Epi-Regular Independence Categories

• Epi-regular independence categories are defined as categorical structures equipped with independent pullbacks and epi-based factorizations that replace traditional pullback geometry.
• They establish a 2-equivalence with dilatory dagger categories, unifying the relational calculus with regular-category notions through precise factorization and coisometry techniques.
• Examples from regular categories, surjections, finite probability spaces, and Hilbert spaces illustrate their practical impact across algebraic, probabilistic, and operator-theoretic frameworks.

Searching arXiv for the supplied papers and closely related work on epi-regular independence categories, regular categories, and categorical independence. Epi-regular independence categories are independence categories equipped with an epi-oriented replacement for the regular-category calculus of pullbacks, images, and relations. In the formal sense established in "Dagger categories of relations: the equivalence of dilatory dagger categories and epi-regular independence categories" (Meglio et al., 2 Aug 2025), they are the categorical structures whose independent squares, independent pullbacks, and span factorisations support a dagger category of relations built from jointly monic spans, and they stand to dilatory dagger categories as regular categories stand to tabular allegories. The term also sits near older categorical independence programs based on effective squares, pullback squares of selected monomorphisms, and accessible-category stability, so the surrounding literature contains both a precise relational definition and several closely related, more model-theoretic or factorization-theoretic usages.

1. Independence-categorical definition

An independence category is a category $C$ equipped with a predicate $\perp$ on commutative squares, called independent squares. The axioms are: commutativity; the identity square is independent; vertical and horizontal pasting stability; symmetry; and a unit law. This is the framework used for "systems of independent pullbacks," and it is the ambient structure in which epi-regular independence categories are defined (Meglio et al., 2 Aug 2025).

Within such a category, an independent pullback of a cospan $(u\colon A\to C, v\colon B\to C)$ is a span $(p_1\colon P\to A, p_2\colon P\to B)$ making an independent square and satisfying a terminal property among independent squares over $(u,v)$. Independent pullbacks yield jointly monic spans. The second basic ingredient is a span factorisation: a span $(f,g)$ factors through a morphism $e$ and a jointly monic span $(m_1,m_2)$, with $e$ strong epic. Here strong epimorphism is formulated by left orthogonality to jointly monic spans rather than to single monomorphisms.

An epi-regular independence category is then an independence category satisfying three axioms: $$\text{E1: every cospan admits an independent pullback,}$$

$\perp$0

$\perp$1

A useful equivalent formulation replaces E2 and E3 by regular-epic conditions: $\perp$2

$\perp$3

where the kernel pair is understood as an independent kernel pair. In this sense the theory is explicitly “an epi-version” of regular-category structure: ordinary pullbacks are replaced by independent pullbacks, and every morphism is regular epic rather than merely admitting a regular-epi/mono image factorisation (Meglio et al., 2 Aug 2025).

2. Relation to regular categories

The closest classical antecedent is the theory of regular categories. A regular category is a finitely complete category in which coequalizers of kernel pairs exist and regular epimorphisms are stable under pullback. Equivalently, every morphism admits a pullback-stable image factorization

$\perp$4

with $\perp$5 regular epi and $\perp$6 mono; the mono $\perp$7 is the image of the morphism. Regular categories therefore supply a robust calculus of kernel pairs, quotients, and images, with regular epimorphisms closed under pullback, composition, and products (Gran, 2020).

Epi-regular independence categories retain the epi-centered part of this picture but weaken finite-limit structure to an independence structure. The precise analogy drawn in the relational/dagger literature is that regular categories correspond to tabular allegories, whereas epi-regular independence categories correspond to dilatory dagger categories (Meglio et al., 2 Aug 2025). The passage from regularity to epi-regular independence is therefore not a minor variation on exactness; it is a replacement of ordinary pullback geometry by a designated class of independent squares with its own universal pullback-like objects.

The regular-category background remains important for examples. For any regular category $\perp$8, the subcategory $\perp$9 of regular epimorphisms is epi-regular independence. This includes the familiar settings where regular epimorphisms are surjective homomorphisms or quotient maps: Set and varieties of universal algebras, quasivarieties and monadic categories over Set, topological algebraic structures such as $(u\colon A\to C, v\colon B\to C)$0, abelian categories, and elementary toposes all belong to the regular side of the story (Gran, 2020).

3. Dagger categories of relations and the $(u\colon A\to C, v\colon B\to C)$1 correspondence

The defining structural result is a 2-equivalence

$(u\colon A\to C, v\colon B\to C)$2

between epi-regular independence categories and dilatory dagger categories. On the dagger side, one starts with a dagger category $(u\colon A\to C, v\colon B\to C)$3, its wide subcategory of coisometries $(u\colon A\to C, v\colon B\to C)$4, and the notion of dilation: a span

$(u\colon A\to C, v\colon B\to C)$5

of coisometries with $(u\colon A\to C, v\colon B\to C)$6. A dilator is a terminal such dilation, and a dagger category is dilatory when every morphism has a dilator. The coisometries then carry a canonical independence predicate: a square of coisometries is independent iff

$(u\colon A\to C, v\colon B\to C)$7

In a dilatory dagger category the single equation $(u\colon A\to C, v\colon B\to C)$8 already implies $(u\colon A\to C, v\colon B\to C)$9. Moreover, every independent square in $(p_1\colon P\to A, p_2\colon P\to B)$0 is a pushout square in both $(p_1\colon P\to A, p_2\colon P\to B)$1 and $(p_1\colon P\to A, p_2\colon P\to B)$2 (Meglio et al., 2 Aug 2025).

On the independence-category side, the dagger category $(p_1\colon P\to A, p_2\colon P\to B)$3 of relations is built from an epi-regular independence category $(p_1\colon P\to A, p_2\colon P\to B)$4. Its objects are those of $(p_1\colon P\to A, p_2\colon P\to B)$5, while morphisms $(p_1\colon P\to A, p_2\colon P\to B)$6 are isomorphism classes of jointly monic spans

$(p_1\colon P\to A, p_2\colon P\to B)$7

Composition is defined by taking an independent pullback of the middle legs, factoring the resulting composed span as a $(p_1\colon P\to A, p_2\colon P\to B)$8 span, and keeping the jointly monic part. The dagger is span reversal: $(p_1\colon P\to A, p_2\colon P\to B)$9 The epi-regular axioms guarantee that this composition is well-defined, associative, and unital.

The comparison functors are explicit. $(u,v)$0 sends $(u,v)$1 to $(u,v)$2. $(u,v)$3 sends a dilatory dagger category to its wide subcategory of coisometries equipped with the canonical independence structure. Coisometries inside $(u,v)$4 are exactly the morphisms $(u,v)$5, and the embedding

$(u,v)$6

is an isomorphism of epi-regular independence categories. Conversely,

$(u,v)$7

is a dagger functor giving the counit of the equivalence. The result extends to a strict 2-adjoint 2-equivalence when 2-cells are taken to be natural coisometries on the dagger side and independent natural transformations on the independence side (Meglio et al., 2 Aug 2025).

4. Representative examples

The most immediate example comes from ordinary regularity: for any regular category $(u,v)$8, the subcategory $(u,v)$9 is epi-regular independence. This captures the idea that the “maps” of the relational theory are quotient-like rather than arbitrary morphisms (Meglio et al., 2 Aug 2025).

A concrete non-classical example is $(f,g)$0, the category of surjections. It arises as the coisometry subcategory of the dagger category $(f,g)$1 of surjective multivalued functions. In $(f,g)$2, a commuting square of surjections is independent iff the induced map from the apex to the pullback is surjective. The resulting epi-regular independence category reconstructs $(f,g)$3 as a dagger category of relations, so multirelations are recovered from surjections plus an independence predicate (Meglio et al., 2 Aug 2025).

A probabilistic example is supplied by $(f,g)$4, the dagger category of finite probability spaces and measure-preserving stochastic matrices. Its coisometries are deterministic stochastic matrices, so $(f,g)$5 is the category of finite probability spaces and deterministic maps. Independence of squares corresponds to conditional independence of random variables, and independent pullbacks are conditional products $(f,g)$6 with joint law

$(f,g)$7

The associated relation category is the category of couplings of finite probability spaces, identified with $(f,g)$8 by the equivalence (Meglio et al., 2 Aug 2025).

An operator-theoretic example is $(f,g)$9, the dagger category of Hilbert spaces and contractions. Dilators exist via minimal unitary dilation, and the resulting coisometry category is epi-regular independence. This gives an abstract relation-theoretic treatment of Hilbert contractions through coisometric spans and dilators rather than through regular pullbacks (Meglio et al., 2 Aug 2025).

5. Algebraic regularity, internal relations, and independence-like behavior

Regular categories provide more than a source of examples; they also supply the relation calculus that motivates the epi-regular viewpoint. In a regular category, relations are jointly monomorphic spans, their composition is defined by pullback followed by image factorization, and the resulting category $e$0 is associative. Kernel pairs are always internal equivalence relations, and the Barr–Kock theorem expresses a strong stability phenomenon linking kernel pairs, regular epimorphisms, and pullbacks (Gran, 2020).

Mal’cev and Goursat conditions sharpen this relation theory. In a regular Mal’cev category, every reflexive relation is an equivalence relation, every relation is difunctional,

$e$1

and equivalence relations $e$2-permute: $e$3 In a regular Goursat category, equivalence relations $e$4-permute: $e$5 and regular images of equivalence relations remain equivalence relations. Goursat pushouts and the denormalized $e$6 lemma characterize this stability in terms of kernel pairs and exact forks (Gran, 2020).

The regular-category introduction explicitly observes that these conditions are structurally close to categorical notions of independence. Regularity provides image factorisations and stable quotients; Goursat categories preserve equivalence relations under regular epimorphic images; Mal’cev categories enforce maximal symmetry and difunctionality. This suggests that independence expressed through quotients, relations, or compatibility of factorisations is governed by the same epis, kernel pairs, and relational compositions that epi-regular independence categories abstract away from ordinary pullback calculus (Gran, 2020).

6. Related programs and terminological landscape

The expression “epi-regular independence category” is not uniform across the broader categorical-independence literature. In one nearby line of work, the phrase is reorganized around the standard $e$7 factorization system. There, pullback squares of a class $e$8 of monomorphisms yield NSOP$e$9-like or simple independence when $(m_1,m_2)$0 is cubic or strongly cubic, continuous, and accessible; with $(m_1,m_2)$1 taken as regular monomorphisms, this motivates a plausible notion of epi-regular independence category built from the subcategory $(m_1,m_2)$2 and its pullback squares (Kamsma et al., 2023).

The model-theoretic AECat program gives a different abstraction level. Independence is a ternary relation on subobjects satisfying invariance, monotonicity, symmetry, transitivity, existence, extension, and union, with stable, simple, and NSOP$(m_1,m_2)$3-like refinements obtained by adding base-monotonicity, club local character, stationarity, and the independence theorem. In that setting, canonicity theorems identify simple independence with isi-dividing and isi-forking, and NSOP$(m_1,m_2)$4-like independence with long Kim-dividing, but all morphisms are monomorphisms, so this is not itself an epi-regular theory in the relational sense (Kamsma, 2021).

The accessible-category stability literature supplies additional neighboring usage. Cellular categories and stable independence are linked by the statement that combinatorial cellular categories are exactly those giving rise to stable independence notions via $(m_1,m_2)$5-effective squares (Lieberman et al., 2019). In the same general framework, induced and higher-dimensional stable independence apply equally when the distinguished class $(m_1,m_2)$6 is epi-like or regular-epi-like, and stable independence in such settings yields higher-dimensional independence and excellence (Lieberman et al., 2020). These works describe an epi-based independence program, but they do not coincide formally with the dagger-theoretic definition of epi-regular independence category.

A further prospective extension appears in quasi-abelian categories. Universal monomorphisms and epimorphisms are defined by stability under pushout and pullback, respectively, and in any quasi-abelian category Axiom $(m_1,m_2)$7 and Axiom $(m_1,m_2)$8 hold: universal monomorphisms are stable under pullback and universal epimorphisms are stable under pushout. In $(m_1,m_2)$9, universal epimorphisms are exactly surjective operators and universal monomorphisms are exactly closed-range injective operators. A plausible implication is that these classes furnish regular-like epi/mono classes for an epi-regular theory beyond the strictly regular setting (Zinchenko, 13 Mar 2026).