Regular category

A regular category has finite limits, image factorizations into regular epimorphisms followed by monomorphisms, and pullback-stable regular epimorphisms.

For a small category that is a regular category, use a sort for each object, an operation for each morphism, and regular theory axioms describing finite limits and regular epimorphisms. Its models are regular functors into the Category of sets. For a proper subobject $m:S\hookrightarrow A$, the sequent asserting $\forall x:A\exists y:S,m(y)=x$ cannot be derivable, since its categorical interpretation would make $m$ a cover and hence invertible. The chase completeness for regular logic supplies a model where it fails. It is essential to allow empty sorts.

A capital regular category, in the convention of Definitions 1.1–1.3, is a regular category whose well-supported objects are well-pointed objects in a category. It does not require arbitrary objects to be detected by global points. A partially ordered set with finite meets, regarded as a category, is capital: its only well-supported object is the top element. Such a category can have many proper subterminal objects, all with no points. This example distinguishes the definition from the stronger convention that the global-sections functor reflects every isomorphism.

Here $I$ is a set regarded as a discrete category. Finite limits and regular epimorphisms are coordinatewise. A well-supported object is a family of nonempty sets. With the axiom of choice, every coordinate element extends to a global point of its product. Any subobject containing all global points must therefore contain every coordinate element, and be the whole family. Thus the category is capital. Families with some empty coordinates explain why global points need not reflect arbitrary isomorphisms.

If some $f_i$ is not injective, extend two distinct equal-image elements to tuples using common choices at all other coordinates; the product is not injective. If some $f_i$ is not surjective, extend a missing target element to a tuple; the product is not surjective. The axiom of choice supplies these extensions. Thus the set-product functor reflects isomorphisms between families whose coordinates are all nonempty.

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.

In a locally small capital regular category, $\Gamma$ is a regular functor and reflects terminal objects. A well-supported object $A$ has a point: otherwise $A\times A$ is well-supported with no points, so its diagonal is invertible by well-pointedness; then $A$ is subterminal and its regular epimorphism to $1$ is invertible, a contradiction. Pulling a regular epimorphism back along a point gives a well-supported object, so the point lifts. This proves preservation of regular epimorphisms, while representability gives limit preservation. If $\Gamma X$ is a singleton, its unique point $1\to X$ is monic and contains all points; $X$ is well-supported, so that point is invertible. Global sections need not reflect arbitrary isomorphisms between objects of proper support.

The support of an object $A$ in a regular category is the subterminal object appearing as the image of a morphism in a regular category of $A\to 1$. A regular functor preserves this construction. In the Category of sets, the support is the empty set if $A$ is empty and a singleton otherwise.

A regular category is almost totally supported if every object is a well-supported object or a strict initial object. The case in which every object is well-supported is allowed. There can be at most one strict initial object up to isomorphism. In the Category of sets, these two cases distinguish nonempty sets from the empty set. A meet-semilattice regarded as a category satisfies the condition precisely when it has at most two objects up to isomorphism.

For a regular category that is a small category, a regular functor to the Category of sets that reflects isomorphisms exists exactly when the category has almost total support for a regular category. Necessity follows because a nonempty image forces the object's support to be terminal, while an empty image makes all projections $A\times X\to A$ invertible and makes all equalizers of maps out of $A$ invertible. This proves that $A$ is strict initial. For sufficiency, use the capitalization theorem to map conservatively and regularly into a capital regular category, then take global sections of a capital regular category. Well-supported objects retain points; proper strict initial objects retain empty point sets. A bijection between nonempty point sets makes the image subobject entire and makes the kernel-pair diagonal entire, by well-pointedness. Thus the image arrow is both a regular epimorphism and a monomorphism, and is invertible. The capitalization theorem is a substantive existence input; this argument does not infer arbitrary large-category representability from small-category hypotheses.

An object of a regular category is well-supported when its support of an object in a regular category is the terminal object, equivalently when $A\to 1$ is a regular epimorphism. Finite products of well-supported objects are well-supported, by pullback and composition stability of regular epimorphisms. A regular epimorphism has the same support as its codomain, and an object with a point $1\to A$ is well-supported.

The regular coverage on a small regular category takes each regular epimorphism as a one-arrow cover. Pullback and composition stability make these a coverage basis. It is subcanonical because maps constant on the kernel pair of a regular epimorphism descend uniquely to its quotient.

If a family of subsheaves covers $yA$, local membership of $1_A$ gives a regular epimorphism $\alpha :B\to A$ belonging to one subsheaf. The sheaf condition descends that section to $1_A$ in the same subsheaf. All maps into $A$ are restrictions of $1_A$, so that subsheaf is $yA$. Thus representable sheaves for the regular coverage are irreducible even for arbitrary unions.

For a subfunctor $F\prime \subseteq F$ of a regular-coverage sheaf, its closure consists of sections whose restriction belongs to $F\prime$ along one covering regular epimorphism. Pullbacks prove that local membership is a subfunctor; composition of witnessing covers proves the sheaf condition. It is the smallest subsheaf containing $F\prime$.

Take two copies of the category of rings, identify their terminal zero rings, and adjoin a strict initial object $\perp$. Same-copy finite categorical limits are ordinary ring limits; products of nonzero objects in different copies are $\perp$. Ring surjections and $1_{\perp }$ are precisely the strong epimorphisms. They are stable under pullback along monomorphisms, so image factorizations exist and Frobenius reciprocity for subobjects holds. But pulling $\mathbb{Z}$ in one copy $\to 0$ back along $\mathbb{Z}[x]$ in the other copy $\to 0$ gives $\perp \to \mathbb{Z}[x]$. This is not epic: evaluations $x\mapsto 0,1$ agree after precomposing with it. Thus the category is not regular, since strong epimorphisms would then be regular and pullback-stable.

Every left-exact reflective subcategory of a regular category is regular. The reflector sends a regular-epimorphism--monomorphism factorization to such a factorization among fixed objects. A morphism between fixed objects is a regular epimorphism precisely when the closure of its image under the closure operation induced by a left-exact reflector is the whole codomain; pullback stability follows from pullback stability of images and of closure.

In a regular category, the image of $f:A\to B$ is the monomorphism in its essentially unique factorizationas a regular epimorphism followed by a monomorphism. Images are stable under pullback.

For a regular category $\mathcal{C}$, the category $\mathbf{Rel}(\mathcal{C})$ has the objects of $\mathcal{C}$ and subobjects of $A\times B$ as relations $A⇝B$. Composition takes the image of the pullback expressing existential quantification over the middle object.

For $\mathcal{C}=\mathbf{Set}$ this is the category of relations (sets); the general construction also applies to other regular categories.

The category of relations on sets has sets as objects and binary relations as morphisms. Relational composition declares $aRc$ when some intermediate $b$ satisfies both component relations; the identity is the diagonal relation.

In a regular category, a relation $R\hookrightarrow A\times B$ is total if its projection to $A$ is a regular epimorphism, and single-valued if two pairs with the same first component have the same second component. The latter makes that projection a monomorphism. A regular epimorphism that is monic is invertible, so $f={\pi }_B{\pi }_A^{-1}$ has graph exactly $R$. No axiom of choice or set-based element selection is used.

The graph of $f:A\to B$ is the relation represented by $(1_A,f):A\hookrightarrow A\times B$. Its converse is right adjoint to it in the pointwise order on relations.