Regular Double Hyperdoctrine

• Regular double hyperdoctrine is a categorical framework that packages reindexing, existential pushforward, Beck–Chevalley, and Frobenius into one unified double-categorical object.
• It employs a lax symmetric monoidal double pseudofunctor mapping spans (with adequate triple structure) to quintets, thereby integrating tight and loose morphism semantics.
• Companion commuter transformations in the framework ensure the invertibility of Frobenius cells, extending classical regular hyperdoctrines to accommodate graphical logic and operadic system specifications.

Searching arXiv for the cited paper and closely related prior work. A regular double hyperdoctrine is a lax symmetric monoidal double pseudofunctor

$$F:\;\mathbb{S}\mathrm{pan}(C,L,R)^{op}\to\mathbb{Q}\mathrm{t}(K)$$

whose monoidal laxators are companion commuter transformations, thereby packaging reindexing, existential quantification, Beck–Chevalley, and Frobenius into a single double-categorical object. In the formulation developed in "Double-functorial representation of regular structures" (Siqueira, 8 Aug 2025), this notion arises from viewing regular hyperdoctrines through the double category of spans and the double category of quintets. The resulting framework extends the classical observation that bicategories of spans capture the Beck–Chevalley relationship underlying existential semantics, and identifies Frobenius as a monoidal compatibility condition expressed by companion commuters (Siqueira, 8 Aug 2025).

1. Classical regular hyperdoctrines and the span semantics

Classically, a regular hyperdoctrine over a category $C$ with finite limits assigns to each object $X$ a fiber $PX$ of predicates, typically a meet-semilattice, and to each arrow $f:X\to Y$ a reindexing map $Pf:PY\to PX$, together with a left adjoint $\exists_f\dashv Pf$. The structure is required to satisfy Beck–Chevalley and Frobenius. In the semilattice setting, Frobenius has the form

$$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$

These data instantiate Lawvere’s dictum that quantifiers are adjoints (Siqueira, 8 Aug 2025).

The span construction gives the standard semantics for existential quantification. In a bicategory of spans, a 1-cell $X\to Y$ is a span $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$, interpreted as substitution along $C$0 followed by existential pushforward along $C$1. Composition is by pullback, and this universal behavior is exactly what enforces Beck–Chevalley. For a pullback square

$C$2

the corresponding isomorphism is

$C$3

The paper’s starting point is that this span-based representation captures Beck–Chevalley but, by itself, does not yet isolate Frobenius as an intrinsic categorical property (Siqueira, 8 Aug 2025).

2. Double categories, adequate triples, and quintets

The double-categorical formulation separates two kinds of morphism. A double category has a category of objects and tight arrows, a category of loose arrows and cells, source and target functors, identities, and horizontal composition, with associativity and unit constraints possibly weak. In the span case, tight arrows are ordinary morphisms of contexts, whereas loose arrows are spans; squares mediate between these two directions (Siqueira, 8 Aug 2025).

To control the existence of pullbacks used in span composition, the framework employs an adequate triple $C$4. Here $C$5 and $C$6 are classes of arrows closed under composition and containing identities, and whenever a cospan has left leg in $C$7 and right leg in $C$8, the relevant pullback exists, with pullback stability conditions for the two classes. In the cartesian case, $C$9 is cartesian monoidal, $X$0 and $X$1 are closed under finite products, and projections belong to them. A $X$2-span is then a span whose left leg lies in $X$3 and right leg lies in $X$4, and these form the double category $X$5 (Siqueira, 8 Aug 2025).

The codomain of the representation is the double category of quintets $X$6 associated to a 2-category $X$7. In $X$8, both tight and loose arrows are 1-morphisms of $X$9, and squares are 2-cells of shape

$PX$0

This choice is significant because it lets a single double pseudofunctor encode both the tight reindexing direction and the loose existential-composite direction. In this sense, adjunctions and Beck–Chevalley appear not as external axioms attached afterward, but as cells internal to the double-categorical semantics (Siqueira, 8 Aug 2025).

3. Double pseudofunctorial representation of regular structures

A double pseudofunctor

$PX$1

consists of maps on objects/tight arrows and on loose arrows/cells, preserving sources and targets, together with coherent unit and composition comparison data. The paper studies lax symmetric monoidal double pseudofunctors from $PX$2 to $PX$3, where $PX$4 is a cartesian monoidal 2-category and the monoidal structure carries the usual pseudofunctorial coherence, including laxators $PX$5 and unit data $PX$6 (Siqueira, 8 Aug 2025).

The central representation theorem states that if $PX$7 is a cartesian adequate triple and $PX$8 is a cartesian monoidal 2-category, then every $PX$9-regular indexed monoidal structure

$f:X\to Y$0

extends to a lax symmetric monoidal double pseudofunctor

$f:X\to Y$1

whose monoidal laxators are companion commuter transformations. Conversely, if $f:X\to Y$2 contains all diagonals, the tight component of any such $f:X\to Y$3 recovers a $f:X\to Y$4-regular indexed monoidal structure in $f:X\to Y$5 (Siqueira, 8 Aug 2025).

The forward construction is explicit. Objects are sent by $f:X\to Y$6. Tight arrows $f:X\to Y$7 are sent to the reindexing morphisms $f:X\to Y$8. A loose arrow given by a span

$f:X\to Y$9

is sent to the composite

$Pf:PY\to PX$0

in $Pf:PY\to PX$1. Morphisms of spans are mapped to pastings built from the unit and counit of $Pf:PY\to PX$2 together with the image under $Pf:PY\to PX$3 of the mediating map. The double-pseudofunctorial composition constraint is obtained from the Beck–Chevalley mate associated to the pullback used in composing spans (Siqueira, 8 Aug 2025).

This packaging has a conceptual consequence: reindexing and existential pushforward are no longer treated as separate semantic operations related by axioms, but as the tight and loose images of one double-functorial semantics. A plausible implication is that regular logic can be studied in settings where the usual fiberwise presentation is less natural but a span-based double category is available.

4. Beck–Chevalley, companions, and Frobenius

Within this framework, Beck–Chevalley is represented by the preservation of span composition along pullbacks. For a $Pf:PY\to PX$4-pullback square, the mate of the reindexing square is denoted $Pf:PY\to PX$5, and invertibility of this mate is required precisely in the pullback cases. The compositor of the double pseudofunctor at a composite of spans is identified with the corresponding Beck–Chevalley isomorphism, which computes the familiar relation

$Pf:PY\to PX$6

Thus, Beck–Chevalley is not merely compatible with the double-categorical formalism; it is built into the meaning of horizontal composition in $Pf:PY\to PX$7 (Siqueira, 8 Aug 2025).

The more distinctive feature is the treatment of Frobenius. In the generalized monoidal setting, for $Pf:PY\to PX$8 the right Frobenius cell is the canonical comparison

$Pf:PY\to PX$9

and regularity requires this cell to be invertible; by cartesian symmetry, the left Frobenius cell is equivalent. The paper’s key claim is that this invertibility is characterized by the monoidal laxators being companion commuter cells (Siqueira, 8 Aug 2025).

For a tight arrow $\exists_f\dashv Pf$0 in a double category, a companion $\exists_f\dashv Pf$1 is a loose arrow $\exists_f\dashv Pf$2 equipped with unit and counit squares exhibiting it as a horizontal shadow of $\exists_f\dashv Pf$3, while a conjoint $\exists_f\dashv Pf$4 is the dual notion. These structures encode horizontal adjunctions; in $\exists_f\dashv Pf$5, a conjoint is an adjunction internal to $\exists_f\dashv Pf$6. A cell between loose arrows with vertical sides admitting companions is a companion commuter if its transpose along those companions is a globular tight isomorphism. The paper shows that when the monoidal laxator cells have this property, the induced Frobenius comparisons are invertible (Siqueira, 8 Aug 2025).

More concretely, the Frobenius cell is obtained by whiskering the monoidal laxator cell with the unit and counit of the adjunction $\exists_f\dashv Pf$7. If the laxator cell is a companion commuter, then its transpose along the companions of $\exists_f\dashv Pf$8 and $\exists_f\dashv Pf$9 is a tight isomorphism, and that is exactly what yields Frobenius. The proposed notion of regular double hyperdoctrine therefore isolates Frobenius as a purely double-categorical monoidal condition rather than an additional fiberwise axiom (Siqueira, 8 Aug 2025).

5. Data and axioms of a regular double hyperdoctrine

The structural data consist of a cartesian adequate triple $$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$0, a cartesian monoidal 2-category $$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$1, and a lax symmetric monoidal double pseudofunctor

$$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$2

that is normal on units and is equipped with monoidal laxators $$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$3, $$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$4, and $$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$5. The tight component $$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$6 gives a pseudofunctor

$$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$7

which is the reindexing part of the doctrine (Siqueira, 8 Aug 2025).

Existential quantification is encoded using conjoints in the source double category. For each $$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$8, the conjoint of $$\exists_{f}(P \wedge f^{*}Q) \;\cong\; (\exists_{f}P) \wedge Q.$$9 in $X\to Y$0 is mapped by $X\to Y$1 to an adjunction internal to $X\to Y$2,

$X\to Y$3

with unit $X\to Y$4 and counit $X\to Y$5. Beck–Chevalley then requires that for every $X\to Y$6-pullback square, the mate of the reindexing square be invertible. Frobenius requires invertibility of the canonical comparison cells

$X\to Y$7

and

$X\to Y$8

In semilattice fibers, this reduces to

$X\to Y$9

Finally, the monoidal laxators themselves must be companion commuter transformations (Siqueira, 8 Aug 2025).

The converse direction has an explicit hypothesis: $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$0 must contain all diagonals. Under this condition, the tight component of such an $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$1 suffices to reconstruct a $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$2-regular indexed monoidal structure in $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$3. This restriction is not incidental. It marks a boundary of the reconstruction theorem and prevents the converse from being stated without further assumptions (Siqueira, 8 Aug 2025).

A common misunderstanding would be to treat the double-categorical reformulation as a notational variant of ordinary hyperdoctrines. The paper’s construction is stronger: it identifies precisely which double-categorical and monoidal cells recover the regular structure, and it shows that Frobenius is controlled by companion-commuting laxators rather than being simply postulated independently. This suggests that the reformulation is substantive at the level of structure, not only of presentation.

6. Examples, graphical logic, and relation to earlier work

Several examples illustrate the scope of the notion. For classical regular logic, one may take $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$4 with finite limits, $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$5, and fibers $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$6 meet-semilattices. For $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$7, the construction recovers ordinary regular hyperdoctrines: a span is interpreted by reindexing followed by existential quantification, Beck–Chevalley holds along all pullbacks, and Frobenius is expressed using $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$8 (Siqueira, 8 Aug 2025).

The Lawvere quantale example takes $X \xleftarrow{x_1} S \xrightarrow{x_2} Y$9 and $C$00 with fibers

$C$01

ordered by $C$02 and tensor $C$03. Reindexing is by precomposition, and the left adjoint is

$C$04

According to the paper, this yields a $C$05-regular indexed monoidal poset in which Beck–Chevalley and Frobenius hold in the quantale semantics (Siqueira, 8 Aug 2025).

The slice-category example takes $C$06 for a category $C$07 with finite limits. The monoidal structure is given by pullback, reindexing is pullback along $C$08, and $C$09 is dependent sum. Beck–Chevalley and Frobenius are then instances of the pullback pasting lemma. The paper further notes that for $C$10, using homotopy categories of slices yields a $C$11-regular indexed monoidal category, and similarly for locally compact Hausdorff spaces with proper maps on the right, fibers may be homotopy categories of chain complexes of sheaves (Siqueira, 8 Aug 2025).

The lens example uses a cartesian fibration $C$12, with $C$13 the vertical maps and $C$14 the cartesian maps. Then $C$15 is the double category of lenses. The paper states that regular double hyperdoctrines over such triples model structured system specifications whose compositionality respects lens structure. This connects the formalism to system-theoretic semantics without changing the basic definition (Siqueira, 8 Aug 2025).

An application emphasized by the paper is graphical regular logic suitable for operadic composition of specifications, including port-plugging systems. In this interpretation, objects are contexts and judgements are spans; composition is existential hiding after substitution, realized by pullback and pushforward. The semantics validates the graphical inference rules because squares in $C$16 are exactly the required naturality, Beck–Chevalley, and Frobenius equalities of cells. The worked schematic identifies the equality of $C$17 and $C$18 with the invertibility of the Frobenius cell, represented diagrammatically by the monoidal laxator commuting with $C$19 and $C$20 (Siqueira, 8 Aug 2025).

In relation to earlier work, the paper situates itself after results of Hermida and Dawson–Paré–Pronk on sinister morphisms and Beck–Chevalley via span functors, and after double-categorical span constructions with strong universal properties due to Shulman and Grandis. Its stated novelty is the extension from the Beck–Chevalley picture to regular logic with monoidal fibers, and specifically the characterization of Frobenius by companion commuter cells, drawing on the companion/conjoint technology of Grandis–Paré and Paré’s retrocells (Siqueira, 8 Aug 2025).

The framework is presented with explicit limitations and directions. It assumes a cartesian adequate triple to guarantee the required pullbacks and products, and a cartesian monoidal 2-category $C$21 to support symmetric pseudomonoids and strong reindexing. The converse reconstruction requires $C$22 to contain all diagonals. The paper identifies as open directions the extension beyond cartesian monoidal $C$23, the incorporation of further logical connectives and modalities, and the study of interactions with operadic systems theory, particularly port-plugging. This suggests that the notion of regular double hyperdoctrine is intended as a stable base for broader double-categorical semantics rather than a terminal formulation (Siqueira, 8 Aug 2025).