seam groupoid

Floor 2, Les sutures · introduced in Chapter 1, Un objet, plusieurs noms

How are all the seams of a family recorded at once?

The groupoid whose vertices are a family of incarnations and whose arrows are their seams; consistency means it is the pair groupoid.

Plate 2.2Two seam groupoids. Left, the object of size 28: one seam between any two incarnations, and every triangle commutes. Right, the object of size 24: three seams between any two, and a vertex group of order 3.

Definition(Seam groupoid)

Let $(Y_i)_{i\in I}$ be a family of incarnations of one object. Its seam groupoid has vertex set $I$ , and the arrows from $i$ to $j$ are the seams $Y_i\to Y_j$ , composed as maps.

The groupoid language is used for one question: whether the seams of a family are consistent, so that passing from one incarnation to another along different routes gives the same map. In groupoid terms the question is whether the seam groupoid is the pair groupoid of $I$ , the groupoid with exactly one arrow from each vertex to each vertex.

Corollary(Coherence)

Let the object have stabilizer class $[H]$ . If it is rigid, the seam groupoid is the pair groupoid: writing $s_{ij}$ for the unique seam $Y_i\to Y_j$ , $s_{ii}=\id$ and $s_{jk}\circ s_{ij}=s_{ik}$ . If it is not rigid, every vertex group is isomorphic to $N_G(H)/H\neq1$ , and there are $|N_G(H):H|$ seams from each $Y_i$ to each $Y_j$ .

Proof

In the rigid case $s_{jk}\circ s_{ij}$ and $s_{ik}$ are both seams $Y_i\to Y_k$ , and there is only one. Otherwise the vertex group at $i$ is $\Aut_G(Y_i)\cong\Aut_G(X)\cong N_G(H)/H$ , and the seams between two vertices form a torsor for it.

Example

For the five incarnations of the object of size 28 the seam groupoid is the pair groupoid on five vertices: there is one seam for each ordered pair, 25 in all, and all 125 composites agree, as was checked by computation. For the object of size 24 each vertex group is cyclic of order 3, and there are three arrows between any two vertices.

Built from: seam
Builds: coherence
Objects: the twenty-eightthe object of size 24
In the Esquisse: 4La monodromie des sutures