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Choose one regular functor witnessing the regular-logic separation of a proper subobject for each proper subobject represented by a monomorphism of the small regular category. Their product is regular and reflects invertibility of monomorphisms. If an arbitrary morphism becomes invertible, its image inclusion becomes invertible and so does the diagonal of its kernel pair. The original morphism is both a regular epimorphism and a monomorphism, hence invertible. The target is a power of sets as a capital regular category. This proves an isomorphism-reflecting regular representation without asserting full faithfulness.

Every epimorphism in an abelian category is the coequalizer of its kernel pair. Its kernel-pair square is a pullback in a category along an epimorphism and therefore also a pushout. A map equalizing the two projections supplies the same map on both pushout legs and so factors uniquely through the original epimorphism. In particular, all such epimorphisms are regular epimorphisms.

Map two disjoint walking arrows to the two mutually inverse arrows of the walking isomorphism. Their images generate the entire target, so the functor is an extremal epimorphism. Its kernel pair only identifies the matching endpoint objects, not the inverse-composition relations. Mapping both arrows to a nonidentity idempotent in a one-object category equalizes that kernel pair but cannot factor through the target groupoid. Thus the functor is not a regular epimorphism, although the category of small categories has finite limits.

A regular category has finite limits, factorizations of every arrow as a regular epimorphism followed by a monomorphism, and pullback-stable regular epimorphisms. Equivalently one may require coequalizers of kernel pairs and pullback stability of the resulting quotients. A regular epimorphism is a coequalizer of some parallel pair; an extremal epimorphism is an epimorphism $e$ such that $e=mh$, with $m$ monic, forces $m$ to be invertible.

First, every regular epimorphism is extremal without assuming regularity of the whole category. Suppose $e:A\to B$ coequalizes $r,s$ and $e=mh$. Since $m$ is monic, $hr=hs$, so the universal property gives $k:B\to \mathrm{dom}(m)$ with $ke=h$. Then $mke=e$; cancellation of the epimorphism $e$ gives $mk=1_B$. Monicity of $m$ now gives $km=1$, so $m$ is an isomorphism. Conversely, in a regular category, factor an extremal $e$ as $mq$ with $q$ regular epic and $m$ monic. Extremality makes $m$ invertible, so $e$ is regular epic. Thus covers and regular epimorphisms coincide.

For a counterexample, work in the category of small categories, which has finite limits constructed on objects and arrows. Let $C$ be two disjoint walking arrows $a\stackrel{u}{\to }b$ and $c\stackrel{v}{\to }d$. Let $D$ be the walking isomorphism, with two objects $0,1$ and inverse arrows $p:0\to 1$, $q:1\to 0$. Define $F:C\to D$ by $a,d\mapsto 0$, $b,c\mapsto 1$, $u\mapsto p$, $v\mapsto q$. These images generate all of $D$, so two functors agreeing after $F$ agree everywhere; thus $F$ is epic.

A monic functor is injective on objects and faithful, as tested by functors from the terminal category and from the walking arrow. If $F$ factors through such a functor $m:D^{\prime }\to D$, its image contains both objects and both arrows $p,q$. Faithfulness forces their lifted composites to be the identities, so the image contains every arrow of $D$ and $m$ is an isomorphism. Hence $F$ is extremal.

It is not regular. Its kernel pair identifies the endpoint pairs $a,d$ and $b,c$ but has no relation forcing the composites of the images of $u,v$ to be identities. Concretely, send all four objects to the one-object category with endomorphism monoid $\{1,e\}$, $e^2=e\ne 1$, and send both $u,v$ to $e$. This functor equalizes the kernel pair: corresponding identities agree, and the only nonidentity arrow pairs in the kernel pair are an arrow paired with itself. It cannot factor through $D$, because images of $p,q$ would have to be inverse, whereas $e^2\ne 1$. In a category with kernel pairs, every regular epimorphism is the coequalizer of its kernel pair: its original coequalized pair factors through that kernel pair, giving the universal property. Therefore $F$ is an extremal but nonregular epimorphism, the promised nonregular extremal epimorphism in the category of categories.

For an idempotent monad, the unit laws give $T\eta =\eta T={\mu }^{-1}$. If $a:TA\to A$ is an algebra, $a{\eta }_A=1$ and naturality giveThus algebras are exactly the objects on which ${\eta }_A$ is invertible, with the unique algebra structure ${\eta }_A^{-1}$. They form a full reflective subcategory with reflector $T$. The assumed preservation of finite limits makes this a left-exact reflective subcategory.

For a subobject $S\hookrightarrow A$, define its closure byusing the unit and the monomorphism $TS\hookrightarrow TA$. This closure operation induced by a left-exact reflector is extensive by unit naturality, stable under pullback and preserves intersections because $T$ preserves finite limits. Applying $T$ to its defining square and using $T\eta$ invertible identifies $T\overline{S}$ with $TS$ over $TA$. Closing a second time therefore gives the same subobject. It is a universal idempotent closure operation, hence defines a local operator $j$ by classifying the closure of $t:1\hookrightarrow \Omega$; closure of any mono is then classified by composing its classifier with $j$.

A mono $s:S\hookrightarrow A$ is $j$-dense exactly when $Ts$ is invertible. One implication follows immediately from the closure square. Conversely if its closure is all of $A$, the unit ${\eta }_A$ factors through $Ts$. Applying $T$ and using the invertible $T{\eta }_A$ makes $Ts$ split epic; since it is also monic, it is invertible. Every fixed object $B$ is consequently a $j$-sheaf, since the adjunction givesand a dense restriction becomes an isomorphism.

Conversely let $B$ be a $j$-sheaf. Its diagonal is closed: the two projections from the closure of the diagonal agree on the dense diagonal, so uniqueness of extension into $B$ makes them agree everywhere. The closure formula identifies this closed diagonal with the kernel pair of ${\eta }_B$, since $T$ takes the diagonal of $B$ to the diagonal of $TB$. Thus ${\eta }_B$ is monic. Also $T{\eta }_B$ is invertible, so this mono is dense. The sheaf property extends $1_B$ across it to a retraction $TB\to B$. Since $TB$ is fixed and hence itself a sheaf, uniqueness across that dense mono also makes the other composite the identity. Therefore ${\eta }_B$ is invertible. We have proved the algebra category is exactly the category of $j$-sheaves.

Its topos structure is concrete. Finite limits are computed in $\mathcal{E}$. For sheaves $A,B$, the ambient exponential $B^A$ is a sheaf: extension across a dense mono reduces, by transposition, to extension into $B$ across its product with $A$, again a dense mono. The classifier is ${\Omega }_j=\mathrm{Eq}(j,1_{\Omega })$ with the induced truth point; it classifies closed subobjects. Colimits are obtained by taking the ambient colimit and applying $T$. These supply an elementary topos, as required.

For the second part, use the square reader monadThe two unit laws select $(a,b)$ again. Each side of associativity selects the entries indexed $(0,0,0)$ and $(1,1,1)$ in an eight-entry array, so the monad laws hold. An algebra is an operation $\ast :A\times A\to A$ satisfyingThese imply associativity, since $(x\ast y)\ast z=(x\ast y)\ast (z\ast z)=x\ast z$ and $x\ast (y\ast z)=(x\ast x)\ast (y\ast z)=x\ast z$. They are the rectangular band laws.

For nonempty $A$, choose $e\in A$, and put $B=\{a\ast e:a\in A\}$, $C=\{e\ast a:a\in A\}$. Define $\varphi (a)=(a\ast e,e\ast a)$ and $\psi (b,c)=b\ast c$. Then $(a\ast e)\ast (e\ast a)=a\ast a=a$, and for $b=x\ast e$, $c=e\ast y$ the laws give $(b\ast c)\ast e=b$ and $e\ast (b\ast c)=c$. Thus $\varphi ,\psi$ are inverse bijections. In these coordinates the operation isConversely any nonempty product with this operation satisfies the algebra laws, proving the requested correspondence with product decompositions. A homomorphism between nonempty products has its first coordinate depending only on the first factor and its second only on the second, as follows by applying it to the displayed operation. Hence it is exactly a pair of factor maps. There is in addition the unique empty algebra, a strict initial object.

This algebra category is Cartesian closed. For nonempty $A=B\times C$ and $D=E\times F$, its exponential is $E^B\times F^C$ with rectangular multiplication: the usual set adjunction in each factor gives the universal property. For the empty cases, $D^0=1$ and $0^A=0$ when $A$ is nonempty; these formulas also give $0^0=1$. They verify the adjunction against empty test objects as well. Thus the algebra category is Cartesian closed.

It is not a topos. If a classifier existed, its global elements would correspond to the two subobjects of the terminal singleton algebra, so its underlying set would have two elements. A two-element rectangular band is either left-zero or right-zero: its two nonempty factors have sizes $(2,1)$ or $(1,2)$. A homomorphism from the four-element band $\{0,1\}\times \{0,1\}$ to a left-zero classifier depends only on the first coordinate, so the preimage of truth cannot be a single point; for a right-zero classifier the same failure occurs in the second coordinate. Yet a single point is a subalgebra and its inclusion is monic. Pullbacks of algebra maps are computed on underlying sets, so this subobject could not be classified. Therefore there is no subobject classifier and the algebra category is not a topos, despite its cartesian closure.

Work in ordinary set theory with the axiom of choice. A regular category has finite limits, image factorizations as regular epimorphisms followed by monomorphisms, and pullback-stable regular epimorphisms. A regular functor preserves finite limits and regular epimorphisms.

A capital regular category is one whose well-supported objects are well-pointed objects in a category. Here well-supported means $A\to 1$ is a regular epimorphism; well-pointed means a subobject of $A$ containing every point $1\to A$ must be the whole object. This is the convention of Definition 1.3. It permits objects of proper support with no points. Replacing this condition by global points detecting every isomorphism would change the assertion: a three-element chain is regular but cannot admit a conservative finite-limit-preserving functor to such a category.

Here is a regular-logic proof sketch of capitalization of a small regular category. Use a sort for every object of $\mathcal{C}$, a function symbol for every morphism, and regular theory axioms for composition, finite limiting diagrams and regular-epimorphism lifting. Its set models are precisely regular functors $\mathcal{C}\to \mathbf{Set}$. Empty sorts must be allowed.

The key separation lemma is the regular-logic separation of a proper subobject. For $m:S\hookrightarrow A$ proper, the sequentis not derivable: interpreting a derivation in $\mathcal{C}$ would make the monomorphism $m$ a regular epimorphism, hence an isomorphism. The chase completeness for regular logic supplies a set model in which it fails. Briefly, start with a named antecedent element, adjoin witnesses for the existential axioms, and quotient by required equalities. Every consequent about the original parameter arising at a finite stage has a finite proof; thus a nonderivable consequent remains false. The filtered union is the desired model. This witness construction, rather than an assumption that every sort is inhabited, is what keeps proper supports distinguishable.

Since $\mathcal{C}$ is small, choose a set-indexed family $(M_i{)}_{i\in I}$ of regular functors, one detecting each proper monomorphism, and formIt is regular by coordinatewise finite limits and surjections. It reflects invertibility of monomorphisms. If $P(f)$ is invertible, apply this fact to the image inclusion of $f$ and to the diagonal of its kernel pair. They are invertible, so $f$ is both a regular epimorphism and a monomorphism, hence invertible. Thus $P$ is a conservative functor.

The target is a power of sets as a capital regular category. A well-supported family has every coordinate nonempty, and choice extends any coordinate element to a global tuple. A subobject containing every tuple must consequently contain every coordinate element. Therefore it is capital, giving the required isomorphism-reflecting regular functor. This is the separation aspect of the regular representation results originating in Barr's embedding work; full faithfulness is unnecessary here.

The stronger target of a single set category has a precise restriction: $\mathcal{C}$ admits a conservative regular functor to the Category of sets exactly when it is an almost totally supported regular category. This means every object is well-supported or a strict initial object. Here are necessity and sufficiency.

Suppose $V:\mathcal{C}\to \mathbf{Set}$ is regular and conservative. If $VA$ is nonempty, its support in sets is $1$. Since $V$ preserves images, it sends the support inclusion $\sigma (A)\hookrightarrow 1$ to an isomorphism; conservativity makes $\sigma (A)=1$, so $A$ is well-supported.

If $VA$ is empty, every projection $A\times X\to A$ is sent to the unique bijection between empty sets, hence is invertible. This supplies a morphism $A\to X$ for every $X$. The equalizer of any two such morphisms is also sent to a bijection, hence is invertible, proving uniqueness. Thus $A$ is initial. Every arrow $Y\to A$ forces $VY$ to be empty and is then sent to a bijection; conservativity makes it invertible. So $A$ is strict initial.

Conversely suppose this support condition holds. Use $P$ above and takeProducts preserve finite limits and, by choice, coordinatewise surjections, so $V$ is regular. A well-supported object has all coordinates nonempty. A proper strict initial object has a subterminal image in every coordinate, with at least one empty coordinate: otherwise its arrow to $1$ would be sent by $P$ to an isomorphism, contradicting conservativity.

If $Vf$ is a bijection between well-supported objects, the conservativity of set products on totally supported families makes each $M_i(f)$ bijective: a missing coordinate preimage or a pair of equal-image coordinate elements extends to a tuple witnessing failure of surjectivity or injectivity. Hence $P(f)$ is invertible and so is $f$. An arrow between strict initial objects is already invertible; an arrow from a strict initial object to a well-supported object cannot become a bijection between empty and nonempty sets. An arrow into a strict initial object is invertible by definition. These cases prove conservativity and the conservative regular representation in sets criterion.

A regular category has finite limits, coequalizers of kernel pairs, and regular epimorphisms stable under pullback. Equivalently, every morphism factors as a regular epimorphism followed by a monomorphism, and these factorizations are stable under pullback. A regular functor preserves finite limits and regular epimorphisms, hence also preserves these image factorizations.

For this question, the capital regular category convention is the one in which every well-supported object is a well-pointed object in a category. Here an object $A$ is well-supported if $A\to 1$ is a regular epimorphism. It is well-pointed if a subobject $m:B\hookrightarrow A$ through which every point $1\to A$ factors must be the whole object. This convention is documented in Definitions 1.1–1.3. It imposes no such point-detection requirement on objects whose support is a proper subobject of $1$. In particular, it should not be replaced here by the stronger assertion that $\mathcal{A}(1,-)$ reflects every isomorphism.

First prove two facts about a capital regular category $\mathcal{A}$. Every well-supported object $A$ has a point. Otherwise $A\times A$ is well-supported and has no points, so its diagonal $A\hookrightarrow A\times A$ contains all points and is invertible. That makes $A$ a subterminal object; the regular epimorphism $A\to 1$ is then also monic and hence invertible, contradicting the absence of a point. If $e:X\to Y$ is a regular epimorphism and $y:1\to Y$ is a point, its pullback in a category is a well-supported object over $1$, so it has a point lifting $y$. Thus the representable functorpreserves regular epimorphisms; it already preserves all existing limits. Hence $\Gamma$ is a regular functor. Second, if $\Gamma X$ is a singleton, its unique point $x:1\to X$ is a monomorphism, $X$ is well-supported, and $x$ contains all points of $X$. Well-pointedness makes $x$ invertible. Thus $\Gamma$ reflects terminal objects, even though it need not reflect arbitrary isomorphisms.

Now let $\mathcal{C}$ be a small category that is a regular category. For each object $B$, its slice category $\mathcal{C}/B$ is small and regular: finite limits, images and the stability of regular epimorphisms are computed using the corresponding constructions in $\mathcal{C}$. By the permitted capitalization result choose a conservative functor that is regular,with ${\mathcal{A}}_B$ a small capital regular category. The base-change functor $B^{\ast }:\mathcal{C}\to \mathcal{C}/B$, given by $X\mapsto (X\times B\to B)$, preserves finite limits and regular epimorphisms. So doesTaking these as the coordinates defines a regular functorThe exponent is a set of indices, so this is a power of the Category of sets, with finite limits and regular epimorphisms computed coordinatewise.

To prove reflection of isomorphisms, suppose $R(f)$ is invertible for $f:A\to B$. In $\mathcal{C}/B$, the object $(A,f)$ is the pullback of $B^{\ast }f:(A\times B\to B)\to (B\times B\to B)$ along the point given by the diagonal $B\to B\times B$. Applying ${\Gamma }_B{J}_B$ shows that ${\Gamma }_B{J}_B(A,f)$ is the fibre of the bijection $R_B(f)$ over this diagonal point, and is a singleton. The preceding lemma gives $J_B(A,f)\cong 1$ through its canonical arrow to $1$. Since $J_B$ reflects isomorphisms, $(A,f)\to (B,1_B)$ is invertible. Its underlying morphism is $f$. Therefore

In the small-category setting of this representation result, the elementary condition for replacing the power by one copy of $\mathbf{Set}$ is:This is almost total support for a regular category. “Strict initial” means initial and every arrow into it is invertible; the condition allows the case in which all objects are well-supported and no initial object exists.

For necessity, let $V:\mathcal{C}\to \mathbf{Set}$ be regular and conservative. If $VA\ne \varnothing$, factor $A\to 1$ through its support of an object in a regular category $S\hookrightarrow 1$. Preservation of images gives $VS=1$. Conservativity makes $S\to 1$ invertible, so $A$ is well-supported. If $VA=\varnothing$, then for every $X$ the projection $A\times X\to A$ is sent to the bijection $\varnothing \to \varnothing$, hence is invertible. Its inverse followed by the other projection supplies an arrow $A\to X$. For any two such arrows, their equalizer is sent to a bijection onto $VA$, hence is invertible, proving uniqueness. Thus $A$ is initial. If $X\to A$ exists, then $VX\to \varnothing$ exists, so $VX=\varnothing$ and that arrow is sent to a bijection. Conservativity makes it invertible. Hence $A$ is strict initial.

For sufficiency, choose the permitted conservative regular functor $J:\mathcal{C}\to \mathcal{A}$ into a small capital regular category, and put $V=\Gamma J$. This is regular. A well-supported object of $\mathcal{C}$ is sent to a well-supported object of $\mathcal{A}$, which has a point. A non-well-supported object is strict initial by the condition, and is consequently a proper subterminal object. The functor $J$ preserves subterminality and reflects the noninvertibility of its arrow to $1$; its image is a proper subterminal object and has no points. Thus $VA$ is empty exactly on the non-well-supported objects.

Suppose $Vf$ is a bijection for $f:A\to B$. If both sets are empty, $A$ and $B$ are strict initial objects and $f$ is invertible. Otherwise $A,B$ are well-supported. Factor $Jf$ as a regular epimorphism followed by a monomorphism $m:I\hookrightarrow JB$. Surjectivity of $Vf$ makes every point of $JB$ factor through $m$; capitality of $\mathcal{A}$ makes $m$ invertible. Hence $Jf$ is a regular epimorphism. Its kernel pair $P=JA{\times }_{JB}JA$ is well-supported, because $P\to JA$ and $JA\to 1$ are regular epimorphisms. Injectivity of $Vf$ implies that every point of $P$ has equal coordinates, and therefore factors through the diagonal $JA\hookrightarrow P$. Since $P$ is well-pointed, this diagonal is invertible. Thus $Jf$ is also monic, hence an isomorphism, and conservativity of $J$ makes $f$ invertible. This proves sufficiency with actual point and image arguments, rather than asserting that global sections of every capital regular category is conservative.

The equality predicate on $A$ is interpreted by the diagonal monomorphism ${\Delta }_A:A\hookrightarrow A\times A$. The antecedent equality of the two $f$-values is the kernel pair $A{\times }_{B}A\hookrightarrow A\times A$. Semantic validity of the implication says that the kernel-pair subobject is contained in the diagonal; the reverse containment always holds. Hence the sequent is valid exactly whenIf $f$ is monic, this equality is its cancellation property expressed by the pullback. Conversely, if $fr=fs$ for $r,s:X\to A$, then $(r,s)$ factors through this kernel pair and therefore through the diagonal, giving $r=s$. Thus the sequent holds exactly when the map is a monomorphism. This argument uses generalized arrows from arbitrary $X$, not a claim that global elements detect maps.

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.

Use the displayed square's notation $f:A\to B$, $h:A\to C$, $g:B\to D$, $k:C\to D$, with $gf=kh$ and $g$ epic. Form the biproduct and the morphismsThe map $q$ is an epimorphism, since its restriction to $B$ is $g$: equality after $q$ implies equality after $g$. The pullback in a category property says exactly that $j$ is a categorical kernel of $q$. Indeed $qj=gf-kh=0$, and a map into $B\oplus C$ killed by $q$ is a pair $(x,y)$ with $gx=ky$, which factors uniquely through $(f,h)$.

Use the standard abelian category property that every epimorphism is the categorical cokernel of its categorical kernel. If $u:B\to X$ and $v:C\to X$ satisfy $uf=vh$, then $[u,-v]j=0$. Hence there is a unique $w:D\to X$ with $wq=[u,-v]$. Restriction to the two summands gives $wg=u$ and $wk=v$. This is the pushout in a category universal property. Thus a pullback of an epimorphism is a pushout in an abelian category.

In this pushout, $g$ is the pushout of $h$ along $f$. The reflection result of part (b) therefore makes $h$ epic. This proves pullback stability of epimorphisms in an abelian category.

Finally, let $e:E\to B$ be an epimorphism and let $p_1,p_2:R\rightrightarrows E$ be its kernel pair. Their pullback square is a pushout by the result just proved. If $u:E\to X$ satisfies $u{p}_1=u{p}_2$, the two copies of $u$ form a pushout cocone. There is a unique $v:B\to X$ with $ve=u$. Hence epimorphisms in an abelian category are coequalizers of their kernel pairs, so they are regular epimorphisms.

A first-order signature specifies sorts, function symbols with specified input and output sorts, and relation symbols with specified input sorts. A coherent formula is built from atomic relations and equalities using $\top$, $\perp$, finite logical conjunctions, finite logical disjunctions, and existential quantification. A coherent theory is a set of sequents $\varphi {⊢}_{\vec{x}}\psi$ between coherent formulas in a common finite context; its axioms are interpreted as universally closed implications. Neither general negation nor universal quantification is allowed inside coherent formulas.

One complete presentation of coherent logic consists of the following axiom and rule schemes, together with the theory's sequents. All displayed formulas have compatible sorts and contexts, and bound variables can be renamed.

Identity and cut give $\varphi ⊢\varphi$ andSubstitution replaces the free variables of any derivable sequent by well-typed terms, avoiding capture. Contexts can be enlarged by unused variables, and permuted or renamed.

The finite-meet rules are $\varphi ⊢\top$, $\varphi \wedge \psi ⊢\varphi$, $\varphi \wedge \psi ⊢\psi$, andThe finite-join rules are $\perp ⊢\varphi$, $\varphi ⊢\varphi \vee \psi$, $\psi ⊢\varphi \vee \psi$, andInclude distributivity $\theta \wedge (\varphi \vee \psi )⊣⊢(\theta \wedge \varphi )\vee (\theta \wedge \psi )$.

Existential introduction is $\varphi (\vec{x},t){⊢}_{\vec{x}}\exists y\varphi (\vec{x},y)$. Existential elimination iswhere $y$ is absent from $\psi$. Equivalently, the quantifier is left adjoint to weakening along the context projection. Include the Frobenius rule in coherent logicIt can also be derived from the usual coherent natural-deduction rules.

Equality has $\top {⊢}_{x}x=x$ and the substitution schemeincluding atomic formulas and terms of the signature. Symmetry, transitivity and congruence for all functions and relations follow. These schemes impose no unintended inhabitedness axiom on a sort.

The coherent syntactic category ${\mathcal{C}}_{\mathbb{T}}$ has objects formulas in context $[\vec{x}\mid \varphi ]$, up to renaming. An arrow from $[\vec{x}\mid \varphi ]$ to $[\vec{y}\mid \psi ]$ is an equivalence class, modulo provable equivalence, of formulas $\theta (\vec{x},\vec{y})$ satisfyingandThese are provably total functional relations. The identity is the equality graph restricted by $\varphi$. If $\theta$ and $\rho$ are consecutive arrows, their composite is represented by $\exists \vec{y}(\theta \wedge \rho )$. Equality, cut and existential rules give the category laws.

A coherent category has finite limits, pullback-stable regular-epi/mono image factorizations, and finite unions of subobjects stable under pullback. We verify these structures syntactically. The terminal object is the empty-context truth formula. Products conjoin formulas in disjoint contexts; equalizers add equality of the two output tuples. For a functional relation $\theta$, its image in the target is represented by $\exists \vec{x}\theta$. Its factor onto that image is regular epic: two arrows out of the image agreeing on the source agree by existential elimination, and the same argument with the kernel pair gives the coequalizer property. Frobenius makes these image factorizations stable under pullback.

Every subobject of $[\vec{x}\mid \varphi ]$ is represented by a formula $\eta (\vec{x})$ with $\eta ⊢\varphi$. Indeed, take the existential image of a monic functional relation; uniqueness makes its map to that image an isomorphism. Subobject inclusion is exactly provable implication. Finite unions are consequently disjunctions, with bottom as the empty subobject; distributivity and substitution make them pullback-stable. Hence ${\mathcal{C}}_{\mathbb{T}}$ is coherent.

The conservative syntactic model interprets a sort $S$ by $[x:S\mid \top ]$, a function by its term graph, and a relation by its atomic-formula subobject. Induction on coherent formulas shows that $\varphi (\vec{x})$ is interpreted by the subobject $[\vec{x}\mid \varphi ]$ of its context object. It satisfies every theory axiom by construction. Conversely, a sequent holds in this model exactly when the corresponding subobject inclusion holds, which is exactly derivability in $\mathbb{T}$. ThereforeThis is conservativity for coherent sequents, not a claim about non-coherent formulas.

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 regular category has finite categorical limits, coequalizers of kernel pairs, and regular epimorphism–monomorphism image factorizations whose regular epimorphisms are stable under every pullback in a category. Equivalently, finite limits, pullback-stable regular epimorphism–monomorphism factorizations suffice. In particular, every strong epimorphism in a regular category is regular.

For the weaker hypotheses here, interpret an image factorization as $f=me$ with $m:I\hookrightarrow B$ the least subobject through which $f$ factors, and $e:A\to I$ a strong epimorphism. With pullback in a category constructions, the least-subobject characterization itself implies that $e$ is strong: in a lifting square against a monomorphism, pull that monomorphism back to $I$; since $e$ factors through this pullback, image minimality forces its mono into $I$ to be invertible, giving the required diagonal. Thus the two descriptions of images agree under the stated hypotheses.

For $a:A^{\prime }\hookrightarrow A$, define ${\exists }_f(A^{\prime })$ as the image factorization subobject of $fa:A^{\prime }\to B$. For $b:B^{\prime }\hookrightarrow B$, the pullback in a category universal property and image minimality giveHenceThis is an adjunction between the posets of subobjects.

For Frobenius reciprocity for subobjects, factor $fa=me$ through its image $m:I\hookrightarrow B$, and put $P=I{\times }_B{B}^{\prime }$. Its mono into $B$ represents ${\exists }_f(A^{\prime })\cap B^{\prime }$. The pullback in a category of $e$ along the monomorphism $P\to I$ has domain canonically $A^{\prime }{\times }_B{B}^{\prime }$, which represents $A^{\prime }\cap f^{\ast }(B^{\prime })$. If strong epimorphisms are stable under pullback along monomorphisms, that pulled-back arrow is strong, so its composite with $P\hookrightarrow B$ is an image factorization. This givesEquality here is equality of subobjects, hence an isomorphism of their representatives.

Conversely, let $e:X\to Y$ be a strong epimorphism and $b:B^{\prime }\hookrightarrow Y$ be monic. Its image is all of $Y$: in any image factorization of $e$, the lifting property makes the mono invertible. Apply Frobenius reciprocity for subobjects to $f=e$ and $A^{\prime }=X$. The induced morphism $p:X{\times }_Y{B}^{\prime }\to B^{\prime }$ has image all of $B^{\prime }$, because the image of $bp$ in $Y$ is $B^{\prime }$. Its image factorization therefore has invertible mono, so $p$ is a strong epimorphism. This proves the converse.

We now use the glued ring categories counterexample to regularity. All rings and ring homomorphisms are unital; the zero ring, where $0=1$, is allowed. It is the terminal object of $\mathbf{Rng}$, and there is no unital ring homomorphism from it to a nonzero ring. Denote the common zero ring by $Z$, and the newly adjoined strict initial object by $\perp$. Nonzero objects in different copies have no morphisms between them. There is one morphism from every object to $Z$ and from $\perp$ to every object, and no morphism into $\perp$ except its identity.

Finite products in a category and equalizers can be described explicitly. The terminal object is $Z$. The product in a category of two nonzero objects in one copy is their ordinary ring product; the product in a category of nonzero objects in different copies is $\perp$, since only $\perp$ can map to both. A product with $Z$ is the other factor, and one with $\perp$ is $\perp$. For parallel morphisms between nonzero objects in the same copy, the ordinary ring equalizer works; it is nonzero because its subring contains distinct $0$ and $1$. All other parallel pairs are equal, and their equalizer is the identity of the domain. Thus the construction of small limits from products and equalizers gives all finite categorical limits.

Within either ring copy, monomorphisms are precisely injective ring homomorphisms: maps from $\mathbb{Z}[x]$ detect unequal elements. The only monos from outside a copy are the maps $\perp \to X$, and the only subobjects of $Z$ are $\perp$ and $Z$. Indeed a nonzero ring $R$ is not subterminal, since $\mathbb{Z}[x]\to R$ can send $x$ to $0$ or to $1$.

Ordinary ring-image factorizations remain image factorizations in the glued category. Their surjective parts remain strong epimorphisms: a lifting square into a nonzero ring stays in one copy; a square into $Z$ with mono $\perp \to Z$ would require a nonexistent map from a noninitial domain to $\perp$. Maps with domain $\perp$ are already monic and have identity strong part. Conversely, a strong epimorphism within a ring copy has surjective ring image, since its mono image part must be invertible. A map $\perp \to X$ with $X\ne \perp$ is monic but noninvertible, and therefore cannot be strong. This classifies the strong epimorphisms as the identities of $\perp$ and the surjective homomorphisms in the copies, including maps to $Z$.

Pullback along a monomorphism within a copy preserves those surjections, using the given regularity of $\mathbf{Rng}$. Pullback along $\perp \to X$ gives $1_{\perp }$, and these exhaust the additional monos, including those into $Z$. Thus strong epimorphisms are stable under pullback along monos, so Frobenius reciprocity for subobjects holds.

However, let $e:{\mathbb{Z}}_{\mathcal{D}}\to Z$ be the surjective ring homomorphism in the first copy, and pull it back along $\mathbb{Z}[x{]}_{\mathcal{E}}\to Z$ in the other copy. The resulting arrow isIt is not even an epimorphism, since the distinct evaluation ring homomorphisms $x\mapsto 0$ and $x\mapsto 1$ to ${\mathbb{Z}}_{\mathcal{E}}$ agree after composing with $\perp \to \mathbb{Z}[x{]}_{\mathcal{E}}$. Since $e$ is strong, it would be a regular epimorphism in a regular category, whose pullbacks must be regular, in particular epic. Therefore the glued category has finite limits and images and satisfies Frobenius reciprocity, but is not regular.

Let $P$ be a tiny object of $[\mathcal{C},\mathbf{Set}]$, and let $0$ be the initial object of $\mathcal{C}$. The representable functor $h_0=\mathcal{C}(0,-)$ is the terminal functor, since each of its values is a singleton. The exponential adjunction and the Yoneda lemma giveSince $P$ is tiny, $E_P:F\mapsto F^P$ is itself a left adjoint, so preserves all colimits. Evaluation at $0$ also preserves colimits, because they are pointwise in a set-valued functor category. Therefore $\mathrm{Nat}(P,-)$ preserves the coproduct construction.

It also preserves epimorphisms. Indeed, every such epimorphism is a pointwise epimorphism in a functor category, hence the coequalizer of its kernel pair, as can be checked pointwise in sets. A left adjoint preserves that coequalizer and therefore sends it to a regular epimorphism, which is an epimorphism; evaluation at zero preserves surjectivity. Thus $P$ is an irreducible projective in a set-valued functor category.

A Cauchy-complete category is one in which every idempotent morphism splits. By the characterization allowed in the question, irreducible projectives are representable functors for such an indexing category. Applying it to $P$ proves the tiny covariant functor on a Cauchy-complete category with an initial object is representable conclusion:

A regular epimorphism $e:A\to B$ is the coequalizer of some parallel pair $r,s:R\rightrightarrows A$. A strong epimorphism has the left lifting property against every monomorphism: in every commutative squarewith $m$ monic, there is $d:B\to C$ satisfying $de=u$ and $md=v$.

If $e$ coequalizes $r,s$, then $mur=ver=ves=mus$. Monicity gives $ur=us$, so the coequalizer property gives a unique $d$ with $de=u$. Now $mde=mu=ve$; every coequalizer is epic, hence $md=v$. This proves that every regular epimorphism has the lifting property of a strong epimorphism.

For an arbitrary $f:A\to B$, form its kernel pair $r,s:R=A{\times }_{B}A\rightrightarrows A$ and let $e:A\to I$ be its coequalizer. There is a unique $m:I\to B$ with $f=me$. To show $m$ monic, suppose $mu=mv$ for $u,v:X\rightrightarrows I$. Pull back $e$ along $u$, obtaining an epic $p:P\to X$ and $a:P\to A$ with $ea=up$. Pull back $e$ along $vp$, obtaining an epic $q:Q\to P$ and $b:Q\to A$ with $eb=vpq$. Thenso $(aq,b)$ factors through the kernel pair of $f$. Since $e$ coequalizes that pair, $eaq=eb$, and hence $upq=vpq$. The composite $pq$ is epic, so $u=v$. This proves the regular-epimorphism-monomorphism factorization from kernel pairs.

If $f$ is strong, apply its lifting property to $f=me$:A diagonal $d:B\to I$ satisfies $md=1_B$. Since $m$ is also monic, $dm=1_I$, so $m$ is an isomorphism and $f$ is regular.

For the category of small categories, let $\mathcal{A}$ have objects $0,1,1^{\prime },2$, nonidentity arrows $a:0\to 1$ and $b:1^{\prime }\to 2$, and no other nonidentity generators. Send it to the ordinal category $[2]=(0\to 1\to 2)$ by identifying $1$ with $1^{\prime }$ and sending $a,b$ to the two generating arrows. The image generates $[2]$, so this is a strong epimorphism in the category of small categories, but it is not full because the composite $0\to 2$ has no preimage; therefore it is not regular.

Every functor factors through the subcategory of its codomain generated by its image. The first functor is strong epic and the inclusion is monic, so strong-epi--mono factorizations always exist in $\mathbf{Cat}$. Regular-epi--mono factorizations do not always exist: the displayed strong nonregular functor would make its monic factor an isomorphism by strongness, forcing the original functor to be regular.

Let $\mathcal{C}$ be a small Cauchy-complete category with initial object $0$. The terminal object of $[\mathcal{C},\mathbf{Set}]$ is $h_0$, so $\mathrm{Nat}(P,F)\cong (F^P)(0)$. If $P$ is tiny, exponentiation by $P$ is itself a left adjoint and preserves all colimits. Evaluation at zero also preserves colimits. Consequently $\mathrm{Nat}(P,-)$ preserves coproducts and epimorphisms, since every epimorphism in a set-valued functor category is the coequalizer of its kernel pair. Thus $P$ is an irreducible projective in a set-valued functor category, hence representable.