TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Finite cocycle-weighted groupoid gauging and gluing

Statement

Let $K = K_{1} \cup_{Σ} K_{2}$ be a finite ordered simplicial complex written as a union of two finite subcomplexes with common subcomplex $Σ$ . Let $F$ , $F_{1}$ , $F_{2}$ , and $S$ be finite background field groupoids for $K$ , $K_{1}$ , $K_{2}$ , and $Σ$ , with restriction functors $r_{i} : F_{i} \to S$ . The homotopy pullback $H = F_{2} \times_{S}^{h} F_{1}$ has objects $(x_{2}, x_{1}, α)$ with $α : r_{2} (x_{2}) \overset{≅}{⟶} r_{1} (x_{1})$ and morphisms $(u_{2}, u_{1})$ satisfying $r_{1} (u_{1}) \circ α = α^{'} \circ r_{2} (u_{2})$ .

Assume that restriction induces an equivalence $J : F \to H$ . Let $W, W_{1}, W_{2}$ be $U (1)$ -valued cocycle/action weights on these groupoids, and assume that under $J$ the weight factors as $W (x_{2}, x_{1}, α) = W_{2} (x_{2}) W_{1} (x_{1})$ . For each object $s$ of $S$ , define the homotopy fiber $(F_{i})_{s}$ to have objects $(x_{i}, η_{i} : r_{i} (x_{i}) \overset{≅}{⟶} s)$ and morphisms $u_{i}$ satisfying $η_{i}^{'} \circ r_{i} (u_{i}) = η_{i}$ , and put

$Z_{i} (s) : = \sum_{[(x_{i}, η_{i})] \in π_{0} ((F_{i})_{s})} \frac{W_{i} (x_{i})}{∣ {Aut}_{(F_{i})_{s}} (x_{i}, η_{i}) ∣} .$

Then the normalized gauging sums are finite, are invariant under replacement of a field by a gauge-isomorphic field, and satisfy

$Z_{K} (W) = \sum_{[s] \in π_{0} (S)} \frac{Z_{2} (s) Z_{1} (s)}{∣ {Aut}_{S} (s) ∣} .$

This is a finite groupoid theorem. A finite cocycle action is covered only when its evaluated weights obey the stated arrow-invariance and gluing-factorization hypotheses.

Statement references

Finite background groupoid and cocycle-weighted gauging data

Finite groupoid cardinality and homotopy-fiber integration

Homotopy pullback of finite groupoids

Facts & Assumptions

Given: The finite groupoids and restriction functors in the Statement, invariant phase weights $W, W_{1}, W_{2}$ , an equivalence $J : F \to H$ , and the stated factorization of weights under $J$ .

[F1]

Finite background groupoid and cocycle-weighted gauging data: a finite background field groupoid retains its automorphism groups; a cocycle/action weight is constant on every arrow; its gauging sum is the sum over isomorphism classes weighted by $1 / ∣ Aut ∣$ .

[F2]

Finite groupoid cardinality and homotopy-fiber integration: for a finite groupoid, weighted integration sums over isomorphism classes with reciprocal automorphism weights; a homotopy fiber over $s$ consists of pairs $(x, η : p (x) ≅ s)$ with arrows commuting with $η$ .

[F3]

Homotopy pullback of finite groupoids: a homotopy-pullback object consists of two source objects and an isomorphism between their images; its morphisms are compatible pairs of source arrows.

Proof

1.1givenF1F2algebra

Each $W_{i}$ is constant on gauge arrows by [F1]. It therefore descends to isomorphism classes, and the finite groupoid integrals defining $Z_{K} (W)$ and $Z_{i} (s)$ are finite by [F2].

1.2givenF1F2construct

An equivalence of groupoids induces a bijection on isomorphism classes and an isomorphism on each object's automorphism group. Since $J$ also preserves the weight by the factorization hypothesis, the normalized sum on $F$ equals the weighted groupoid cardinality of $H$ .

1.3givenF3construct

Fix representatives $x_{2} \in F_{2}$ and $x_{1} \in F_{1}$ . The finite set of possible third entries of homotopy-pullback objects represented by this pair is ${Iso}_{S} (r_{2} (x_{2}), r_{1} (x_{1}))$ . The group $Aut (x_{2}) \times Aut (x_{1})$ acts on it by $(u_{2}, u_{1}) \cdot α = r_{1} (u_{1}) \circ α \circ r_{2} (u_{2})^{- 1}$ . Its orbits are exactly the isomorphism classes of triples with this pair of source classes, and the stabilizer of $α$ is the automorphism group of $(x_{2}, x_{1}, α)$ by the compatibility equation in [F3].

1.4givenF2algebra

For a fixed boundary representative $s$ , the automorphism group of $(x_{i}, η_{i})$ in $(F_{i})_{s}$ is the subgroup of ${Aut}_{F_{i}} (x_{i})$ preserving $η_{i}$ . The action of ${Aut}_{F_{i}} (x_{i})$ on the finite set ${Iso}_{S} (r_{i} (x_{i}), s)$ has these stabilizers. Orbit-stabilizer therefore gives $Z_{i} (s) = \sum_{[x_{i}] \in π_{0} (F_{i})} W_{i} (x_{i}) ∣ {Iso}_{S} (r_{i} (x_{i}), s) ∣ / ∣ {Aut}_{F_{i}} (x_{i}) ∣$ .

2.1F1step 1.3algebra

By orbit-stabilizer, summing the reciprocal stabilizer orders over the orbits in step 1.3 gives $\sum_{[α]} 1 / ∣ {Aut}_{H} (x_{2}, x_{1}, α) ∣ = ∣ {Iso}_{S} (r_{2} (x_{2}), r_{1} (x_{1})) ∣ / (∣ {Aut}_{F_{2}} (x_{2}) ∣ ∣ {Aut}_{F_{1}} (x_{1}) ∣)$ . Multiplying by $W_{2} (x_{2}) W_{1} (x_{1})$ and summing over $[x_{2}]$ and $[x_{1}]$ gives the weighted sum on $H$ .

3.1givenF2step 1.4step 2.1algebra

Expanding $\sum_{[s]} Z_{2} (s) Z_{1} (s) / ∣ {Aut}_{S} (s) ∣$ by step 1.4 reduces its coefficient for fixed $[x_{2}], [x_{1}]$ to $\sum_{[s] \in π_{0} (S)} ∣ Iso (r_{2} (x_{2}), s) ∣ ∣ Iso (r_{1} (x_{1}), s) ∣ / ∣ {Aut}_{S} (s) ∣$ . If the two boundary objects are not isomorphic, every summand is zero. If they are isomorphic, there is exactly one contributing isomorphism class $[s]$ ; each isomorphism set in the numerator is a torsor for ${Aut}_{S} (s)$ , so the displayed sum equals $∣ {Aut}_{S} (s) ∣$ , which is also the cardinality of ${Iso}_{S} (r_{2} (x_{2}), r_{1} (x_{1}))$ . This is the coefficient in step 2.1.

4.1givenF1step 1.1step 1.2step 2.1step 3.1∎

Steps 1.2, 2.1, and 3.1 identify the normalized sum on $F$ with the stated boundary sum. Step 1.1 proves gauge invariance of every summand and finiteness of all sums.

Gluing diagram

The comparison functor $J$ sends a global background to its two restrictions and their boundary isomorphism. The displayed square is homotopy cartesian by the equivalence hypothesis in the Statement. The proof above uses the explicit homotopy-pullback objects and their automorphism groups, not the drawing.

Remarks

The theorem proves a finite weighted gluing law under an explicit groupoid-equivalence and cocycle-factorization hypothesis. It does not establish that a continuum gauge theory exists or that a proposed higher-group cochain model realizes a physical symmetry. The full finite two-group specialization retains vertex gauge-of-gauge transformations and uses the separate higher gauging theorem; the present ordinary-groupoid theorem remains valid with exactly its stated hypotheses.