Dilatory Dagger Categories

• Dilatory dagger categories are dagger categories in which every morphism has a universal coisometric span presentation, called a dilator.
• They replace traditional map-based relational frameworks with structures built on coisometries and independent pullbacks to model anomalous examples.
• Their construction unifies diverse examples including surjective multivalued functions, injective partial functions, finite probability spaces, and Hilbert spaces.

Dilatory dagger categories are dagger categories in which every morphism admits a universal coisometric span presentation, called a dilator. They were introduced to account for several categories that behave like categories of relations but do not fit the established theory of relations in regular categories, including the category of surjective multivalued functions, the category of injective partial functions, the category of finite probability spaces and stochastic matrices, and the category of Hilbert spaces and linear contractions (Meglio et al., 2 Aug 2025). Their structural role is explicitly relation-theoretic: just as regular categories correspond to tabular allegories, epi-regular independence categories correspond to dilatory dagger categories, with the equivalence sending an epi-regular independence category to its associated dagger category of relations and a dilatory dagger category to its wide subcategory of coisometries (Meglio et al., 2 Aug 2025).

1. Origins in relation theory

The classical theory of relations explains many “relation-like” categories by passing from a regular category to its category of jointly monic spans. On the abstract side, this is captured by tabular allegories, and the standard correspondence is an adjoint equivalence

$\begin{tikzcd}[column sep=large] \LexReg \ar[r, "\Rel", shift left=2] \ar[r, left adjoint] {} & \TabAll \ar[l, "\Map", shift left=2] \end{tikzcd}$

between finitely complete regular categories and unitary tabular allegories (Meglio et al., 2 Aug 2025).

The motivating problem is that several natural categories appear relational without being explained by that paradigm. In the anomalous examples, morphisms can still be represented by spans or cospans with specially constrained legs, but those legs do not arise as the maps or comaps selected by the existing allegorical frameworks. The obstruction is structural: the wide subcategory of “function-like” morphisms is typically not a regular category, and may fail to have the pullbacks needed for ordinary relational composition. The paper’s response is to replace regularity by epi-regularity, ordinary pullbacks by independent pullbacks, and tabularity by a dagger-theoretic universality condition expressed through dilators (Meglio et al., 2 Aug 2025).

2. Dagger-theoretic substrate

A dilatory dagger category is first of all a dagger category. Thus, for every morphism $r \colon A \to B$, there is a morphism

$r^\dagger \colon B \to A$

such that

$$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$

Within any dagger category, a morphism $f$ is isometric when

$f^\dagger f = 1,$

coisometric when

$ff^\dagger = 1,$

and unitary when both hold. The corresponding wide subcategories are denoted

$\Isom(D), \qquad \Coisom(D).$

The coisometries play the role of map-like morphisms in the dilatory theory (Meglio et al., 2 Aug 2025).

This shift from maps to coisometries is decisive. In the classical regular-category story, relations are composed through spans of maps. In the dilatory setting, relations are encoded by spans of coisometries and, dually, by cospans of isometries. A basic lemma underscores this interpretation: if a coisometry $f$ has a dilator, then $(1,f)$ is a dilator of $r \colon A \to B$0. In that sense, coisometries are precisely the morphisms already presented in universal relational form (Meglio et al., 2 Aug 2025).

3. Dilations, codilations, and universal span presentations

For a morphism $r \colon A \to B$1 in a dagger category, a dilation of $r \colon A \to B$2 is a span $r \colon A \to B$3 from $r \colon A \to B$4 to $r \colon A \to B$5 such that $r \colon A \to B$6 and $r \colon A \to B$7 are coisometries and

$r \colon A \to B$8

A dilator of $r \colon A \to B$9 is a terminal dilation: a dilation $r^\dagger \colon B \to A$0 such that every other dilation $r^\dagger \colon B \to A$1 factors uniquely through it by a coisometry. Dually, a codilation of $r^\dagger \colon B \to A$2 is a cospan $r^\dagger \colon B \to A$3 from $r^\dagger \colon B \to A$4 to $r^\dagger \colon B \to A$5 with $r^\dagger \colon B \to A$6 and $r^\dagger \colon B \to A$7 isometries and

$r^\dagger \colon B \to A$8

and a codilator is an initial codilation (Meglio et al., 2 Aug 2025).

The dagger exchanges the two notions. A pair $r^\dagger \colon B \to A$9 is a dilator of $$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$0 if and only if $$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$1 is a codilator of $$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$2. Consequently, a dagger category is dilatory precisely when every morphism has a dilator, equivalently when every morphism has a codilator. Chosen dilators are written

$$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$3

with corresponding chosen codilator legs $$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$4 and $$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$5. Dilators and codilators are unique up to isometric isomorphism, so the universal factorization is canonical at the dagger-theoretic level (Meglio et al., 2 Aug 2025).

Conceptually, the dilator is the direct analogue of a tabulator in an allegory. The difference is that the ambient involution is now the dagger, and the representing span is required to use coisometries rather than ordinary maps.

4. Independence and epi-regularity

The map-side counterpart of a dilatory dagger category is an independence category. This is a category equipped with a predicate $$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$6 on commutative squares

$$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$7

satisfying axioms $$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$8–$$1^\dagger = 1, \qquad (sr)^\dagger = r^\dagger s^\dagger, \qquad r^{\dagger\dagger} = r.$$9: independence implies commutativity, identity squares are independent, independence pastes, it is symmetric, and diagonal identity squares are independent (Meglio et al., 2 Aug 2025).

An independent pullback of a cospan $f$0 is then a span $f$1 forming an independent square and terminal among all spans doing so. This replaces the ordinary pullback as the operation by which relation-like morphisms compose. An independence category is epi-regular when:

1. every cospan has an independent pullback;
2. every span has a $f$2 factorization;
3. every morphism is strong epic.

The resulting category of such structures is denoted

$f$3

The paper also gives an alternative characterization: an independence category is epi-regular iff every cospan has an independent pullback, every span has a $f$4 factorization, and every morphism is the coequaliser of its kernel pair (Meglio et al., 2 Aug 2025).

These conditions are tailored to the motivating examples. They retain enough exactness to support relational calculus, while abandoning the requirement that the map-like morphisms form a regular category in the ordinary sense.

5. The equivalence with categories of relations

Given an epi-regular independence category $f$5, the paper constructs a dagger category $f$6. Its objects are those of $f$7, and a morphism $f$8 is an isomorphism class $f$9 of jointly monic spans

$f^\dagger f = 1,$0

The identity is

$f^\dagger f = 1,$1

and the dagger is span reversal,

$f^\dagger f = 1,$2

Composition is defined by first taking an independent pullback of the middle legs and then a $f^\dagger f = 1,$3 factorization of the induced outer span (Meglio et al., 2 Aug 2025).

Conversely, given a dilatory dagger category $f^\dagger f = 1,$4, the wide subcategory $f^\dagger f = 1,$5 becomes an epi-regular independence category by declaring a square

$f^\dagger f = 1,$6

independent exactly when

$f^\dagger f = 1,$7

In a dilatory dagger category, the second condition already implies commutativity, so independence can be checked by

$f^\dagger f = 1,$8

The paper proves that $f^\dagger f = 1,$9 is epi-regular (Meglio et al., 2 Aug 2025).

The main theorem is then the advertised equivalence

$ff^\dagger = 1,$0

between epi-regular independence categories and dilatory dagger categories. On one side, a map-category is sent to its dagger category of relations; on the other, a dilatory dagger category is sent to its wide subcategory of coisometries (Meglio et al., 2 Aug 2025).

6. Examples and broader significance

The motivating examples all fit the pattern that each morphism is represented by a span or cospan with special isometric or coisometric legs. Surjective multivalued functions are represented by jointly monic spans of surjective functions. Injective partial functions are represented by jointly epic cospans of injective functions. Finite probability spaces with measure-preserving stochastic matrices admit representations by spans of deterministic surjective maps, i.e. couplings. Hilbert spaces and linear contractions admit factorizations as cospans of isometries, reflecting the paper’s explicit connection with dilation theory (Meglio et al., 2 Aug 2025).

What unifies these examples is not that they are merely dagger categories, but that their “function-like” subcategory is governed by universal dilations rather than by ordinary pullback-stable maps. The relation calculus is therefore encoded by coisometries and independent pullbacks, not by maps and regular pullbacks. This makes dilatory dagger categories a precise abstraction of relation categories whose representing legs are epi-like or coisometric rather than map-like in the classical allegorical sense.

The broader literature contains other constructions of dagger categories of relations. In particular, symmetric comparison relations and tame relations inside an involutive monoidal category yield dagger categories whose examples include partial injections, locally bifinite relations, bifinite multirelations, formal distributions, and Hilbert spaces with bounded linear maps (Jacobs, 2011). This suggests that dilatory dagger categories belong to a wider program in which dagger structure is extracted from relation calculi rather than imposed as an isolated involution. Within that program, the distinct contribution of dilatory dagger categories is the exact analogue they provide to the classical regular-category/tabular-allegory correspondence.