Double Groupoid-Crossed Braided Bicategory

• Double Groupoid-Crossed Braided Bicategory is a bicategorical structure featuring two independent crossed gradings by groupoids, extending traditional braided monoidal frameworks.
• It employs a relative center construction within biactegories and op-monoidal functors to capture coherent half-braidings and tensorial interactions.
• The framework generalizes Turaev’s group-crossed braided categories by integrating algebraic and coalgebraic data to support generalized Yetter–Drinfeld modules.

Searching arXiv for the cited paper and closely related work to support the article. First, locating the main source paper by title. A double groupoid-crossed braided bicategory is a bicategorical braided structure equipped with two independent but compatible crossed gradings, one by a groupoid $H$ and one by a groupoid $G$, together with corresponding left and right actions by strong monoidal equivalences on endohom categories. In the formulation introduced in "Generalized Yetter-Drinfeld modules, the center of bi-actegories and groupoid-crossed braided bicategories" (Aziz et al., 11 Jul 2025), the notion arises as the global organizational framework for generalized Yetter–Drinfeld modules associated with bialgebras, bicomodule algebras, and bimodule coalgebras. It extends Turaev’s group-crossed braided monoidal categories in two directions at once: from monoidal categories to bicategories, and from a single group action to two groupoid actions, one on each side of the bicategorical structure.

1. Relative centers and the biactegorial setting

The construction is rooted in the notion of a $(\mathcal{C},\mathcal{D})$-biactegory $\mathcal{M}$. Such an $\mathcal{M}$ is simultaneously a left $\mathcal{C}$-actegory and a right $\mathcal{D}$-actegory, with coherent left unit and associativity constraints, right unit and associativity constraints, and a middle associativity isomorphism

$\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$

natural in $V\in\mathcal{C}$, $M\in\mathcal{M}$, and $G$0. The interchanger $G$1 is part of the standard coherence data for biactegories (Aziz et al., 11 Jul 2025).

An op-monoidal functor $G$2 is a functor equipped with structure morphisms

$G$3

subject to the unit and associativity compatibilities stated in Mac Lane-coherent form. Given a center datum $G$4, the lax $G$5-center $G$6 has as objects pairs $G$7, where $G$8 and

$G$9

is a natural transformation in $(\mathcal{C},\mathcal{D})$0, called an $(\mathcal{C},\mathcal{D})$1-half-braiding, satisfying the heptagon identity

$(\mathcal{C},\mathcal{D})$2

Morphisms are those maps in $(\mathcal{C},\mathcal{D})$3 compatible with the half-braidings. The strong $(\mathcal{C},\mathcal{D})$4-center $(\mathcal{C},\mathcal{D})$5 is the full subcategory in which all $(\mathcal{C},\mathcal{D})$6 are natural isomorphisms.

Under mild assumptions, $(\mathcal{C},\mathcal{D})$7 inherits a $(\mathcal{C},\mathcal{D})$8-biactegory structure. Its left and right actions are given explicitly by

$(\mathcal{C},\mathcal{D})$9

with the half-braidings defined by the formulas in Lemma 2.6. This relative-center mechanism is the categorical origin of the later crossed braided bicategorical structure.

2. Generalized Yetter–Drinfeld modules as a relative center

The principal specialization takes $\mathcal{M}$0, $\mathcal{M}$1, $\mathcal{M}$2, where $\mathcal{M}$3 and $\mathcal{M}$4 are bialgebras over a commutative ring $\mathcal{M}$5, $\mathcal{M}$6 is an $\mathcal{M}$7-bicomodule algebra, and $\mathcal{M}$8 is a $\mathcal{M}$9-bimodule coalgebra. In this setting,

$\mathcal{M}$0

The functor $\mathcal{M}$1 is op-monoidal if and only if $\mathcal{M}$2 is a $\mathcal{M}$3-bimodule coalgebra, with structure maps

$\mathcal{M}$4

The actegory $\mathcal{M}$5 becomes a $\mathcal{M}$6-biactegory by the $\mathcal{M}$7-linear tensor product action

$\mathcal{M}$8

A generalized Yetter–Drinfeld module over the YD datum $\mathcal{M}$9 is a left $\mathcal{C}$0-module $\mathcal{C}$1 with right $\mathcal{C}$2-coaction

$\mathcal{C}$3

such that

$\mathcal{C}$4

When $\mathcal{C}$5 is Hopf with bijective antipode $\mathcal{C}$6, this is equivalent to

$\mathcal{C}$7

The central structural result is Theorem 3.4: $\mathcal{C}$8 The equivalence is explicit. From an $\mathcal{C}$9-half-braiding $\mathcal{D}$0, the component at the regular $\mathcal{D}$1-module determines the coaction by

$\mathcal{D}$2

Conversely, if $\mathcal{D}$3 is a generalized Yetter–Drinfeld module, then for any $\mathcal{D}$4,

$\mathcal{D}$5

The half-braiding heptagon is exactly encoded by the coalgebra comultiplication in $\mathcal{D}$6, so the relative center recovers generalized Yetter–Drinfeld modules, rather than merely resembling them (Aziz et al., 11 Jul 2025).

3. Braided biactegories and the transport of Yetter–Drinfeld structure

The relative center formalism not only identifies generalized Yetter–Drinfeld modules; it also explains their tensorial and braided behavior. For two YD data $\mathcal{D}$7 and $\mathcal{D}$8, if $\mathcal{D}$9 is pure in $\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$0 and $\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$1 is a $\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$2-bimodule coalgebra, then the ordinary tensor product lifts to

$\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$3

The lifted $\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$4-action and $\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$5-coaction on $\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$6 are

$\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$7

The YD compatibility is verified componentwise, together with the balancing relations appearing in Proposition 4.1.

The crossed braiding enters sharply when $\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$8 is a $\eta:(V\cdot M)\cdot X \to V\cdot(M\cdot X),$9-bi-Galois co-object. In that case $V\in\mathcal{C}$0 is a strong monoidal equivalence, and if $V\in\mathcal{C}$1 denotes the inverse bi-Galois co-object, then the Morita maps $V\in\mathcal{C}$2 and $V\in\mathcal{C}$3 and the coalgebra map $V\in\mathcal{C}$4 control transport of YD structures. If $V\in\mathcal{C}$5, then

$V\in\mathcal{C}$6

with right coaction

$V\in\mathcal{C}$7

For $V\in\mathcal{C}$8 and $V\in\mathcal{C}$9, the natural map

$M\in\mathcal{M}$0

is $M\in\mathcal{M}$1-linear and right $M\in\mathcal{M}$2-colinear. Lemma 4.3 identifies the corresponding heptagon and hexagon identities, and Theorem 4.8 shows that $M\in\mathcal{M}$3 becomes a $M\in\mathcal{M}$4-prebraided $M\in\mathcal{M}$5-biactegory; in the Hopf/bi-Galois case it is $M\in\mathcal{M}$6-braided. Corollary 4.7 adds that

$M\in\mathcal{M}$7

is an equivalence, and in particular

$M\in\mathcal{M}$8

is an equivalence (Aziz et al., 11 Jul 2025).

4. Groupoids of Galois data and bicategorical grading

The bicategorical organization uses two groupoids built from Galois-type data. On the coalgebra side, one has the category whose objects are bialgebras and whose morphisms $M\in\mathcal{M}$9 are isomorphism classes of $G$00-bimodule coalgebras, composed by balanced tensor product $G$01. Restricting to Hopf algebras and bi-Galois co-objects yields the groupoid $G$02. Dually, on the algebra side, morphisms $G$03 are isomorphism classes of $G$04-bicomodule algebras, composed by cotensor product $G$05 under the stated flatness or purity conditions; restricting to bi-Galois objects gives the groupoid $G$06 (Aziz et al., 11 Jul 2025).

The paper also defines the paired category of YD data, whose morphisms are pairs $G$07 making $G$08 a YD datum, with composition by cotensor on the algebra side and reversed balanced tensor on the coalgebra side. This paired category is the natural grading target for the full bicategory of generalized Yetter–Drinfeld modules.

More generally, if $G$09 is a category, an $G$10-graded bicategory $G$11 is specified by a 2-functor

$G$12

where $G$13 is regarded as a discrete bicategory. Then for objects $G$14 of $G$15,

$G$16

and horizontal composition respects grading: $G$17

The central example is the bicategory denoted $G$18, whose objects are bialgebras and whose 1-cells $G$19 are generalized Yetter–Drinfeld modules over some YD datum $G$20 from $G$21 to $G$22. The paper further exhibits restricted sub-bicategories such as $G$23, graded over the pair of groupoids of bi-Galois objects and bi-Galois co-objects. This grading is not decorative: it records precisely which Galois algebra and co-object data control the crossed braidings and their transport across hom-categories.

5. Definition of the double crossed braided structure

Before introducing the double version, the paper formulates a single-groupoid notion. A right $G$24-crossed braided bicategory $G$25 is $G$26-graded via $G$27, equipped with a groupoid opmorphism

$G$28

with $G$29, so that for $G$30 there is a strong monoidal functor

$G$31

satisfying $G$32, $G$33, and

$G$34

A braiding is then given by 2-cells

$G$35

for $G$36 and $G$37, subject to the heptagon and hexagon identities written in Section 5.

A double groupoid-crossed braided bicategory is defined from two groupoids $G$38 and $G$39. A bicategory $G$40 is $G$41-double crossed braided if it is $G$42-graded via

$G$43

and carries opmorphisms

$G$44

with $G$45 and $G$46. These actions preserve the gradings by

$G$47

The structure includes two braiding families. For

$G$48

one has a left-handed family

$G$49

For

$G$50

one has a right-handed family

$G$51

These satisfy heptagon and hexagon axioms of the same pattern as the single-groupoid case, together with the left-handed counterparts for $G$52 (Aziz et al., 11 Jul 2025).

The paper also gives a more general variant in which the grading targets are categories $G$53, together with functors $G$54 and $G$55. In that form, the bicategory is $G$56-double graded and $G$57-double crossed, with the same formal pattern of constraints but grading identities expressed through $G$58 and $G$59.

6. Realization for generalized Yetter–Drinfeld modules, reductions, and examples

The organizational theorem states that the bicategory of generalized Yetter–Drinfeld modules $G$60 is a $G$61-double graded $G$62-double crossed braided bicategory. Its bi-Galois restriction $G$63 is a $G$64-double crossed braided bicategory. On endohom categories, the two actions are given by

$G$65

on the left, for bi-Galois objects, and

$G$66

on the right, for bi-Galois co-objects. The braidings $G$67 and $G$68 are induced by the formulas established in Theorems 4.8 and 4.9, including

$G$69

The relation to Turaev’s group-crossed braided monoidal categories is direct. Turaev’s setting involves a single group $G$70 acting by strict monoidal automorphisms on a $G$71-graded monoidal category $G$72, with braiding

$G$73

The present construction generalizes that picture by allowing multiple objects and 1-cells in a bicategory and by replacing the single group with two groupoids acting from opposite sides. If one fixes a single object $G$74, restricts to the endohom category $G$75, lets the groupoids reduce to groups, and trivializes one side, then one recovers Turaev’s right or left group-crossed braided monoidal category. This addresses a frequent point of confusion: the new notion is designed as an enlargement of Turaev’s framework, not as a disconnected alternative (Aziz et al., 11 Jul 2025).

The paper records several specializations. In the classical case $G$76, $G$77, $G$78 with regular actions and coactions, the category $G$79 is braided monoidal, with action

$G$80

coaction

$G$81

and braiding

$G$82

When $G$83 is Hopf with bijective antipode, this coincides with the Drinfeld center $G$84.

For $G$85-Yetter–Drinfeld modules over a Hopf algebra $G$86, the data $G$87 yield generalized YD modules satisfying

$G$88

recovering standard and anti-YD cases in the indicated specializations. The group algebra case $G$89 similarly specializes the generalized YD condition and the braiding to the familiar $G$90-graded crossed braiding when $G$91.

The framework is formulated over a commutative base ring $G$92. For certain dual center arguments, modules are assumed projective as $G$93-modules. Strong-center and weak-center coincidence requires additional hypotheses, such as the isomorphism of suitable Galois canonical maps or sufficient faithfulness and flatness conditions. The double crossed braided structure is described as most transparent on the bi-Galois restricted sub-bicategory $G$94; in the fully general $G$95 setting, grading by all YD data is retained through functors from the Galois groupoids. Within those assumptions, the construction provides a categorical mechanism in which generalized Yetter–Drinfeld modules appear simultaneously as relative-center objects and as 1-cells in a bicategory equipped with two crossed braidings.