eta }_{c^{\prime }}f$. Uniqueness proves the identity and composition laws, making $F$ a functor. Initiality also gives bijectionsThe definition of $Ff$ proves naturality in $c$, and postcomposition proves naturality in $d$. These bijections are the required adjunction $F⊣G$.

For necessity, if $F⊣G$, the single arrow ${\eta }_c:c\to GFc$ is a solution set: the transpose of any $u:c\to Gd$ gives its factorization through ${\eta }_c$. If $d$ is a limiting cone vertex of a small diagram $D_j$ in $\mathcal{D}$, then, naturally in $c$,These are the canonical cone-factorization bijections, so $Gd$ with its image projections is a categorical limit of $G{D}_j$. Thus $G$ preserves all small categorical limits, completing both directions of the theorem.

With $T=GF$, unit $\eta$ and multiplication $G\epsilon F$, the Eilenberg-Moore comparison functor isHere $G{\epsilon }_d:T(Gd)=GFGd\to Gd$ is the monad algebra action, and the underlying functor satisfies $UK=G$. The triangle identities and counit naturality provide the monad algebra laws and the compatibility of $Gh$, as required for this definition.

For each algebra for a monad $(A,a)$, consider the specific pair in $\mathcal{D}$Then the exact condition isEach pair is reflexive, with common section $F{\eta }_A$: the two composite identities follow from $a{\eta }_A=1_A$ and ${\epsilon }_{FA}F{\eta }_A=1_{FA}$. Thus existence of all reflexive coequalizers is sufficient, but the displayed specified pairs give the necessary-and-sufficient condition.

To prove it, take $h:FA\to d$ and its adjoint transpose $x=Gh{\eta }_A:A\to Gd$. The transpose of $hFa$ is $xa$, by naturality of $\eta$. The transpose of $h{\epsilon }_{FA}$ isMoreover counit naturality and a triangle identity giveBecause transposition is bijective, the coequalizing equation is therefore equivalent toThe right side says exactly that $x:(A,a)\to Kd$ is a monad algebra morphism.

For sufficiency, choose a coequalizer $q_a:FA\to L(A,a)$ for every monad algebra. Its universal property and the equivalence just proved give natural bijectionsTo see explicitly that the chosen objects form a functor, an monad algebra morphism $u:(A,a)\to (B,b)$ makes $q_{b}Fu$ coequalize the pair for $a$: use $ua=bTu$ and naturality of the counit. Hence there is a unique $Lu$ with $Lu{q}_a=q_{b}Fu$. Uniqueness proves its identity and composition laws and the naturality of the displayed bijections. Thus $L⊣K$.

Conversely, suppose $L⊣K$ with unit ${\lambda }_{(A,a)}:(A,a)\to KL(A,a)$. Transpose its underlying arrow $A\to GL(A,a)$ through $F⊣G$ to obtain $q_a:FA\to L(A,a)$. Since the unit is an monad algebra morphism, the equivalence above shows that $q_a$ coequalizes the specified pair. Any coequalizing $h:FA\to d$ transposes to an monad algebra map $(A,a)\to Kd$, which factors uniquely through the unit by $L⊣K$. Transposing back gives a unique factorization of $h$ through $q_a$. Thus $q_a$ is the required coequalizer. This proves both directions of the Left adjoint to the Eilenberg-Moore comparison functor criterion.

Use the elementary, or weak, topos axioms: finite limits and a power object $PA$ for every object $A$, naturally representing $\mathrm{Sub}(A\times X)$. No arbitrary products or coproducts are assumed. In particular $\Omega =P1$ is the subobject classifier. Inverse image of the membership relation defines $Pf:PB\to PA$ for $f:A\to B$.

Interchanging the factors in a relation gives natural bijectionsThus the contravariant power-object functor $P:{\mathcal{E}}^{\mathrm{op}}\to \mathcal{E}$ has left adjoint $P^{\mathrm{op}}$. Its adjunction unit ${\eta }_A:A\to PPA$ sends a generalized element $a$ to the predicate on $PA$ given by membership of $a$.

This unit is monic. If $x,y:T\to A$ have the same image under ${\eta }_A$, membership of $x$ and $y$ agrees in every $T$-parameterized subobject of $A$. Apply this to the graph of $x$ in $A\times T$: membership of $x$ is true, so membership of $y$ is true, which means $y=x$. This argument uses parameterized relations, not an assumption that global elements detect morphisms.

The functor $P$ reflects isomorphisms. Suppose $Pf$ is invertible. Then $PPf$ is invertible, and naturality $PPf{\eta }_A={\eta }_{B}f$ together with monicity of ${\eta }_A$ shows that $f$ is monic. Regard $f$ as a subobject of $B$. Its inverse image along itself is the whole of $A$, just as for the whole subobject of $B$. Those two subobjects correspond to maps $1\to PB$; since $Pf$ is monic, they coincide. Therefore the subobject $f$ is all of $B$, and $f$ is an isomorphism.

We now prove the preservation condition for the crude monadicity theorem. A reflexive pair in ${\mathcal{E}}^{\mathrm{op}}$ is a coreflexive pair $f,g:B\rightrightarrows A$ in $\mathcal{E}$, with $rf=rg=1_B$ for some $r:A\to B$. Let $e:E\hookrightarrow B$ be its equalizer. For any monomorphism $m$, define ${\exists }_m$ on power objects by composing a parameterized subobject with $m$; no general image construction is needed because $m$ is monic. Inverse image followed by this extension obeys $Pm{\exists }_m=1$.

Here both $f,g$ are monic. For a parameterized subobject $S$ of $B$, an element $b$ belongs to $g^{-1}(f(S))$ exactly when $gb=f{b}^{\prime }$ for some $b^{\prime }\in S$. Applying $r$ gives $b=b^{\prime }$, so this condition is precisely $b\in S$ and $fb=gb$. Consequently, as morphisms of power objects,If $h:PB\to Z$ equalizes $Pf,Pg$, these identities giveIt therefore factors uniquely through $Pe$, which is a split epimorphism with section ${\exists }_e$. Since $fe=ge$, $Pe$ also equalizes $Pf,Pg$. Hence $Pe$ is their coequalizer. This proves explicitly that power objects turn coreflexive equalizers into coequalizers.

Finite equalizers in $\mathcal{E}$ give all coequalizers of reflexive pairs in ${\mathcal{E}}^{\mathrm{op}}$, and we have shown that $P$ preserves them and reflects isomorphisms. The crude monadicity theorem now yields

Fix a set $I$ of cardinality $\kappa$. If $\mathcal{E}$ has $I$-indexed products in a category, its Eilenberg-Moore category for the double-power monad has them, since the forgetful functor creates existing limits. The displayed equivalence then gives $I$-products in ${\mathcal{E}}^{\mathrm{op}}$, hence $I$-coproducts in a category in $\mathcal{E}$.

For the converse, suppose those coproducts exist. The diagonal $\Delta :\mathcal{E}\to {\mathcal{E}}^I$ is a logical functor, because all the elementary topos structure in ${\mathcal{E}}^I$ is pointwise, and it has left adjoint ${\coprod }_I$. Apply power-object monadicity to both toposes. Under their monadic identifications, ${\Delta }^{\mathrm{op}}$ is the lift of $\Delta$ to the double-power Eilenberg-Moore categories. The base functor $\Delta$ is a right adjoint to ${\coprod }_I$, so the adjoint lifting theorem for monad algebra functors gives a left adjoint to its algebra lift. Its required reflexive coequalizers exist in ${\mathcal{E}}^{\mathrm{op}}$, since they are opposites of finite equalizers in $\mathcal{E}$. Taking opposites gives a right adjoint to $\Delta$, which is exactly the $I$-product functor. Thus the products and coproducts of fixed cardinality in a topos satisfy

Let $(G,\epsilon ,\delta )$ be a Cartesian comonad, so its endofunctor preserves finite limits. A coalgebra for a comonad is $(A,a)$ with $a:A\to GA$, ${\epsilon }_{A}a=1_A$ and ${\delta }_{A}a=Gaa$. Its morphisms satisfy $bf=Gfa$. Write $U:{\mathcal{E}}^G\to \mathcal{E}$ for the forgetful functor and $R(X)=(GX,{\delta }_X)$ for the cofree coalgebra. Transposition isThe forgetful functor creates finite limits, since $G$ preserves them. We construct both exponential objects and the subobject classifier.

For coalgebras $(A,a),(B,b)$, set $D=B^A$ and $W=(GB{)}^A$ in the ambient topos. Define $p,q:GD\to W$ by transposingwhere $GD\times GA\cong G(D\times A)$. Their cofree transposes are coalgebra morphismsLet $H$ be their coalgebra equalizer. For a coalgebra $(C,c)$, an arrow $C\to R(D)$ corresponds to a map $h:C\times A\to B$. Its evaluation respects coalgebra structure exactly whenUnder the displayed transpositions this is precisely the condition that it equalize $\hat{p},\hat{q}$. Thus ${\mathcal{E}}^G(C,H)\cong {\mathcal{E}}^G(C\times A,B)$ naturally, and $H$ is the required exponential in a coalgebra topos.

Let $\kappa :G\Omega \to \Omega$ classify $G\top :G1\cong 1\hookrightarrow G\Omega$. In $R(\Omega )$ take the equalizerThe truth map into $R(\Omega )$ is $G\top$ and factors through this equalizer. To verify its universal property, take an underlying subobject $m:S\hookrightarrow A$ of a coalgebra. It carries a compatible coalgebra structure exactly when $a$ maps $S$ into $GS$. Such a structure, if it exists, is unique because $Gm$ is monic; its laws follow by cancellation of $m,Gm,G^{2}m$. Moreover $a^{-1}(GS)\subseteq S$ always: applying ${\epsilon }_A$ to a point of $GS$ lands in $S$. Thus stability is equivalent to $S=a^{-1}(GS)$.

If $\chi :A\to \Omega$ classifies $S$, this last equality is exactlyThe cofree transpose $\hat{\chi }=G\chi a:A\to R(\Omega )$ satisfies this equation precisely when it factors through ${\Omega }_G$. Indeed transposing the equality $\hat{\chi }=G\kappa {\delta }_{\Omega }\hat{\chi }$ gives the equation above. Pulling back its truth map recovers $S$, because $\kappa$ classifies $G\top$. Hence ${\Omega }_G$ is the subobject classifier of a coalgebra topos. We have obtained finite limits, exponentials and a classifier, proving ${\mathcal{E}}^G$ is a topos.

Now let $f:\mathcal{H}\to \mathcal{E}$ be a geometric morphism, and put $L=f^{\ast }$, $V=f_{\ast }$, with unit $\eta$. Both functors preserve finite limits, so $G=LV$ is a Cartesian comonad on $\mathcal{H}$. Put $\mathcal{D}={\mathcal{H}}^G$. Its comparison functoris left exact and has right adjointTo check the adjunction, transpose an underlying map $LX\to A$ to $X\to VA$. The coalgebra-morphism equation becomes exactly the equality of the two composites into $VLVA$, so the transposed arrow factors uniquely through the displayed equalizer.

The counit $KJ(A,a)\to (A,a)$ is invertible. Applying $L$ to the defining equalizer gives the equalizer of $Ga,{\delta }_A:GA\rightrightarrows G^{2}A$. The structure map $a:A\to GA$ is exactly that equalizer: it equalizes by coassociativity; if $t:T\to GA$ equalizes, apply ${\epsilon }_{GA}$ to obtain $t=a{\epsilon }_{A}t$, giving its unique factorization through $a$. Thus the underlying comparison counit is an isomorphism, and the forgetful functor reflects isomorphisms. Therefore $J$ is full and faithful.

The adjunctions $U⊣R$ and $K⊣J$ defineHere $p^{\ast }=U$ is faithful, so $p$ is a surjective geometric morphism; $i_{\ast }=J$ is full and faithful, so $i$ is a geometric embedding. Their inverse-image composite is $UK=L$, establishing the surjection-embedding factorization of a geometric morphism.

For uniqueness, let $S$ be the class of arrows in $\mathcal{E}$ inverted by $f^{\ast }$. In any surjection-embedding factorization $f=i^{\prime }{p}^{\prime }$, conservativity of $p^{\prime \ast }$ shows that $S$ is also the class inverted by $i^{\prime \ast }$. The essential image of the fully faithful $i_{\ast }^{\prime }$ consists exactly of the $S$-local objects $Y$, namely those for which $\mathcal{E}(B,Y)\to \mathcal{E}(A,Y)$ is bijective for every $A\to B$ in $S$. One direction follows from the adjunction. For the converse, the reflector unit $Y\to i_{\ast }^{\prime }{i}^{\prime \ast }Y$ belongs to $S$; locality gives a retraction, and locality of the reflected object makes that retraction also a right inverse. Thus the unit is invertible. The image subcategory is consequently determined by $f$, independently of the factorization. The two intermediate toposes are equivalent over $\mathcal{E}$, and their inverse-image composites with the embeddings determine the surjection legs. This proves uniqueness up to equivalence.

A signature $\Sigma$ specifies sorts, finite-arity function symbols and finite-arity relation symbols. A coherent formula is built from atomic relations and equality using $\top ,\perp$, finite conjunction, finite disjunction and existential quantification. A coherent theory consists of coherent sequents $\varphi {⊢}_{\vec{x}}\psi$ in finite typed contexts. The context expresses universal quantification of the free variables; unrestricted universal quantification or negation is not a formula constructor here.

Construct the coherent syntactic category ${\mathcal{C}}_{\mathbb{T}}$ as follows. Objects are formulas in context $\{\vec{x}.\varphi \}$, modulo renaming and provable equivalence. An arrow to $\{\vec{y}.\psi \}$ is a formula $\rho (\vec{x},\vec{y})$, modulo provable equivalence, for which the theory provesThese say that the relation is supported on the specified objects, total and single-valued. The identity is equality on the context, and composition isThe logical rules give associativity and the identity laws. The empty-context true formula is terminal; conjunction in disjoint contexts gives products; conjunction with equality of the two functional relations gives equalizers. Existential quantification gives images, and finite disjunction gives unions of subobjects. Substitution preserves these constructions, so images and unions are stable under pullback. This makes ${\mathcal{C}}_{\mathbb{T}}$ a coherent category.

Let $M$ be a $\mathbb{T}$-model in a topos $\mathcal{E}$. Interpret each context as a finite product of its sort objects and each formula as its corresponding subobject. DefineThe interpretation of an arrow relation $\rho$ is a total single-valued relation between these objects. Its projection to the domain is both an epimorphism, by totality, and a monomorphism, by single-valuedness, hence invertible in a topos. Its other projection therefore determines a unique morphism. Composition agrees with relational composition, so $F_M$ is a functor. Finite limits interpret equality and conjunction; images interpret existential quantification; finite unions interpret disjunction, including the empty union for $\perp$. Consequently $F_M$ is a coherent functor.

Conversely there is a canonical syntactic model $U$ in ${\mathcal{C}}_{\mathbb{T}}$. Interpret a sort $s$ by $\{x:s.\top \}$, a function symbol by its term graph, and a relation symbol by its atomic formula-in-context subobject. Induction on formulas identifies the interpretation of every $\varphi$ in $U$ with its formula-in-context object. Every axiom is then an inclusion of the associated subobjects, because it is provable in $\mathbb{T}$.

For a coherent functor $F:{\mathcal{C}}_{\mathbb{T}}\to \mathcal{E}$, set $M_F=F(U)$. Preservation of finite limits, images and unions proves inductively that $F$ preserves all coherent interpretations, so every axiom holds in $M_F$. The same induction supplies natural isomorphismscompatible with the functional relations defining arrows. Thus $F\cong F_{M_F}$; directly on the sorts, operations and relations, $M_{F_M}\cong M$.

Finally these constructions agree on morphisms. A model homomorphism $h:M\to N$ induces maps on finite contexts. Atomic relations are preserved by definition; finite conjunction and disjunction and existential witnesses then show that each formula object maps to the corresponding formula object. Compatibility with functional relations makes these maps a natural transformation $F_M\to F_N$. Conversely a natural transformation between coherent functors restricts to its components on sort objects; naturality for term graphs preserves operations, and naturality for relation inclusions preserves relations. It is therefore a model homomorphism. The two correspondences are inverse. We have proved the categorical, not merely objectwise, equivalence

Work internally with the Heyting algebra $\Omega$ and the truth value $u$ of $U\hookrightarrow 1$. For $o(u)(p)=u\Rightarrow p$, the identitiesprove that it is an open local operator.

For a monomorphism $m:S\hookrightarrow X$ with characteristic map $\chi$, density means $u\Rightarrow \chi =\top$, equivalently $u\le \chi$. This says exactly that $U\times X$ is contained in $S$, or that the pullback mono $U\times S\to U\times X$ is invertible. Consequently

An object $X$ is an $o(U)$-sheaf exactly when the canonical map ${\eta }_X:X\to X^U$, taking a value to its constant function on $U$, is invertible. Indeed $T\times U\hookrightarrow T$ is dense for every $T$. The sheaf condition makes $\mathcal{E}(T,X)\to \mathcal{E}(T\times U,X)\cong \mathcal{E}(T,X^U)$ bijective for every $T$, which proves ${\eta }_X$ invertible by the Yoneda lemma. Conversely, if ${\eta }_X$ is invertible and $m:S\hookrightarrow T$ is dense, the isomorphism $S\times U\cong T\times U$ givesso $X$ is a sheaf.

The pullback functor $L:\mathcal{E}\to \mathcal{E}/U$ has right adjoint ${\Pi }_U$, sending $A\to U$ to $A^U$. Every map $X\times U\to A$ is automatically over $U$, since $U$ is subterminal object, proving this adjunction directly. Its counit $U\times A^U\to A$ is evaluation and is invertible. The inverse sends $a\in A$ to its image in $U$ together with the constant function $U\to A$ of value $a$; all elements of $U$ agree, so the composites are identities. The right adjoint is therefore full and faithful, and its image consists of objects fixed by $X\to X^U$. The sheaf characterization above proves

For a general local operator $j$, its dense monomorphism classifier is the ordered subobjectA mono is j-dense exactly when its characteristic map factors through $J$. If $j=o(u)$, then $J=\{p:u\le p\}$, which has least element $u$. This is least internally over every parameter object, not just among global points.

Conversely suppose a global $u:1\to J$ is internally least. Upward closure of $J$ gives $jp=\top$ exactly when $u\le p$, in every context. For $S\hookrightarrow X$ let $\overline{S}$ be its $j$-closure, with characteristic predicate $jp$. The mono $S\hookrightarrow \overline{S}$ is dense: restricting $jp$ to $\overline{S}$ gives truth. Its characteristic map therefore lands in $J$, so leastness of $u$ gives $U\cap \overline{S}\subseteq S$. Internally this isOn the other hand $ju=\top$, and applying $j$ to $u\wedge (u\Rightarrow p)\le p$ gives $j(u\Rightarrow p)\le jp$. Inflationarity then gives $u\Rightarrow p\le jp$. Both inequalities yieldThis proves the least dense truth characterizes an open local operator criterion without replacing its internal order condition by the weaker condition on global truth values.

Use one ring sort with $0,1,+,-,\cdot$ and the usual unital commutative ring equations. Write $\mathrm{Inv}(x)$ for the coherent formula $\exists y(xy=1)$. Add the coherent sequentsThis presents the coherent theory of local rings. For an ordinary nonzero ring satisfying these axioms, the nonunits form an ideal: a multiple of a nonunit cannot be a unit; and if nonunits $a,b$ had unit sum, multiplying by $(a+b{)}^{-1}$ would make two nonunits sum to $1$, contradicting the second axiom. They are also closed under additive inverses. This proper ideal contains every proper ideal, since a proper ideal contains no unit, and its quotient is a field; it is the unique maximal ideal. Conversely in a local ring, $x$ and $1-x$ cannot both belong to the unique maximal ideal, so one is a unit. The first sequent excludes the zero ring. In a topos, the disjunction asserts that these unit alternatives hold locally.

Let $\mathcal{C}$ be a small skeleton of the opposite of the category of finitely presented unital commutative rings; write $C_A$ for the object corresponding to $A$. The Zariski coverage on finitely presented rings is generated byand the empty cover of $C_0$. Localization is finitely presented, and its compatibility with base change proves the covering axioms.

Here is the classifying argument. For an internal commutative ring $M$ and a presentation $A=\mathbb{Z}[x_1,\dots ,x_n]/(p_1,\dots ,p_r)$, define $F_M(C_A)$ as the subobject of $M^n$ satisfying the polynomial equations $p_i=0$. Ring maps induce maps of these solution objects in the site direction. This is independent of presentation, and it preserves finite limits: finite colimits of finitely presented rings correspond to joining finite generators and equations, whose solution objects are finite limits. Conversely a left-exact functor $F:\mathcal{C}\to \mathcal{E}$ reconstructs a commutative ring $F(C_{\mathbb{Z}[t]})$ using the polynomial maps for addition, multiplication, constants and additive inverse. Presentations by generators and equations then show $F\cong F_M$. Thus flat functors on $\mathcal{C}$ correspond to internal commutative rings.

Under $F_M$, the localization arrow above identifies its domain with the locus where $f_i$ evaluates to a unit; the inverse witness is unique. The family is jointly epimorphic exactly when these unit loci cover the solution object for $A$. The locality axiom implies this for every unit-ideal family: from a finite sum $\sum f_i{g}_i=1$, repeatedly apply the two-summand unit alternative, rescaling a unit partial sum when necessary; some $f_i{g}_i$, hence some $f_i$, is a unit locally. Conversely the cover of $\mathbb{Z}[t]$ by its localizations at $t$ and $1-t$ gives precisely the locality axiom, while the empty cover of the zero ring gives $0=1⊢\perp$. Therefore continuous flat functors are exactly models of the displayed coherent theory. The Diaconescu equivalence for geometric morphisms now identifiesnaturally in $\mathcal{E}$. This proves the asserted classifying topos identification.

Since the site is subcanonical, the generic local ring is the representable sheafIts ring operations are induced by polynomial substitution. Its infinitesimal subobject is likewise representable:Products of representables correspond to tensor products of their rings. For every finitely presented $A$, exponential adjunction and the Yoneda lemma giveEvery element of this ring has a unique expression $a+b\epsilon$, because $1,\epsilon$ form an $A$-basis. This holds even when $A$ has nilpotents or zero divisors. The expression is natural under every ring homomorphism $A\to B$, which sends its two coefficients to their images. The canonical morphism $R\times R\to R^D$, transposed from $(a,b,d)\mapsto a+bd$, is consequently a bijection at every stage $C_A$, with a natural inverse extracting the coefficients. ThereforeThis proves the Kock-Lawvere axiom for the generic local ring. The uniqueness is an internal sheaf assertion tested at all finitely presented rings, including the ring with its generic square-zero element; restricting the test to fields would lose that element and would not prove uniqueness.

Throughout, rings are commutative with identity and ring homomorphisms preserve identity. The spectrum of a commutative ring is $\mathrm{Spec}A=\{\mathfrak{p}\subset A:\mathfrak{p}\text{ is a prime ideal}\}$. For an ideal $I$, put $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$; these are the closed sets of the Zariski topology. Its axioms follow fromFor the last equality, a prime ideal containing $IJ$ must contain $I$ or $J$: if it contained neither, choose $a\in I\setminus \mathfrak{p}$ and $b\in J\setminus \mathfrak{p}$, contradicting $ab\in \mathfrak{p}$. The other equalities follow directly from containment. Thus arbitrary intersections and finite unions of closed sets are closed, proving that this is a topology.

The Zorn lemma says that a nonempty partially ordered set in which every chain has an upper bound has a maximal element. For $A\ne 0$, apply it to proper ideals ordered by inclusion. The zero ideal is proper. The union of a nonempty chain is an ideal and is proper, because containing one would put one in a member of the chain; the empty chain has the zero ideal as an upper bound. There is therefore a maximal ideal $\mathfrak{m}$. If $a\notin \mathfrak{m}$, then $\mathfrak{m}+(a)=A$, so the class of $a$ in $A/\mathfrak{m}$ is invertible. This quotient ring is a field, making $\mathfrak{m}$ a prime ideal. Hence $A\ne 0⟹\mathrm{Spec}A\ne \varnothing$.

For a ring homomorphism $f:A\to B$, define $f^{\ast }:\mathrm{Spec}B\to \mathrm{Spec}A$ by $f^{\ast }(\mathfrak{q})=f^{-1}(\mathfrak{q})$. The inverse image is proper because $f(1)=1$, and primality follows from $f(ab)=f(a)f(b)$. Moreoverwhere $IB$ is the ideal generated by $f(I)$. This proves continuity. Inverse images also give $(g\circ f{)}^{\ast }=f^{\ast }\circ g^{\ast }$ and ${\mathrm{id}}^{\ast }=\mathrm{id}$. Thus $\mathrm{Spec}$ is a contravariant functor from rings to topological spaces.

For the specified inclusion, the fundamental theorem of algebra gives the points $0$ and $(x-z)$ of $\mathrm{Spec}\mathbb{C}[x]$. Over $\mathbb{R}$, the nonzero prime ideals are generated either by $x-r$ with $r\in \mathbb{R}$, or by $(x-a{)}^2+b^2$ with $a\in \mathbb{R}$, $b>0$; all irreducible real polynomials have degree one or two. Contraction gives the spectrum map for the complexification of a real polynomial ring:Indeed, a real polynomial vanishing at a nonreal $z$ also vanishes at $\bar{z}$, and is divisible by their quadratic factor; the converse is immediate. This map is surjective. Its fibers contain one point over a real linear factor, two conjugate points over an irreducible quadratic factor, and only the generic point over zero. It identifies complex-conjugate pairs. Since each proper Zariski-closed set in either one-variable polynomial ring is a finite set of closed points, images of closed sets are closed; hence the Zariski topology on the target is the quotient topology for this identification.

Here the usual homological algebra convention is that the right-exact functor $F$ is additive. Module categories have enough projective modules, because any module is a quotient of a free module. Choose a projective resolutionApply $F$ to the unaugmented chain complex and define the left derived functors byAdditivity ensures that $F$ preserves zero maps, sums of maps and homotopy identities, so the transformed differentials still compose to zero.

For a module homomorphism $u:M\to N$, lift it to a chain map between chosen projective resolutions. In degree zero, lift through $Q_0\twoheadrightarrow N$ using projectivity of $P_0$. In each higher degree, the already constructed map lands in the next kernel; exactness of $Q_{\bullet }$ and projectivity of $P_i$ supply the next lift. Any two lifts are chain homotopic: their degree-zero difference lands in $\ker (Q_0\to N)=\mathrm{im}{d}_1^Q$, so it lifts to $h_0:P_0\to Q_1$. Inductively lift the remaining difference after subtracting the preceding homotopy term. This givesApplying $F$ preserves this identity, so the lifts induce the same homology map. Identity lifts and composable lifts show that $L_{i}F$ is a functor. Comparison lifts between two resolutions over the identity of $M$ are inverse up to chain homotopy; hence the construction is independent of the chosen resolution up to its canonical natural isomorphism.

Right exactness identifies $L_{0}F(M)=\mathrm{coker}(F(P_1)\to F(P_0))$ naturally with $F(M)$. For a projective module $P$, the length-zero resolution gives $L_{i}F(P)=0$ for $i>0$. For a short exact sequence $0\to M^{\prime }\to M\to M^{\prime \prime }\to 0$, the horseshoe lemma builds a degreewise split short exact sequence of projective resolutions. Concretely, lift the projective generators covering $M^{\prime \prime }$ into $M$ and combine them with generators covering $M^{\prime }$; repeat on the kernels. The middle projective term in each degree is the direct sum of the two outer ones. An additive functor preserves these split sequences, and lifting cycles, taking their differentials and returning to the subcomplex produces the connecting maps inThis is the natural long exact derived-functor sequence. In particular, if $0\to K\to P\to M\to 0$ has $P$ projective, then $L_{i}F(M)\cong L_{i-1}F(K)$ for $i\ge 2$, and $L_{1}F(M)=\ker (F(K)\to F(P))$. These are dimension-shifting formulas. Positive degrees can be killed by the surjection $P\to M$. If $F$ is exact, it preserves every resolution's exactness and all its positive left derived functors vanish. Conversely, if $L_{1}F$ vanishes on all modules, the displayed long exact sequence makes $F$ exact. A natural transformation between additive right-exact functors induces transformations on the left derived functors by applying it termwise.

For extension of scalars $F(M)=B{\otimes }_{A}M$, these derived functors are the Tor functors ${\mathrm{Tor}}_i^A(B,M)$. When $B=A/(a)$ and $a$ is a non-zero-divisor, there is the length-one free resolutionAlthough the construction above resolves $M$, one may instead use this resolution of $B$. To justify that calculation, tensor a free resolution of $M$ with the two-term resolution of $B$ to form a double complex. Taking homology first along the resolution of $B$ leaves $B{\otimes }_A{P}_{\bullet }$; taking it first along the resolution of $M$ leaves $M\stackrel{a}{\to }M$. The free terms preserve exactness under tensoring. The double complex is first-quadrant with finitely many terms on each total diagonal, so both filtrations compute the same total homology. This proves the required balancing directly.

The kernel and cokernel of multiplication by $a$ now give Tor for a principal non-zero-divisor quotient:These are naturally $B$-modules. For $M=A$, the degree-one group is zero; for $M=B$, multiplication by $a$ is zero and both degrees zero and one are $B$. Thus, when $B\ne 0$, degree one explicitly detects the failure of this base-change functor to be exact.