Definition. Arrows of selected backgrounds [fdip-005C]
Definition. Arrows of selected backgrounds [fdip-005C]
The groupoid \(\mathsf {Bg}\) has selected backgrounds as objects. An arrow \(b\to b'\) is a pair of bijections \(p:V\to V'\), \(q:E\to E'\) satisfying \[\begin {aligned} s'q&=ps,&h'q&=ph,\\ w'q&=w,&m'&=m,\\ P'&=UPU^{-1},&&(U\phi )(p(v))=\phi (v). \end {aligned}\] Here \(U:H\to H'\) is the vertex counting isometry, distinct from the selector \(P\). For coordinate selectors the last equation is equivalent to \(pR=R'\), so these are precisely the exact relabelings.
The larger groupoid \(\mathsf {Bg}^{\pm }\) allows signed endpoint arrows: an additional sign \(\varepsilon (e)\in \{+,-\}\) specifies whether \(q(e)\) has endpoints \((ps(e),ph(e))\) or \((ph(e),ps(e))\). Weights, mass and the selector obey the same equations. The signed edge map is \((Ja)(q(e))=\varepsilon (e)a(e)\), where the sign acts as \(\pm 1\). Then \(d'U=Jd\) and \(W'J=JW\). Consequently \[Q'=UQU^{-1},\qquad G'=UGU^{-1}.\] The endpoint condition is retained even for loops of zero incidence. Signs are part of an arrow, not merely an equation between Laplacians.
Compose vertex and edge bijections, and for signed arrows use \[\varepsilon _{21}(e)=\varepsilon _2(q_1(e))\varepsilon _1(e).\] The identity has identity bijections and positive signs; the inverse has inverse bijections and sign \(\varepsilon (e)\) at \(q(e)\). The endpoint-arrow proof checks all graph equations. Selector transport also composes because \(P''=U_2U_1P(U_2U_1)^{-1}\). It inverts by conjugating back. These operations therefore give groupoids on the stated tuples.
Restrict \(U\) to the isometry \(U_X:X\to X'\) of selected spaces. Then the selected laws satisfy \[\begin {aligned} C_{P'}&=U_XC_PU_X^{-1},\\ S_{P'}&=U_XS_PU_X^{-1},\\ (U_X)_*\nu _b&=\nu _{b'}. \end {aligned}\] For the first identity insert \(P'=UPU^{-1}\) and \(G'=UGU^{-1}\); inversion gives the second. The full-law transport gives \(U_*\mu _Q=\mu _{Q'}\), while \(P'U=U_XP\). Composition of Borel pushforwards proves the third. In particular, for an integrable retained observable \(F'\) on \(X'\), \[\int _{X'}F'\,d\nu _{b'}=\int _X F'\circ U_X\,d\nu _b.\] The equality for \(|F'|\), allowing infinity, first establishes the matching integrability condition.
These arrows carry incidence, not only the precision and projector. Replacing one edge of weight \(w\) by two parallel edges with the same endpoints and positive weights summing to \(w\) leaves \(Q\) unchanged, but changes \(|E|\); it admits no arrow in either groupoid. Reversing the orientation of a specified edge gives an identity-on-vertices signed arrow with negative sign on that edge. That particular correspondence is not directed. Another directed isomorphism may exist if the graph has additional symmetries.