Example. The groupoid of enriched four-cycle backgrounds [fdip-000X]
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October 18, 2026

A classification of Gaussian backgrounds must retain the maps that transport their observables. Consider the alternating four-cycle with vertices $V=\mathbb Z/4\mathbb Z$, forward edges $e_i:i\to i+1$, and weights $(a,b,a,b)$, where $a,b>0$. Choose $m>0$ and one of $R_0=\{0,2\}$, $R_1=\{1,3\}$ as retained vertices. Write $x=(a,b,m,R_j)$ and let $X$ be the set of all these tuples. The tuple specifies a finite Gaussian background, together with its named half-turn.

More explicitly, retain the carrier, counting reference pairing, weight matrix $W_x=\operatorname {diag}(a,b,a,b)$, and \[L_x=B^TW_xB,\qquad Q_x=L_x+m^2I.\] Here $(B\phi )_i=\phi _{i+1}-\phi _i$. For $\phi \ne 0$, \[\begin {aligned} \phi ^TQ_x\phi &=\sum _iw_i(\phi _{i+1}-\phi _i)^2\\ &\quad +m^2\|\phi \|^2>0. \end {aligned}\] The normalized Gaussian $\mu _{Q_x}$ is determined by this precision and coordinate Lebesgue measure. Also retain the measurable coordinate projection $\pi _{R_j}:\mathbb R^V\to \mathbb R^{R_j}$ and the named action $\rho (h)\delta _i=\delta _{i+2}$ of $C_2=\{1,h\}$. The vertex basis $\delta _i(v)=\mathbf 1_{v=i}$ is distinct from the edges $e_i$ and the spectral vectors $u_a$. These structures are specified or determined, rather than additional freely variable coordinates of $X$.

Allow rotations and reflections of the underlying cycle. Put \[D_4=\langle r,s\mid r^4=s^2=1,\ srs=r^{-1}\rangle ,\] where $r(i)=i+1$ and $s(i)=-i$. A reflection reverses the forward orientation. Thus these arrows use the signed endpoint convention and extend the earlier strictly directed exact relabelings. For $g=r^ks^\varepsilon $, with $k\in \mathbb Z/4$ and $\varepsilon \in \{0,1\}$, set \[\begin {aligned} g(i)&=k+(-1)^\varepsilon i,\\ P_g\delta _i&=\delta _{g(i)},\\ \tau _g(i)&=\begin {cases}k+i&\varepsilon =0,\\k-i-1&\varepsilon =1,\end {cases}\\ U_g\eta _i&=(-1)^\varepsilon \eta _{\tau _g(i)}. \end {aligned}\] Here $\eta _i$ is the edge-field coordinate basis. All indices are modulo $4$. The 8 vertex maps are distinct because their values at $0,1$ determine $k,\varepsilon $.

The edge indexed by $\tau _g(i)$ has endpoints $g(i),g(i+1)$, in forward order for a rotation and reverse order for a reflection. Consequently, for every field $\phi $, \[\begin {aligned} (BP_g\phi )_{\tau _g(i)}&=(-1)^\varepsilon (B\phi )_i,\\ BP_g&=U_gB. \end {aligned}\] If $k'$ denotes a second group element, endpoint transport and multiplication of the orientation signs give \[P_{k'g}=P_{k'}P_g,\qquad U_{k'g}=U_{k'}U_g.\] For an explicit group-law check, composition of $(k,\varepsilon )$ followed by $(\ell ,\eta )$ gives \[\begin {aligned} (\ell ,\eta )(k,\varepsilon ) &=(\ell +(-1)^\eta k,\eta +\varepsilon ). \end {aligned}\] The second coordinate is taken modulo two. Both coordinate maps are orthogonal, and their inverses are indexed by $g^{-1}$.

Transport weights by $W_{gx}=U_gW_xU_g^T$ and the retained set by $R\mapsto gR$. The signs cancel in the diagonal weight matrix. The weights swap exactly when $k+\varepsilon $ is odd, while $R_j\mapsto R_{j+k\bmod 2}$. In particular, \[\begin {aligned} r(a,b,m,R_j)&=(b,a,m,R_{j+1}),\\ s(a,b,m,R_j)&=(b,a,m,R_j). \end {aligned}\] The family is closed under these maps, with mass unchanged. Using $BP_g^T=U_g^TB$ gives \[\begin {aligned} L_{gx}&=B^TU_gW_xU_g^TB=P_gL_xP_g^T,\\ Q_{gx}&=P_gQ_xP_g^T. \end {aligned}\] Moreover $g(i+2)=g(i)+2$, so every arrow commutes with the same named half-turn: $P_g\rho (h)=\rho (h)P_g$.

Form the action groupoid $D_4\ltimes X$ by setting \[\operatorname {Hom}(x,y)=\{g\in D_4:g x=y\}.\] The group element remains part of an arrow. Compose $g:x\to y$ and $k:y\to z$ as $kg:x\to z$; the action law gives its target. The identity is $1:x\to x$, the inverse is $g^{-1}:gx\to x$, and the two inverse identities and associativity follow from the group laws. Thus every arrow is invertible, as in [stacks2017stacks, Definition 4.2.5], Tag 0018. The source, target and composition follow the set-level pattern in [stacks2017stacks, Lemma 78.15.1], Tag 0444; that lemma is stated for algebraic spaces, whereas the axioms here were checked directly for sets. $X$ is infinite, but each orbit and its restricted groupoid is finite.

The stabilizer of the weights at $a\ne b$ is $\{1,r^2,rs,r^3s\}\cong C_2\times C_2$; preserving weights requires $k+\varepsilon $ even. The stabilizer of either retained partition is $\{1,r^2,s,r^2s\}\cong C_2\times C_2$, since preserving the partition requires $k$ even. Intersecting these conditions gives the following automorphism groups. When $a\ne b$, \[\operatorname {Aut}(x)=\{1,r^2\}\cong C_2.\] When $a=b$, \[\begin {aligned} \operatorname {Aut}(x)&=\{1,r^2,s,r^2s\}\\ &\cong C_2\times C_2. \end {aligned}\] At equal weights the first condition disappears. In each four-element subgroup the two indicated generators are distinct commuting involutions. No further condition comes from the named half-turn, which commutes with all 8 elements. The kernel $\{1,r^2\}$ of the parameter action is retained as distinct arrows: $P_{r^2}\delta _0=\delta _2\ne \delta _0$.

For unequal weights the orbit consists of both weight orders and both partitions, at fixed mass. Write $x_j=(a,b,m,R_j)$ and $y_j=(b,a,m,R_j)$. The drawing shows selected generator arrows; it does not draw all arrows or isotropy loops.

$\text{[math]}$

At $a=b$, put $z_j=(a,a,m,R_j)$. There are two objects, each with automorphism group $C_2\times C_2$:

$\text{[math]}$

The generators $s,r$ reach every displayed object; there are exactly 4 objects when $a\ne b$ and two when $a=b$. It follows in both directions that the coarse isomorphism classes are $(m,\{a,b\})$. The coarse orbit set forgets the isotropy just computed, as in Example [fgap-000P]. On the strict slice $(1,t,m,R_0)$, isomorphism requires the same mass and $\{1,t\}=\{1,t'\}$, whose products give $t=t'$. Thus reciprocal parameters are not identified there unless $t=1$. This is a set-level groupoid, within the boundary before stacks.

Mathlib's CategoryTheory.ActionCategory constructs the generic action category; a group action makes it a groupoid. CategoryTheory.ActionCategory.stabilizerIsoEnd identifies the stabilizer monoid with endomorphisms, which are invertible in the group case. MulAction.orbitRel and MulAction.stabilizer provide the generic coarse relation and subgroup. DihedralGroup.card gives 8 elements at $n=4$, and DihedralGroup.center_eq_closure_of_even_ne_two gives the central half-turn. In its constructor convention our $r^ks$ corresponds to $\texttt {sr}(-k)$, while $r^k$ corresponds to $\texttt {r}(k)$. These declarations support the group constructions; the particular parameter action, signed incidence and stabilizer intersections are the calculations above.