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A semi-additive category has commutative-monoid enrichment: each hom-set is a commutative monoid, and composition in a category preserves addition and zero in each variable. In the finite-biproduct convention, which we use here, it also has finite products in a category and coproducts in a category. A preadditive category has abelian groups as its hom-sets, with composition additive in each variable; this definition alone does not require finite products in a category or a zero object.

Let $P={\prod }_{i=1}^n{A}_i$ with projections $p_i$. Define $i_j:A_j\to P$ by $p_i{i}_j=1_{A_j}$ for $i=j$ and $p_i{i}_j=0$ otherwise. Bilinearity givesso product uniqueness gives ${\sum }_j{i}_j{p}_j=1_P$. For a family $f_j:A_j\to X$, set $f={\sum }_j{f}_j{p}_j$. Then $f{i}_j=f_j$. If $h:P\to X$ has the same restrictions, $h=h{\sum }_j{i}_j{p}_j={\sum }_j{f}_j{p}_j=f$. Thus the product in a category is also the coproduct in a category, with its canonical injections. Conversely, the analogous construction from a finite coproduct in a category gives projections and the same identities, proving that it is a product in a category.

For the empty family, a terminal object $T$ has a singleton endomorphism monoid, so $1_T=0$. Every map $T\to X$ equals its composite with $1_T=0$, hence equals the zero map; this map exists by enrichment. Therefore $T$ is also an initial object, a zero object. The dual argument applies to an initial object. Consequently finite products in a category and coproducts in a category coincide canonically as biproducts, including the nullary case.

A representation of a functor $(A,x)$ for $G$ says precisely that for every object $(B,y)$ of $(1\downarrow G)$ there is a unique arrow $f:A\to B$ with $Gf(x)=y$. This is exactly the initial object property for $(A,x)$ in that comma category. Conversely, an initial object supplies these unique arrows, hence the representing bijections $\mathcal{C}(A,B)\cong GB$.

An initial object by itself is a weakly initial set. For the converse, let $(W_i{)}_{i\in I}$ be a small weakly initial family and form its product in a category $W$. Given $X$, choose an arrow $W_i\to X$; its composite with the projection $W\to W_i$ shows that $W$ is weakly initial.

Local smallness makes $\mathrm{End}(W)$ a set, so completeness supplies the simultaneous equalizer $e:E\to W$ of all endomorphisms of $W$ with $1_W$. Thus $he=e$ for every $h\in \mathrm{End}(W)$. The object $E$ is weakly initial because it maps to $W$.

For parallel arrows $a,b:E\to X$, take their equalizer $j:Y\to E$. Weak initiality of $W$ supplies $t:W\to Y$. The endomorphism $ejt$ of $W$ satisfies $(ejt)e=e$, and monicity of $e$ gives $jte=1_E$. Therefore $j$ is both a monomorphism and a split epimorphism, hence an isomorphism. Since $aj=bj$, we obtain $a=b$. There is already at least one arrow from $E$ to every $X$, so $E$ is initial. This proves the initial-object lemma for complete categories with a weakly initial set, with smallness used exactly where the endomorphisms are equalized.

If $G$ is representable, it is a limit-preserving functor, and its universal element gives an initial object of $(1\downarrow G)$, hence a weakly initial singleton.

Conversely, a limit-preserving $G$ makes $(1\downarrow G)$ a complete category. For a small diagram $(A_j,x_j)$, take $L={\lim }_j{A}_j$ in $\mathcal{C}$. Its distinguished elements form a compatible family in ${\lim }_{j}G{A}_j$. Categorical limit preservation gives a unique $x\in GL$ with $G{p}_j(x)=x_j$. The underlying categorical limit factorization of a categorical cone preserves this element, proving the comma-category universal property. For the empty diagram, this uses $G1\cong 1$. This is the construction of limits in a comma category of a limit-preserving functor.

The comma category is locally small since its arrows are subsets of the hom-sets in $\mathcal{C}$. By the previous part, its weakly initial set therefore yields an initial object. Part (b) then yields a representation of $G$. Thus representability from a solution set follows with all small categorical limits, rather than finite categorical limits alone.

A projective object in a category is an object $P$ such that, for every epimorphism $e:X\to Y$ and every $u:P\to Y$, there is $v:P\to X$ with $ev=u$.

Let $P={\coprod }_{i\in I}{P}_i$ be a coproduct in a category of projective objects in a category, with injections ${\iota }_i$. Given $e:X\to Y$ epic and $u:P\to Y$, projectivity supplies $v_i:P_i\to X$ with $e{v}_i=u{\iota }_i$. Choose these lifts for the set-indexed family. The coproduct in a category supplies a unique $v:P\to X$ satisfying $v{\iota }_i=v_i$. Since $ev{\iota }_i=u{\iota }_i$ for every $i$, its universal property gives $ev=u$.

Thus coproducts of projective objects are projective. For an empty family, $P$ is the initial object, and the lifting assertion follows directly from its unique maps. The family-of-lifts step uses the usual axiom of choice.

For a set $S$, let ${\iota }_s:1\to S$ select $s$. Given any family $f_s:1\to X$, define $f:S\to X$ by $f(s)=f_s(\ast )$. Then $f{\iota }_s=f_s$, and these equations determine every value of $f$, so it is unique.

This is precisely the universal property of a coproduct in a category, givingThe empty set gives the empty coproduct in a category, namely the initial object of the Category of sets.

For $A\in \mathcal{C}$, the comma category $(A\downarrow G)$ has objects $(D,u)$ with $D\in \mathcal{D}$ and $u:A\to GD$. A morphism $(D,u)\to (D^{\prime },u^{\prime })$ is $h:D\to D^{\prime }$ satisfying $Ghu=u^{\prime }$. Identities and composition are those of $\mathcal{D}$, and functoriality of $G$ verifies the condition under composition.

The Freyd general adjoint functor theorem states: if $\mathcal{D}$ is a locally small category with all small categorical limits, and $\mathcal{C}$ is locally small, then $G:\mathcal{D}\to \mathcal{C}$ has a left adjoint if and only if it preserves small limits and satisfies the solution-set condition. The latter means that for each $A$ there is a set-indexed family $u_i:A\to G{D}_i$ such that every $u:A\to GD$ equals $Gh{u}_i$ for some $i$ and some $h:D_i\to D$.

For necessity, use the standard result that a right adjoint preserves categorical limits. If $F⊣G$, the singleton family containing the unit of an adjunction ${\eta }_A:A\to GFA$ is a solution set, since transposition gives $u=Gh{\eta }_A$ for a unique $h:FA\to D$.

For sufficiency, use the following standard limit fact: if $\mathcal{D}$ is complete and $G$ preserves limits, the projection $(A\downarrow G)\to \mathcal{D}$ creates small limits. Indeed, a compatible family $A\to G{D}_j$ induces a unique arrow into $G(\lim D_j)$, and this makes the underlying limit a limit in the comma category. The comma category is locally small because each of its hom-sets is a subset of a hom-set of $\mathcal{D}$. Its solution family is a weakly initial set.

We prove the remaining initial-object lemma for complete categories with a weakly initial set. In any locally small category $\mathcal{K}$ with all small limits and a weakly initial set $(K_i)$, form $W={\prod }_i{K}_i$. It is weakly initial: for any $X$, some $K_i\to X$ exists and may be composed with the projection $W\to K_i$. The empty family cannot be weakly initial in a nonempty complete category, which has a terminal object.

The set $\mathrm{End}(W)$ is small. Form a simultaneous equalizer $e:E\to W$ of every endomorphism of $W$ and $1_W$; thusThis equalizer exists by completeness, for example as the equalizer of two maps $W\rightrightarrows W^{\mathrm{End}(W)}$. The object $E$ is still weakly initial, since it maps to $W$.

Given $a,b:E\to X$, take their equalizer $j:Y\to E$. Weak initiality of $W$ gives $t:W\to Y$. Since $ejt$ is an endomorphism of $W$, we have $ejte=e$, and cancellation of the monomorphism $e$ gives $jte=1_E$. Thus $j$ is a split epimorphism as well as a monomorphism, so it is an isomorphism. From $aj=bj$ follows $a=b$. There is at least one map $E\to X$ by weak initiality, so $E$ is initial.

Apply this lemma to every $(A\downarrow G)$ and choose its initial object $(FA,{\eta }_A)$. For $v:A\to A^{\prime }$, initiality gives the unique $Fv:FA\to F{A}^{\prime }$ satisfying $GFv{\eta }_A={\eta }_{A^{\prime }}v$. Uniqueness proves the functor laws. The same initiality gives natural bijectionsHence $F⊣G$, completing the theorem without invoking another adjoint functor theorem.

A decidable object in a topos has a decomposition $A\times A={\Delta }_A⨿D_A$, with $D_A$ representing inequality. In the internal logic of a topos, equality on $A$ is decidable. These categorical complements behave well under pullback.

For a subobject $B\hookrightarrow A$, pull back the displayed decomposition along $B\times B\hookrightarrow A\times A$. The diagonal pulls back to ${\Delta }_B$, while $D_A$ pulls back to its complement. Hence every subobject of a decidable object is decidable; the subobject itself need not be complemented in $A$.

For two decidable objects, the diagonals of $A$ and $B$ give four disjoint summands of $(A\times B{)}^2$, according as each coordinate pair is equal or unequal. The both-equal summand is ${\Delta }_{A\times B}$, and the other three give its complement. The terminal object is decidable, so induction gives closure under finite products, including the empty product.

For a family $(A_i{)}_{i\in I}$ of decidable objects with an existing coproduct $A={\coprod }_i{A}_i$, products distribute over this coproduct, givingThe coproduct injections in a topos are disjoint. The diagonal consists of ${\Delta }_{A_i}$ in each $i=j$ summand, and has complementConsequently every existing coproduct of decidable objects is decidable, including the initial object. In a Grothendieck topos all small coproducts exist. No assertion that arbitrary products preserve decidability is used.

Work in the domain-classifying site of part (ii), with generic domain $K=a_{J}U$. For finitely presented rings $A$, the object $a_{J}yA$ is an available stage. Tuples of sections of $K$ are locally represented by tuples of actual elements of $A$, because $U\to a_{J}U$ is locally surjective and sheafification preserves finite products. It is therefore enough to prove the requested implication on such representatives.

We first note the empty-cover criterion for the domain-classifying site: $a_{J}yB$ is initial if and only if the finitely presented ring $B$ is the zero ring. One direction is the generating empty cover. Conversely, any nonzero ring has a maximal ideal and hence a homomorphism to a field. That field is a set-based integral domain and defines a point of the classifying topos. The inverse image of $a_{J}yB$ at this point is ${\mathrm{Hom}}_{\mathrm{Ring}}(B,L)$, which is nonempty for the chosen field $L$. It therefore cannot be the inverse image of an initial object. This uses only the ordinary maximal-ideal existence principle externally, not excluded middle in the internal logic.

Suppose $a_1,\dots ,a_n\in A$ represent a tuple lying in the negation of the all-units subobject. Write $t=a_1\cdots a_n$ and consider the finitely presented localization of a ringEvery $a_i$ is invertible in $B$, since $({\prod }_{j\ne i}{a}_j)z$ is an inverse. But the pulled-back tuple still lies in the negation of the all-units subobject. Thus the whole stage $a_{J}yB$ maps into both that subobject and its negation, forcing this stage to be initial. The empty-cover criterion gives $B=0$.

The localization $A[1/t]$ is zero precisely when $t^N=0$ in $A$ for some integer $N\ge 1$, by the equality criterion in localization. Repeated use of the generating zero-product covers now gives a covering familyIndeed, splitting $t^N=t{t}^{N-1}=0$ and inducting first forces $t=0$ locally; splitting the product $a_1\cdots a_n=0$ then forces one $a_i=0$ locally. More formally the two inductions are coherent derivations from the zero-product axiom, so their quotient families belong to the generated topology. The sheaf semantics of disjunction consequently givesThis is the finite-tuple weak-field property of the generic integral domain. The argument is intuitionistically valid inside the topos, although its description of the site uses ordinary external set theory. There is no appeal to preservation of negation by an arbitrary geometric inverse image.

For the converse assertion, let $R$ be any internally nontrivial commutative unital ring satisfying the $n=2$ implication. Assume $xy=0$. If $x$ and $y$ were both units, multiplying by their two inverses would give $0=1$, contradicting nontriviality. ThereforeThe assumed two-variable implication then gives $x=0\vee y=0$. Together with nontriviality this is exactly the internal integral-domain axiom. Hence a nontrivial ring satisfying the two-variable case is an integral domain.

For example, the ordinary integral domain $\mathbb{Z}$ fails the one-variable implication at $2$. This does not contradict the generic result: the displayed implication uses negation and is non-coherent, so it need not survive the inverse image that classifies an arbitrary set-based domain.

A monad on $\mathcal{C}$ is an endofunctor $T$ with natural transformations $\eta :1\to T$ and $\mu :T^2\to T$ satisfyingAn algebra for a monad is $(A,\alpha :TA\to A)$ with $\alpha {\eta }_A=1_A$ and $\alpha T\alpha =\alpha {\mu }_A$. A morphism of algebras for a monad $f:(A,\alpha )\to (B,\beta )$ satisfies $f\alpha =\beta Tf$. These form the Eilenberg-Moore category ${\mathcal{C}}^T$. Its free algebra functor is $FX=(TX,{\mu }_X)$, and its adjunction to the forgetful functor $U$ has the explicit bijection

Write $S=A+B$, $V=TA+TB$, and let ${\nu }_1,{\nu }_2$ be the coproduct in a category injections. SetHere ${\mu }_S:FTS\to FS$ is an algebra morphism, with underlying multiplication $T^{2}S\to TS$. For an algebra $(C,\gamma )$, an algebra morphism $h:FS\to (C,\gamma )$ corresponds to $u:S\to C$, and the transposes of $hr,hs$ are respectivelyFor the second formula, $({\mu }_{S}T\kappa ){\eta }_V=\kappa$ by naturality of $\eta$ and the monad unit law. Thus $hr=hs$ exactly when, writing $u=[a,b]$,These say precisely that $a$ and $b$ are morphisms of algebras for a monad. Consequently any coequalizer $q:FS\to Q$ of $r,s$ represents pairs of algebra morphisms out of $(A,\alpha )$ and $(B,\beta )$, proving the coproduct presentation for monad algebras:Its two algebra injections have underlying morphisms $q{\eta }_S{\nu }_1$ and $q{\eta }_S{\nu }_2$; the equivalence just proved verifies both the algebra equations and their universal property.

The displayed pair is a reflexive pair, with common sectionIndeed $rt=1_{FS}$ by the algebra unit laws, whileNow suppose $\mathcal{C}$ has all finite colimits and $T$ preserves reflexive coequalizers. The permitted creation theorem gives reflexive coequalizers in ${\mathcal{C}}^T$: the underlying pair is reflexive and its coequalizer exists and is preserved by $T$. The preceding construction therefore gives binary algebra coproducts in a category. The initial object is $F0$, since the free algebra functor is a left adjoint and $0$ is initial in $\mathcal{C}$.

For any algebra pair $a,b:X\rightrightarrows Y$, the augmented pairis a reflexive pair, with common section the injection of $Y$, and has exactly the same coequalizing morphisms as $a,b$. Hence all coequalizers exist in ${\mathcal{C}}^T$. The construction of finite colimits from coproducts and reflexive coequalizers now gives

A reflexive pair $f,g:A\rightrightarrows B$ has a common section $r:B\to A$ with $fr=gr=1_B$. Form the pushout in a category of $f$ and $g$, and write its two maps from $B$ as $j_1,j_2$. Its relation $j_{1}f=j_{2}g$, composed with $r$, gives $j_1=j_2=q$. Thus $qf=qg$. Any $h:B\to C$ with $hf=hg$ gives the compatible pair $(h,h)$ for this pushout and therefore factors uniquely through $q$. This proves the coequalizer of a reflexive pair from a pushout construction.

There is a genuine transcription difference: the original PDF says finite products in the finite-colimit assertion, while the TeX says finite coproducts. The PDF assertion is false with products. Even retaining a pushout-existence assumption, the full subcategory of the Category of sets on nonempty sets has finite products, pushouts, and all coequalizers, but no initial object, hence no empty colimit. Regard the poset with elements $0,a,b,u_0,u_1,\dots$, orderand incomparable $a,b$ as a category. It has finite meets and top $u_0$, hence finite products in a category. Every reflexive pair in a poset is an equal pair and has its identity as a coequalizer. But $a,b$ have no least upper bound, so they have no coproduct in a category.

For the corrected construction using finite coproducts, let $D:\mathcal{J}\to \mathcal{C}$ be a finite diagram in a category. PutDefine $s,t:Y\rightrightarrows X$ on the summand indexed by $a:i\to j$ as ${\nu }_i$ and ${\nu }_{j}D(a)$. Maps $h:X\to Z$ with $hs=ht$ are exactly cocone under a diagram data on $D$. Hence their universal coequalizer is its colimit. Replace $s,t$ by the reflexive pairwhose common section is the second injection and whose coequalizer is unchanged. This proves construction of finite colimits from coproducts and reflexive coequalizers, including the empty diagram via the empty coproduct:

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:

For a locally small category $\mathcal{C}$, a representation of a functor $F:\mathcal{C}\to \mathbf{Set}$ is an object $A$ and an element $a\in F(A)$ such thatis a bijection for every $B$, naturally in $B$. Equivalently, $\theta :\mathcal{C}(A,-)\cong F$ is a natural isomorphism.

Suppose $(A,a)$ and $(A^{\prime },a^{\prime })$ are two representations. Universality gives unique mapssuch that $F(u)(a)=a^{\prime }$ and $F(v)(a^{\prime })=a$. Then $F(vu)(a)=a$, and uniqueness applied to the element $a$ gives $vu=1_A$. Similarly $uv=1_{A^{\prime }}$. Thus the representing objects are uniquely isomorphic in a way carrying one universal element of a set-valued functor to the other.

For $F:\mathcal{C}\to \mathcal{D}$ and $B\in \mathcal{D}$, the comma category $(B\downarrow F)$ has objects $(A,u)$ with $u:B\to F(A)$. A morphism $(A,u)\to (A^{\prime },u^{\prime })$ is a map $h:A\to A^{\prime }$ satisfyingThe universal arrow from an object to a functor criterion says that $F$ has a left adjoint exactly when $(B\downarrow F)$ has an initial object for every $B$. Indeed, an initial $(LB,{\eta }_B)$ represents the functor $A\mapsto \mathcal{D}(B,FA)$, and uniqueness makes $B\mapsto LB$ functorial.

When $F:\mathcal{C}\to \mathbf{Set}$ and $1$ is a singleton, an arrow $1\to F(A)$ is just an element $a\in F(A)$. Hence $(1\downarrow F)$ is the category of elements, and its initial objects are exactly the representations of $F$. This proves

If $F$ has a left adjoint, the universal-arrow criterion immediately makes it representable. Conversely, suppose $\mathcal{C}$ is cocomplete and $F\cong \mathcal{C}(A,-)$. For a set $S$, form the copowerThe coproduct in a category universal property gives natural bijectionsso $L⊣F$.

Cocompleteness cannot be omitted. Let $\mathcal{C}$ be the category of ordinals in reverse order: there is one arrow $\alpha \to \beta$ exactly when $\alpha \ge \beta$ in the ordinary ordering. This large poset is locally small and complete. For a set-indexed family $({\alpha }_i)$, its product in the reversed order is the ordinary supremum ${\sup }_i{\alpha }_i$, and equalizers in a poset are automatic. But $\mathcal{C}$ has no initial object, since that would be a largest ordinal. Any representable functor $\mathcal{C}(A,-)$ is therefore the requested example: if it had a left adjoint $L$, thenwould be a singleton for every $B$, making $L(\varnothing )$ initial, a contradiction.

A functor admitting a further left adjoint must itself be a right adjoint and preserve all existing categorical limits. A functor admitting a further right adjoint must preserve all existing colimits. Failure to preserve a single terminal object, equalizer, initial object or coproduct therefore rules out the corresponding extension. In the adjoint chain for the objects of a category, connected components fail an equalizer and the indiscrete category functor fails a binary coproduct.

An initial object $0$ is strict when every morphism into $0$ is invertible. Adjoining a new strict initial object to a category means adding one map from it to every object and no map to it from any old object. Its only endomorphism is the identity.

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.

A universal arrow from $B$ to $F:\mathcal{C}\to \mathcal{D}$ is an initial object $(A,{\eta }_B:B\to F(A))$ of $(B\downarrow F)$. The functor $F$ has a left adjoint exactly when such a universal arrow exists for every $B$; the universal property makes the chosen objects functorial.