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Flat-connection groupoids satisfy boundary descent

Statement

Let $M_{1}$ and $M_{2}$ be compact triangulated $d$ -bordisms glued along an identified closed triangulated boundary $Σ$ , with collars and compatible triangulations. The finite flat-connection groupoid on the glued manifold is equivalent to the homotopy pullback groupoid

$C_{G} (M_{2} \cup_{Σ} M_{1}) ≃ C_{G} (M_{2}) \times_{C_{G} (Σ)}^{h} C_{G} (M_{1}) .$

Writing $r_{i} : C_{G} (M_{i}) \to C_{G} (Σ)$ for restriction, a pullback object is $(A_{2}, A_{1}, α : r_{2} (A_{2}) ≅ r_{1} (A_{1}))$ . A morphism $(u_{2}, u_{1})$ to $(A_{2}^{'}, A_{1}^{'}, α^{'})$ satisfies $r_{1} (u_{1}) \circ α = α^{'} \circ r_{2} (u_{2})$ .

Statement references

Finite flat connection groupoid on a cell complex

Finite gauge state spaces and bordism spans

Homotopy pullback of finite groupoids

Facts & Assumptions

Given: The compatible triangulated bordisms $M_{1}, M_{2}$ , their common boundary $Σ$ , and the finite flat-connection groupoids from Finite flat connection groupoid on a cell complex.

[F1]

Finite flat connection groupoid on a cell complex: objects are flat edge labels and arrows are vertex gauge transformations; gauge transformations preserve flatness.

[F2]

Finite gauge state spaces and bordism spans: boundary restriction is a functor of finite flat-connection groupoids.

[F3]

Homotopy pullback of finite groupoids: a homotopy-pullback object includes an isomorphism $α$ between the two images, and its arrows satisfy $q (v) \circ α = α^{'} \circ p (u)$ .

Proof

1.1givenF1F2F3construct

A global flat labeling on $M_{2} \cup_{Σ} M_{1}$ restricts to flat labelings on both pieces whose boundary labels agree. The identity boundary isomorphism makes this a triple in the homotopy pullback defined by [F3]. Restriction of a global vertex gauge transformation gives a compatible pair, so restriction defines a functor from the global groupoid.

2.1givenF1F3step 1.1construct

Conversely, let $(A_{2}, A_{1}, α)$ be a homotopy-pullback object as in [F3], where $α : r_{2} (A_{2}) \to r_{1} (A_{1})$ is a boundary gauge isomorphism. Its finite vertex values on the boundary extend to the vertices of $M_{2}$ by setting every non-boundary value to the identity. Applying this gauge transformation to $A_{2}$ makes its boundary edge labels equal to those of $A_{1}$ . The two edge labelings then glue to a unique labeling on the union; every 2-cell lies in one of the pieces or their common boundary, so each flatness equation remains satisfied. This proves essential surjectivity.

3.1F1F2F3step 2.1construct

By [F3], a morphism in the homotopy pullback is a pair of gauge transformations on $M_{2}$ and $M_{1}$ satisfying the boundary compatibility equation. After the boundary representatives are aligned as in step 2.1, this equation says exactly that the two vertex gauge values agree at every identified boundary vertex. They therefore glue to a unique gauge transformation on $M_{2} \cup_{Σ} M_{1}$ . Conversely, every global gauge transformation restricts to such a compatible pair. Thus the restriction functor is fully faithful and, with step 2.1, an equivalence.

4.1givenF1F2F3algebra∎

The boundary isomorphism $α$ is load-bearing. For the one-object groupoid $B G$ of a finite group $G$ , an object of the homotopy pullback of two identity functors is an element $α \in G$ . By [F3], a morphism $(u, v) : α \to α^{'}$ satisfies $v α = α^{'} u$ , so every $α$ is isomorphic to $1$ and the automorphisms of $1$ are the diagonal pairs $(u, u)$ . Thus the homotopy pullback has one isomorphism class with automorphism group isomorphic to $G$ and cardinality $1 / ∣ G ∣$ . The ordinary set-theoretic fiber product has one object with trivial automorphism group and cardinality $1$ . Omitting $α$ therefore discards the stabilizer factor required by the gluing measure.

Gluing diagram

Write $r_{i} : C_{G} (M_{i}) \to C_{G} (Σ)$ for restriction to the common boundary and ${r e s}_{i}$ for restriction from the glued manifold to $M_{i}$ . The restriction square below commutes with the identity boundary isomorphism. It is a homotopy pullback: the comparison functor to $C_{G} (M_{2}) \times_{C_{G} (Σ)}^{h} C_{G} (M_{1})$ , sending a global field to its two restrictions and their identity boundary isomorphism, is the equivalence proved by descent. The proof is independent of the drawing.

\mathcal C_G(M_2\cup_\Sigma M_1) \arrow[r,"\mathrm{res}_1"] \arrow[d,"\mathrm{res}_2"'] &
\mathcal C_G(M_1) \arrow[d,"r_1"] \\
\mathcal C_G(M_2) \arrow[r,"r_2"'] & \mathcal C_G(\Sigma)

Flat-connection groupoids satisfy boundary descent

Statement

Statement references

Facts & Assumptions

Proof

Gluing diagram

Depends on

Dependency tree · 2 levels

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