Gallery fractions, fundamental groupoid and asphericity

The gallery congruence agrees with Deligne's by F1. Their formal groupoid completions therefore agree by the localisation universal property, so Proposition 1.27 and the imported labelled-cover construction apply to this $P$ and $G$.

Fix a starting chamber $A$ and its basepoint $a_A$. In the imported cover $p_A:{\tilde{Y}}_A\to Y$, choose the lift of $a_A$ labelled by $1_A$. For a chamber $B$ the fibre over $a_B$ has labels ${\mathrm{Hom}}_G(A,B)$, with the signed crossing rule of F99. The covering space is contractible, hence path connected and has trivial fundamental groups by F3.

Assign to a path from $a_A$ to $a_B$ the endpoint of its unique lift starting at the chosen lift of $a_A$. Path lifting and endpoint invariance in F2 make this a function on path classes. If two lifts have the same endpoint, concatenate the first with the reverse of the second. This is a loop upstairs and is endpoint-fixed nullhomotopic by F3. Projecting that homotopy makes the projected loop trivial; cancellation in the fundamental groupoid gives equality of the original path classes. Conversely, any point of the fibre is reached by a path upstairs because the cover is path connected. Its projection lifts to that path by uniqueness. The endpoint function is therefore a bijection.

The cover is path connected and surjective, so $Y$ is path connected. Deligne's introductory theorem applies under precisely the same finite central simplicial hypotheses and gives asphericity. This last assertion is imported directly, without an additional local inference about higher homotopy lifting.

Represent an arrow of $G$ by a signed gallery word and concatenate its wall-crossing paths. Its lift ends at its gallery label by F99. Equal words in $G$ have equal labels and hence equal path classes by step 3.1, so the assignment is well-defined. Different labels give different classes, and every class occurs by the same endpoint bijection. Concatenation of gallery words concatenates paths; explicitly, for traversal products, $\mathrm{\Phi }(gh)=\mathrm{\Phi }(h)\circ \mathrm{\Phi }(g)$. Identities and inverses are respected. Thus $\mathrm{\Phi }$ is an isomorphism from $G$ to the full fundamental subgroupoid on the $a_C$.

To exhibit the groupoid equivalence, order the finitely many chambers and retain the chosen real points $a_C$. For $z=x+iy\in Y$, take the first $C$ with $z\in U_C$. A path ${\lambda }_z:z\to a_C$ is obtained by three successive linear interpolations: change the imaginary coordinate from $y$ to $a_C$ at fixed real coordinate $x$, then change the real coordinate from $x$ to $a_C$ at fixed imaginary coordinate $a_C$, then change the imaginary coordinate from $a_C$ to $0$ at fixed real coordinate $a_C$. The first part avoids every $H_{\mathbb{C}}$ because whenever $x\in H$, both $y$ and $a_C$ are on the same open side of $H$ by F98. The second part has imaginary coordinate on no hyperplane, and the third has real coordinate on no hyperplane. At a selected point $z=a_D$ instead set $C=D$ and take the constant path. Define a functor $R$ to the chosen full subgroupoid by $R(z)=a_C$ and $R([\alpha ]:z\to w)=[\overline{{\lambda }_z}\ast \alpha \ast {\lambda }_w]$. The fundamental groupoid laws show that it preserves classes, identities and composition. If $I$ is inclusion, then $RI$ is the identity, and the arrows $[\overline{{\lambda }_z}]:IR(z)\to z$ give a natural isomorphism $IR\Rightarrow 1$. These are equivalence data; composed with step 4.1 they prove the asserted equivalence for $G$. The finite ordering and explicit interpolations require no arbitrary indexed selection of paths.