Example. isotropy survives the coarse quotient [fgap-000P]
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    Let \(C_2=\{1,s\}\) act on \(X=\{-1,0,1\}\) by \(s\mathbin {\cdot }x=-x\). The action groupoid \(C_2\ltimes X\) has the points of \(X\) as objects and an arrow \((h,x):x\to h\mathbin {\cdot }x\) for every \(h\in C_2\). Omitting identity arrows, its shape is

    Thus \[ \operatorname {Aut}(0)\cong C_2,\qquad \operatorname {Aut}(1)=\operatorname {Aut}(-1)=1. \]

    The coarse quotient remembers only the two orbits:

    It has forgotten the nonidentity automorphism at \(0\). By contrast, the action groupoid has one component equivalent to the terminal groupoid and one component equivalent to \(BC_2\). Therefore \[ \mathsf {Set}^{(C_2\ltimes X)^{\mathrm {op}}} \simeq \mathsf {Set}\times \mathsf {B}C_2, \] while sheaves on the discrete coarse quotient form \(\mathsf {Set}\times \mathsf {Set}\).

    The retained information is already the isotropy arrow in the action groupoid. Passing to its presheaf category organizes families of such data, but does not create that arrow. This test supports the groupoid bridge and does not, by itself, admit a topos layer.