Coloured Braid Groupoid

• The coloured braid groupoid is a structure where objects are branch-point labellings by transpositions and morphisms are isotopy classes of braids that follow the Hurwitz conjugation rule.
• It extends the classical lifting homomorphism to all coloured braids, enabling a functorial lift into a mapping-class groupoid of covering surfaces.
• Its construction using Artin generators, graphical moves, and CW-complexes offers effective computational algorithms and deeper categorical insights into branched cover theory.

The coloured braid groupoid, in the sense developed for simple branched covers of the disc, is a groupoid whose objects are branch-point labellings by transpositions and whose morphisms are isotopy classes of braids carrying those labels through the Hurwitz conjugation rule. For fixed integers $n$ and $d \ge 3$, and fixed total monodromy

$\mu=t_1t_2\cdots t_n \in S_d$

with each $t_i$ a transposition, one works with

$\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$

The principal role of the groupoid is that it enlarges the classical domain of the lifting homomorphism: instead of being defined only on the liftable braid subgroup $LB_{n,d}\subset B_n$, the lift extends to every coloured braid as a morphism in a mapping-class groupoid of the covering surfaces (Licata et al., 7 Aug 2025).

1. Algebraic definition

An object of the $n$-strand $d$-coloured braid groupoid $\ColB_{n,d}(\mu)$ is a labelling

$\tau=(t_1,\dots,t_n)\in \T(\mu),$

that is, an ordered $d \ge 3$0-tuple of transpositions in $d \ge 3$1 whose product is $d \ge 3$2. A morphism

$d \ge 3$3

is an isotopy class of braids on $d \ge 3$4 strands in the cylinder $d \ge 3$5 whose initial strand-labels are $d \ge 3$6 and whose final strand-labels are $d \ge 3$7 (Licata et al., 7 Aug 2025).

The labelling rule at a crossing is the defining feature. If the overstrand has label $d \ge 3$8 and the understrand has label $d \ge 3$9, then the overstrand keeps its label $\mu=t_1t_2\cdots t_n \in S_d$0, while the understrand label is conjugated: $\mu=t_1t_2\cdots t_n \in S_d$1 Consequently, once the initial labelling $\mu=t_1t_2\cdots t_n \in S_d$2 is fixed, every crossing determines uniquely how the labels evolve along the braid diagram.

The groupoid operations are the standard ones for braid-like path categories. Composition is defined by stacking braids: $\mu=t_1t_2\cdots t_n \in S_d$3 for $\mu=t_1t_2\cdots t_n \in S_d$4 and $\mu=t_1t_2\cdots t_n \in S_d$5. The identity at $\mu=t_1t_2\cdots t_n \in S_d$6 is the trivial straight braid $\mu=t_1t_2\cdots t_n \in S_d$7. Every braid is invertible by horizontal reflection, and the inverse carries the final labels back to the initial ones.

The same structure admits an Artin-presentation description. The groupoid is generated by the usual Artin generators $\mu=t_1t_2\cdots t_n \in S_d$8 for $\mu=t_1t_2\cdots t_n \in S_d$9, but each generator now acts together with the label-conjugation rule. The only relations are the usual braid relations,

$t_i$0

together with the requirement that $t_i$1-words compose only when the intermediate labellings match.

2. Branched covers and the classical lifting problem

For each labelling $t_i$2, one fixes a $t_i$3-fold simple branched covering

$t_i$4

branched at marked points $t_i$5, with monodromy around $t_i$6 equal to the transposition $t_i$7 (Licata et al., 7 Aug 2025).

A concrete construction is specified as follows. One chooses a basis of arcs $t_i$8 from a boundary basepoint, labels them by the $t_i$9, cuts the disc along auxiliary cuts $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$0, takes $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$1 copies, and regluess them so that the sheet-exchange along $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$2 is exactly $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$3. The result is a compact surface $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$4 with boundary. Its number of boundary components equals the number of cycles in the product $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$5.

The classical notion of liftability appears when a braid preserves the labelling. If a coloured braid $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$6 satisfies $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$7, then $\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$8 is liftable in the classical sense, and it has a unique lift

$\T(\mu)=\bigl\{\tau=(t_1,\dots,t_n)\in (S_d)^n \mid t_1\cdots t_n=\mu\bigr\}.$9

characterised by

$LB_{n,d}\subset B_n$0

The set of such liftable braids is a subgroup $LB_{n,d}\subset B_n$1.

In this classical setting, the lifting homomorphism is only partially defined: it exists only on the proper subgroup of braids that preserve the branch labelling. The coloured braid groupoid is designed precisely to remove this restriction.

3. Extension to a mapping-class groupoid

When $LB_{n,d}\subset B_n$2, the covering recipe still produces a surface $LB_{n,d}\subset B_n$3, homeomorphic to $LB_{n,d}\subset B_n$4 but with its base-arcs permuted. This leads to the mapping-class groupoid $LB_{n,d}\subset B_n$5, whose objects are all surfaces $LB_{n,d}\subset B_n$6 for $LB_{n,d}\subset B_n$7, each equipped with the fixed numbering of boundary basepoints, and whose morphisms are isotopy classes of orientation-preserving homeomorphisms preserving those numbered basepoints (Licata et al., 7 Aug 2025).

For any coloured braid $LB_{n,d}\subset B_n$8, its lift is defined to be the unique morphism

$LB_{n,d}\subset B_n$9

satisfying

$n$0

Existence and uniqueness follow from the usual covering-homotopy argument and from the fact that $n$1 permutes branch points exactly by the Hurwitz rule.

This reformulation changes the ambient category rather than the local lifting equation. The classical lift sits inside the new framework as the special case $n$2, where $n$3 is an endomorphism of $n$4, hence an element of $n$5. The conceptual shift is from a partially defined homomorphism on a subgroup to a functor defined on a full groupoid of labelled braids.

4. Graphical model and explicit lifting functor

The lifting is made explicit through a graph-decorated intermediary. For each $n$6, one considers the pair

$n$7

where

$n$8

is the union of the two lifts of each basis arc $n$9 meeting at the unique branch point above $d$0. The arcs $d$1 form a tree-like spine cutting $d$2 into $d$3 disks (Licata et al., 7 Aug 2025).

For each coloured Artin generator $d$4 acting on a labelling $d$5, one defines a combinatorial move

$d$6

by arc-sliding $d$7 over $d$8 at each shared endpoint, followed by swapping labels $d$9. There are three local configurations—no shared endpoint, two shared endpoints, and one shared endpoint—and in each case the resulting set of arcs again cuts the surface into $\ColB_{n,d}(\mu)$0 disks with the correct new monodromy labelling.

This produces a graphical groupoid $\ColB_{n,d}(\mu)$1, whose objects are the graphical objects $\ColB_{n,d}(\mu)$2 and whose morphisms are generated by the $\ColB_{n,d}(\mu)$3 subject only to the braid relations. If $\ColB_{n,d}(\mu)$4 denotes the $\ColB_{n,d}(\mu)$5-tuple of transpositions recovered from the endpoints of the arcs $\ColB_{n,d}(\mu)$6, then

$\ColB_{n,d}(\mu)$7

so the graph move exactly records the Hurwitz action of $\ColB_{n,d}(\mu)$8 on the labelling.

Two functors are then defined: $\ColB_{n,d}(\mu)$9 On a braid word $\tau=(t_1,\dots,t_n)\in \T(\mu),$0, the graphical functor is

$\tau=(t_1,\dots,t_n)\in \T(\mu),$1

and it is well defined independently of the braid word. The lift $\tau=(t_1,\dots,t_n)\in \T(\mu),$2 is obtained by applying $\tau=(t_1,\dots,t_n)\in \T(\mu),$3 to the decorated spine and then using the unique homeomorphism that takes the resulting decorated spine back to the standard spine of $\tau=(t_1,\dots,t_n)\in \T(\mu),$4.

The functoriality statement is strict: $\tau=(t_1,\dots,t_n)\in \T(\mu),$5 and the lifting equation

$\tau=(t_1,\dots,t_n)\in \T(\mu),$6

holds on the nose for every coloured braid $\tau=(t_1,\dots,t_n)\in \T(\mu),$7. When $\tau=(t_1,\dots,t_n)\in \T(\mu),$8, $\tau=(t_1,\dots,t_n)\in \T(\mu),$9 recovers the classical lift $d \ge 3$00 (Licata et al., 7 Aug 2025).

The same construction gives an explicit combinatorial algorithm: factor $d \ge 3$01 into Artin generators, replace each $d \ge 3$02 by the corresponding arc-slide $d \ge 3$03, and then pass back to the standard decorated spine. The exposition describes this as an effective way to compute the lift of an arbitrary $d \ge 3$04, in contrast with the classical theory, where one first had to check liftability and then factor in the liftable subgroup.

5. Examples and geometric behaviour

A basic example occurs for $d \ge 3$05, $d \ge 3$06, and branch-labelling

$d \ge 3$07

In this case $d \ge 3$08 is an annulus. The half-twist $d \ge 3$09 satisfies

$d \ge 3$10

and $d \ge 3$11 is the smallest positive power of $d \ge 3$12 that fixes the labelling (Licata et al., 7 Aug 2025).

In the graphical picture, there are three distinct objects

$d \ge 3$13

and the composite $d \ge 3$14 returns the decorated graph to itself by three successive arc-slides. Hence

$d \ge 3$15

is the identity on $d \ge 3$16.

A second example uses a $d \ge 3$17-fold simple cover branched at three points with labelling

$d \ge 3$18

For the braid

$d \ge 3$19

tracking the action of each $d \ge 3$20 on the ribbon-graph spine shows that the net effect on $d \ge 3$21 is a single positive Dehn twist around the simple closed curve projecting to the path joining the two $d \ge 3$22-labels. Thus $d \ge 3$23 is that Dehn twist in $d \ge 3$24.

These examples illustrate two distinct geometric outcomes. A braid may be nontrivial in the braid groupoid while inducing the identity on the covering surface, as in the annulus example; alternatively, a braid word may lift to a standard mapping-class-theoretic generator, namely a Dehn twist.

6. Significance, CW models, and related uses of the term

The principal significance of the construction is categorical. By passing from the braid group to the full coloured braid groupoid, one can lift every coloured braid, including braids whose endpoints have different labellings, into a mapping-class groupoid of the covering surface. The restriction to the classical liftable subgroup $d \ge 3$25 is therefore removed (Licata et al., 7 Aug 2025).

The same work also describes CW-complexes built from the groupoids. These provide a graphical universal-cover model for the mapping class group of the $d \ge 3$26-fold cover. One attaches $d \ge 3$27-cells corresponding to the usual braid relations and to the minimal liftable powers, called Type 2 and Type 3 arcs, as well as to the Dehn-twist relations generating $d \ge 3$28. The resulting pair of CW-complexes

$d \ge 3$29

exhibits $d \ge 3$30 as the universal cover of $d \ge 3$31. Potential further directions identified there include a purely groupoid-theoretic presentation of the $d \ge 3$32-cells, extensions to non-simple branched covers, and applications to monodromy factorizations in Lefschetz fibrations.

The term “coloured braid groupoid” also appears in other, structurally different settings. In "Divisor braids" (Bökstedt et al., 2016), a related groupoid arises from configuration spaces on a closed, connected, oriented surface $d \ge 3$33: its objects are $d \ge 3$34-tuples of coloured configurations

$d \ge 3$35

its morphisms are divisor braids $d \ge 3$36 up to homotopy and allowed crossings, and its endomorphism groups are the divisor braid groups $d \ge 3$37. There the colouring data are controlled by a graph $d \ge 3$38, where an edge $d \ge 3$39 means that strands of colours $d \ge 3$40 and $d \ge 3$41 are forbidden to intersect.

A different construction appears in "Invariants of the Colored Braid Groupoid" (Rohozhkin, 18 Jun 2026). There, a braid is treated as a dynamical system of points in the plane, the states are Delaunay triangulations, and an abstract groupoid $d \ge 3$42 is used to represent a coloured braid groupoid $d \ge 3$43. That framework defines homomorphisms

$d \ge 3$44

together with an algorithm for computing the associated rational and orthogonal matrix invariants.

Taken together, these constructions show that the coloured braid groupoid is not a single isolated object but a recurring groupoid-valued formalism in which braid-like dynamics, colour data, and geometric constraints are encoded at the level of objects as well as morphisms. In the simple-branched-cover setting, its distinctive feature is the strict lifting functor to a mapping-class groupoid, which makes the lift of every coloured braid explicit and functorial (Licata et al., 7 Aug 2025).