Finite-group BF state sums, gauge groupoids and gluing

The groupoid cardinality and integral in the Definition are finite sums by [A1]. Fix $d\in \mathcal{D}$ and one representative $c$ from each source isomorphism class. The set of isomorphisms $p(c)\cong d$ is empty or a torsor for the finite group ${\mathrm{Aut}}_{\mathcal{D}}(d)$. Every object of the homotopy fiber is isomorphic to one with such a chosen source representative, so the fiber has finitely many isomorphism classes. The automorphisms of $(c,\alpha )$ are the $u\in {\mathrm{Aut}}_{\mathcal{C}}(c)$ satisfying $\alpha \circ p(u)=\alpha$, a subgroup of a finite group by [A1]. Thus every homotopy fiber is finite and each pushforward sum is finite.

If $\eta :d\to d^{\mathrm{\prime }}$ is an isomorphism in $\mathcal{D}$, the rule $(c,\alpha )\mapsto (c,\eta \circ \alpha )$ is an equivalence ${\mathcal{C}}_d\to {\mathcal{C}}_{d^{\mathrm{\prime }}}$ with inverse given by ${\eta }^{-1}$. It preserves isomorphism classes, automorphism groups, and the values of $f([c])$, so the fiber integral is independent of the representative of $[d]$.

If $c\to c^{\mathrm{\prime }}$ is an isomorphism in $\mathcal{C}$, then $p(c)\to p(c^{\mathrm{\prime }})$ is an isomorphism in $\mathcal{D}$, so $p^{\ast }g$ is invariant; step 1.2 proves the same for $p_{!}f$. Consequently $b_{!}{a}^{\ast }$ is a well-defined linear map, since its sums are finite and both operations preserve linear combinations.

Identity pairs satisfy the compatibility equation. If $(u,v)$ and $(u^{\mathrm{\prime }},v^{\mathrm{\prime }})$ are composable compatible pairs, then $q(v^{\mathrm{\prime }}\circ v)\circ \alpha =q(v^{\mathrm{\prime }})\circ q(v)\circ \alpha =q(v^{\mathrm{\prime }})\circ {\alpha }^{\mathrm{\prime }}\circ p(u)={\alpha }^{\mathrm{\prime }\mathrm{\prime }}\circ p(u^{\mathrm{\prime }}\circ u)$, so their composite is compatible. Inverting the compatibility equation shows $(u^{-1},v^{-1})$ is compatible with the reversed objects. Thus the stated data form a groupoid.

Choose one representative from each of the finitely many isomorphism classes of $\mathcal{C}$ and $\mathcal{E}$. If $u:c\to c_0$ and $v:e\to e_0$ are isomorphisms to representatives, then $(c,e,\alpha )$ is isomorphic to $(c_0,e_0,q(v)\circ \alpha \circ p(u{)}^{-1})$. For each representative pair $(c_0,e_0)$, the possible isomorphisms $p(c_0)\cong q(e_0)$ are either absent or form a set on which postcomposition by ${\mathrm{Aut}}_{\mathcal{D}}(q(e_0))$ acts freely and transitively; that set has finite size by [F1]. Thus there are finitely many isomorphism classes of triples. The automorphism group of a triple is the subgroup of ${\mathrm{Aut}}_{\mathcal{C}}(c_0)\times {\mathrm{Aut}}_{\mathcal{E}}(e_0)$ satisfying the compatibility equation, and is finite by [F1]. Hence the homotopy pullback is a finite groupoid.

Reversing an edge replaces $g_e$ by $g_e^{-1}$, and the gauge formula sends the reverse label to $(h_{t(e)}{g}_e{h}_{s(e)}^{-1}{)}^{-1}=h_{s(e)}{g}_e^{-1}{h}_{t(e)}^{-1}$. For a closed attaching word based at a vertex $v$, the transformed holonomy is $h_v\mathrm{Hol}(w)h_v^{-1}$. Thus the inverse-edge rule and the flatness equations are preserved by every vertex gauge transformation.

For $h,k\in G^{K_0}$, applying $k$ and then $h$ sends an edge label to $h_{t(e)}{k}_{t(e)}{g}_e{k}_{s(e)}^{-1}{h}_{s(e)}^{-1}=(hk{)}_{t(e)}{g}_e(hk{)}_{s(e)}^{-1}$. The identity vertex labeling acts as the identity, so this is a group action and its action groupoid has the composition and inverses stated in the Definition.

There are at most $\mathrm{\mid }G{\mathrm{\mid }}^{\mathrm{\mid }E(K)\mathrm{\mid }}$ edge assignments and exactly $\mathrm{\mid }G{\mathrm{\mid }}^{\mathrm{\mid }{K}_0\mathrm{\mid }}$ gauge transformations, so the action groupoid is finite. Products over the finitely many connected components are finite groupoids as well; the empty product is terminal. By [F1], the connected groupoid cardinality is the orbit sum of reciprocal stabilizer sizes.

For an orbit represented by $g$, the map $G^{K_0}\mathrm{/}\mathrm{Stab}(g)\to G^{K_0}\cdot g$, $h\mathrm{Stab}(g)\mapsto h\cdot g$, is a bijection: equality of images is equivalent to $h_2^{-1}{h}_1\in \mathrm{Stab}(g)$. Hence $\mathrm{\mid }{G}^{K_0}\cdot g\mathrm{\mid }\mathrm{/}\mathrm{\mid }{G}^{K_0}\mathrm{\mid }=1\mathrm{/}\mathrm{\mid }\mathrm{Stab}(g)\mathrm{\mid }$. Summing over the disjoint orbits gives the final ratio in the Definition.

For fixed $z\in \mathcal{Z}$, an object of the iterated fiber is $(y,\beta :q(y)\to z,x,\alpha :p(x)\to y)$. The functor $(y,\beta ,x,\alpha )\mapsto (x,\beta \circ q(\alpha ))$ maps it to $(\mathcal{X}{)}_z$. A quasi-inverse sends $(x,\gamma :q(p(x))\to z)$ to $(p(x),\gamma ,x,{\mathrm{id}}_{p(x)})$. The composite on $(\mathcal{X}{)}_z$ is the identity. For any iterated object, the isomorphism $\alpha :p(x)\to y$ and ${\mathrm{id}}_x$ give a natural isomorphism from the quasi-inverse image back to that object. Hence the functor is an equivalence.

Fix an isomorphism class $[e]$ of $\mathcal{E}$, set $y=b(e)$, and write $A={\mathrm{Aut}}_{\mathcal{Y}}(y)$, $B={\mathrm{Aut}}_{\mathcal{E}}(e)$, $I=\mathrm{im}(B\to A)$, and $N=\ker (B\to A)$. The classes in the homotopy fiber ${\mathcal{E}}_y$ represented by this $[e]$ are the orbits of $I$ on the $A$-torsor of isomorphisms $b(e)\cong y$, so there are $\mathrm{\mid }A\mathrm{\mid }\mathrm{/}\mathrm{\mid }I\mathrm{\mid }$ of them; each has automorphism group $N$. Their contribution after the outer factor $1\mathrm{/}\mathrm{\mid }A\mathrm{\mid }$ is $f([e])\mathrm{/}(\mathrm{\mid }I\mathrm{\mid }\mathrm{\mid }N\mathrm{\mid })=f([e])\mathrm{/}\mathrm{\mid }B\mathrm{\mid }$, by the finite homomorphism counting identity $\mathrm{\mid }B\mathrm{\mid }=\mathrm{\mid }I\mathrm{\mid }\mathrm{\mid }N\mathrm{\mid }$. Summing over $[e]$ proves ${\int }_{y\in \mathcal{Y}}{b}_{!}f(y)={\int }_{e\in \mathcal{E}}f(e)$.

Compose spans $\mathcal{X}\stackrel{a}{\leftarrow }\mathcal{E}\stackrel{b}{\to }\mathcal{Y}$ and $\mathcal{Y}\stackrel{c}{\leftarrow }\mathcal{F}\stackrel{d}{\to }\mathcal{Z}$, and let $P=\mathcal{E}{\times }_{\mathcal{Y}}^h\mathcal{F}$ with projections ${\pi }_E,{\pi }_F$. For fixed $z\in \mathcal{F}$, the homotopy fiber of ${\pi }_F$ over $z$ has objects $((e,f^{\mathrm{\prime }},\alpha ),\tau :f^{\mathrm{\prime }}\cong z)$. The functor to ${\mathcal{E}}_{c(z)}$ sends this object to $(e,c(\tau )\circ \alpha )$ and sends an arrow $(u,v)$ to $u$. Such an arrow satisfies ${\tau }^{\mathrm{\prime }}\circ v=\tau$ and $c(v)\circ \alpha ={\alpha }^{\mathrm{\prime }}\circ b(u)$, so its image satisfies ${\beta }^{\mathrm{\prime }}\circ b(u)=\beta$. Conversely, for any such $u$ between images $(e,\beta )$ and $(e^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }})$, the only possible lift has $v=({\tau }^{\mathrm{\prime }}{)}^{-1}\circ \tau :f^{\mathrm{\prime }}\to f^{\mathrm{\prime }\mathrm{\prime }}$; the equation ${\beta }^{\mathrm{\prime }}\circ b(u)=\beta$ implies $c(v)\circ \alpha ={\alpha }^{\mathrm{\prime }}\circ b(u)$, so this lift exists and is unique. The functor is fully faithful; every object $(e,\beta )$ is its image from $((e,z,\beta ),{\mathrm{id}}_z)$, so it is essentially surjective. It is therefore an equivalence. By [F1], equivalent fibers have equal finite integrals, giving $c^{\ast }{b}_{!}g=({\pi }_F{)}_{!}{\pi }_E^{\ast }g$ for every invariant function $g$ on $\mathcal{E}$. Applying step 1.1 to ${\pi }_F$ and $d$ now gives $d_{!}{c}^{\ast }{b}_{!}{a}^{\ast }f=(d\circ {\pi }_F{)}_{!}(a\circ {\pi }_E{)}^{\ast }f$, which is the push-pull map of the homotopy-pullback composite span.

Put $\varphi ([e]):=\overline{f([a(e)])}g([b(e)])$, which is invariant on ${\pi }_0(\mathcal{E})$. Expanding the left inner product by the definition of $b_{!}$ gives $⟨b_{!}{a}^{\ast }f,g{⟩}_{\mathcal{Y}}={\sum }_{[y]}\frac{1}{\mathrm{\mid }{\mathrm{Aut}}_{\mathcal{Y}}(y)\mathrm{\mid }}{\sum }_{[(e,\eta )]\in {\pi }_0({\mathcal{E}}_y)}\frac{\overline{f([a(e)])}g([y])}{\mathrm{\mid }{\mathrm{Aut}}_{{\mathcal{E}}_y}(e,\eta )\mathrm{\mid }}$. Since $\eta :b(e)\cong y$ and $g$ is invariant, $g([y])=g([b(e)])$, so this is ${\int }_{y\in \mathcal{Y}}{b}_{!}\varphi (y)={\int }_{e\in \mathcal{E}}\varphi (e)={\sum }_{[e]}\frac{\varphi ([e])}{\mathrm{\mid }{\mathrm{Aut}}_{\mathcal{E}}(e)\mathrm{\mid }}$ by step 1.2. On the other side, expanding $a_{!}{b}^{\ast }g$ and using $\xi :a(e)\cong x$ to replace $\overline{f([x])}$ by $\overline{f([a(e)])}$ gives $⟨f,a_{!}{b}^{\ast }g{⟩}_{\mathcal{X}}={\sum }_{[x]}\frac{1}{\mathrm{\mid }{\mathrm{Aut}}_{\mathcal{X}}(x)\mathrm{\mid }}{\sum }_{[(e,\xi )]\in {\pi }_0({\mathcal{E}}_x)}\frac{\varphi ([e])}{\mathrm{\mid }{\mathrm{Aut}}_{{\mathcal{E}}_x}(e,\xi )\mathrm{\mid }}={\int }_{x\in \mathcal{X}}{a}_{!}\varphi (x)={\int }_{e\in \mathcal{E}}\varphi (e)$, again by step 1.2. Both inner products therefore equal the same finite weighted sum, proving the adjoint identity.

Step 2.2 proves that the adjoint of $b_{!}{a}^{\ast }$ is $a_{!}{b}^{\ast }$, and steps 1.1 and 2.1 prove composition through homotopy-pullback spans.

For each connected component $K_j$, start with any flat assignment and set the root gauge value at $v_j$ to the identity. Along the unique tree paths in $T_j$ from $v_j$, choose the gauge value at each new vertex so that the transformed label on the traversed tree edge is the identity; when that edge is traversed against its chosen orientation, use its inverse label. This recursively produces a tree-normalized assignment on every component. It is unique among gauges whose value at each $v_j$ is the identity, since the condition that each transformed tree label remain $1$ determines each next vertex value.

In a tree-normalized assignment on $K_j$, every edge of $T_j$ has label $1$. Assign to each generator $x_e$ of ${\pi }_1(K_j,v_j)$ the inverse of the label on its oriented non-tree edge. By [F4], this is a homomorphism for the traversal-first product on loops: holonomy of a concatenation is the reverse-ordered product, and reversing the path in the Statement reverses that order. Flatness says exactly that every relator $r_f$ maps to $1$. By the presentation in [F2], the normalized assignments on $K_j$ are in bijection with homomorphisms $\rho :{\pi }_1(K_j,v_j)\to G$.

For a connected subcomplex $L_i\subseteq K_{j(i)}$ and base data as in the Statement, the pointed inclusion at $v_i$ induces a well-defined homomorphism on fundamental groups by [F5]. The base-path change map $c_{{\gamma }_i}^{K_{j(i)}}:{\pi }_1(K_{j(i)},v_i)\to {\pi }_1(K_{j(i)},w_{j(i)})$, given by $[\alpha ]\mapsto [{\gamma }_i\ast \alpha \ast {\bar{\gamma }}_i]$, is a well-defined isomorphism: endpoint-fixed homotopies remain endpoint-fixed after adjoining ${\gamma }_i,{\bar{\gamma }}_i$, insertion of the retraced path ${\bar{\gamma }}_i\ast {\gamma }_i$ contracts in a product of loop classes, and $c_{{\bar{\gamma }}_i}^{K_{j(i)}}$ is its inverse. Thus ${\mathrm{ȷ}}_{{\gamma }_i,\ast }=c_{{\gamma }_i}^{K_{j(i)}}\circ {\mathrm{ȷ}}_{\ast }$ is well defined. For a flat assignment $A$ and an edge loop $\mathrm{\ell }$ in $L_i$, the edge-product convention gives ${\mathrm{\Phi }}_{K_{j(i)}}(A)({\mathrm{ȷ}}_{{\gamma }_i,\ast }[\mathrm{\ell }])=H_A({\bar{\gamma }}_i)H_A(\bar{\mathrm{\ell }})H_A({\gamma }_i)=H_A({\bar{\gamma }}_i){\mathrm{\Phi }}_{L_i}(A{\mathrm{\mid }}_{L_i})([\mathrm{\ell }])H_A({\bar{\gamma }}_i{)}^{-1}$. If a gauge arrow $h:A\to A^{\mathrm{\prime }}$ is applied, its path-product rule gives $H_{A^{\mathrm{\prime }}}({\bar{\gamma }}_i)=h(w_{j(i)})H_A({\bar{\gamma }}_i)h(v_i{)}^{-1}$; this is exactly the naturality condition for the stated conjugation arrows. Hence inverse path holonomy intertwines restriction with the componentwise precomposition functor up to that natural isomorphism.

On a fixed component $K_j$, a gauge transformation taking one tree-normalized assignment to another must have equal values at the two endpoints of every edge of $T_j$, because both edge labels are $1$. Since $T_j$ is connected, it is constant on all vertices of $K_j$. Conversely, each constant gauge value $a\in G$ preserves tree normalization and conjugates every generator label by $a$. Thus the full subgroupoid on tree-normalized objects over $K_j$ is identified with $\mathrm{Hom}({\pi }_1(K_j,v_j),G)\mathrm{/}\mathrm{/}G$. Step 1.1 makes its inclusion into the full flat-connection groupoid essentially surjective, and the inclusion is fully faithful because it is full. The root-one gauge used in step 1.1 leaves inverse path holonomy at $v_j$ unchanged, so the object map ${\mathrm{\Phi }}_{K_j}$ in the Statement agrees with normalization followed by this identification. For any gauge arrow $h:A\to A^{\mathrm{\prime }}$, the gauge factors telescope along each based edge loop $\delta$ at $v_j$ to give $H_{A^{\mathrm{\prime }}}(\bar{\delta })=h(v_j)H_A(\bar{\delta })h(v_j{)}^{-1}$, so ${\mathrm{\Phi }}_{K_j}(A^{\mathrm{\prime }})=h(v_j){\mathrm{\Phi }}_{K_j}(A)h(v_j{)}^{-1}$. Conversely, after normalizing $A$ and $A^{\mathrm{\prime }}$ by root-one gauges, each conjugating arrow between their representations is the unique constant gauge arrow between the normalized assignments; conjugating back gives the unique gauge arrow between $A$ and $A^{\mathrm{\prime }}$ with that value at $v_j$. Hence the stated functor is fully faithful as well as essentially surjective, and is an equivalence.

For each connected component $K_j$, the subgroup of vertex gauges equal to $1$ at $v_j$ acts freely and transitively on each fiber of the map from flat assignments to their tree-normalized representative, by step 1.1 and its uniqueness. Its order is $\mathrm{\mid }G{\mathrm{\mid }}^{\mathrm{\mid }(K_j{)}_0\mathrm{\mid }-1}$, so $\mathrm{\mid }\mathrm{Flat}(K_j,G)\mathrm{\mid }=\mathrm{\mid }G{\mathrm{\mid }}^{\mathrm{\mid }(K_j{)}_0\mathrm{\mid }-1}\mathrm{\mid }\mathrm{Hom}({\pi }_1(K_j,v_j),G)\mathrm{\mid }$. Substituting this count into [F1] gives the connected normalization. Equivalently, the action-groupoid cardinality in step 4.1 sums $1\mathrm{/}\mathrm{\mid }\mathrm{Stab}(\rho )\mathrm{\mid }$ over conjugacy classes; no freeness of the conjugation action is assumed.

For $K=K_1\bigsqcup \cdots \bigsqcup K_m$, the defining edge-label and vertex-gauge data split componentwise, so ${\mathcal{C}}_G(K)={\prod }_j{\mathcal{C}}_G(K_j)$. Applying step 4.1 on each component gives the product equivalence ${\mathrm{\Phi }}_K:{\mathcal{C}}_G(K)\simeq {\mathcal{R}}_G(K)$, and applying step 3.1 componentwise gives the stated restriction naturality. Automorphism orders multiply in a product groupoid and its isomorphism classes are componentwise tuples, so its groupoid cardinality is the product of the connected cardinalities from step 5.1. When $m=0$, the empty product is terminal with cardinality $1$.

Each boundary and bulk triangulation has finitely many cells, and $G$ is finite. Thus the field sets and gauge groups in [F1] are finite, the boundary state spaces are finite-dimensional complex vector spaces, and the sums defining their pairings and amplitudes are finite. The displayed inner product is positive definite because every coefficient $1\mathrm{/}\mathrm{\mid }\mathrm{Aut}(A)\mathrm{\mid }$ is a positive real number.

Restricting the labels and vertex gauge transformations to a boundary transports each attaching word and its flatness equation, so the resulting edge assignment is still flat. If a boundary identification reverses a chosen edge direction, the transported label is inverted, and $(h_{t(e)}{g}_e{h}_{s(e)}^{-1}{)}^{-1}=h_{s(e)}{g}_e^{-1}{h}_{t(e)}^{-1}$ is exactly the gauge action on that reversed edge. Thus the reindexed restrictions $r_0,r_1$ are functors. Pullback and homotopy-fiber pushforward therefore define $Z_M$ by [F2]; both are linear because they are substitution and finite weighted summation.

If ${\mathrm{\Sigma }}_0={\mathrm{\Sigma }}_1=\mathrm{\varnothing }$, the restriction span has the terminal groupoid at each end. Its induced map sends $1$ to ${\int }_{A\in {\mathcal{C}}_G(M)}1=\mathrm{\mid }{\mathcal{C}}_G(M)\mathrm{\mid }$ by [F2], which is the stated closed amplitude.

A global flat labeling on $M_2{\cup }_{\mathrm{\Sigma }}{M}_1$ 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.

Conversely, let $(A_2,A_1,\alpha )$ be a homotopy-pullback object as in [F3], where $\alpha :r_2(A_2)\to r_1(A_1)$ 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.

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 $M_2{\cup }_{\mathrm{\Sigma }}{M}_1$. 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.

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

By [F4], with the homotopy-pullback groupoid explicitly given by [F5], the groupoid on the glued bordism is equivalent to the homotopy-pullback composite of the restriction spans for $M_1$ and $M_2$.

Write $\mathrm{\Sigma }={⨆}_{j=1}^m{\mathrm{\Sigma }}_j$ and let $C_j={\mathrm{\Sigma }}_j\times [0,1]$. For each $j$ and $ϵ\in \{0,1\}$, choose $v_j\in {\mathrm{\Sigma }}_j$ and base the end inclusion $i_{ϵ}:{\mathrm{\Sigma }}_j\to C_j$ at $(v_j,ϵ)$. The projection $p_j:C_j\to {\mathrm{\Sigma }}_j$ satisfies $p_j{i}_{ϵ}=\mathrm{id}$, and $i_{ϵ}{p}_j$ is based-homotopic to the identity through $H_s^{(0)}(x,t)=(x,(1-s)t)$ for $ϵ=0$ and $H_s^{(1)}(x,t)=(x,(1-s)t+s)$ for $ϵ=1$. By [F6], these maps induce inverse isomorphisms on fundamental groups. Precomposition with a group isomorphism is an equivalence of representation action groupoids, with inverse precomposition by its inverse and the identity map on conjugating arrows. The componentwise representation equivalence and restriction naturality in [F3] therefore make both end restriction functors ${\mathcal{C}}_G(C_j)\to {\mathcal{C}}_G({\mathrm{\Sigma }}_j)$ equivalences; taking products gives the same conclusion for disconnected $\mathrm{\Sigma }$. Choose an edge path in each connected cylinder 1-skeleton joining its two end base vertices. The base-path naturality in [F3] gives a natural isomorphism between the two restriction functors. Hence the cylinder span is equivalent to the identity span. Each homotopy fiber of either end equivalence has one isomorphism class and trivial automorphism group, while the natural isomorphism identifies the two boundary values; its weighted push-pull map is therefore the identity on $V_G(\mathrm{\Sigma })$.

For a disjoint union, a flat assignment and a vertex gauge transformation are exactly independent assignments on its components. Hence the bulk and boundary groupoids are products. Since the component sets are finite, the indicator functions of pairs of isomorphism classes form the tensor-product basis of the functions on their product. The automorphism weights multiply and the finite fiber sums factor, so the state spaces and maps identify with tensor products.

By the corrected orientation convention in [F1], reversing the manifold orientation leaves each fixed directed edge label and its gauge action unchanged; it only exchanges the incoming and outgoing restriction functors. Thus the span of $\bar{M}$ is the transposed span with the same underlying finite groupoids, so [F2] gives $Z_{\bar{M}}=Z_M^{†}$ on those function spaces. Every coefficient of these push-pull maps in the isomorphism-class indicator bases is a finite sum of positive reciprocal automorphism orders, hence is real; the identity on underlying groupoids therefore yields $Z_{\bar{M}}\circ c_{{\mathrm{\Sigma }}_1}=c_{{\mathrm{\Sigma }}_0}\circ Z_M^{†}$. Pointwise conjugation is the only anti-linear operation, since this model has no cocycle phase.

The push-pull composition identity in [F2] identifies the linear map of that composite span with $Z_{M_2}\circ Z_{M_1}$. Equivalent finite groupoids give the same weighted fiber sums, so this map is $Z_{M_2{\cup }_{{\mathrm{\Sigma }}_1}{M}_1}$.

For empty boundary the amplitude is $\mathrm{\mid }{\mathcal{C}}_G(M)\mathrm{\mid }$ by [F1]. Applying [F3] to connected $M$ gives $Z_G(M)=\mathrm{\mid }\mathrm{Hom}({\pi }_1(M),G)\mathrm{\mid }\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$, with all conjugation stabilizers retained. The gluing, identity, disjoint-union, and orientation statements above establish the asserted finite state-sum rules.