Morphisms of Groupoid Actions and Recurrence

Theorem 4.14 claims Ξ̃_N^M ⋄ ρ′(g(M)) ⊂ (Ξ̃′)_{g(N)}^{g(M)} without extra hypotheses and argues: for any element ξ⋄ρ′(g(σ)) with ξ ∈ Ξ̃_N^M and σ ∈ M, one has [ξ⋄ρ′(g(σ))]•′g(σ) = g(ξ•σ) ∈ g(N). The last inclusion implicitly assumes ξ•σ ∈ N for that σ, but Definition 2.4 gives only the existence of some σ′ ∈ M with ξ•σ′ ∈ N, not that the property holds for all σ with ρ(σ)=d(ξ). This is an existential-versus-universal slip. A concrete counterexample (discrete groupoids and actions) shows the inclusion fails as stated. The corrected, quantifier-accurate inclusion is { ξ⋄ρ′(g(σ)) | ξ∈Ξ̃_N^M, σ∈M, ρ(σ)=d(ξ), ξ•σ∈N } ⊂ (Ξ̃′)_{g(N)}^{g(M)}. If one strengthens assumptions so that ρ′∘g is constant on ρ-fibers, the paper’s inclusion becomes valid; with Sat⋄[ρ′(g(Σ))]=Ξ′ and injective g, the reverse inclusion holds and equality follows. See the paper’s Definition 2.4 for Ξ̃_N^M, the actor equivariance (4.6)–(4.7), and the statement/proof of Theorem 4.14; the flaw is visible where g(ξ•σ) is asserted to lie in g(N) from ξ∈Ξ̃_N^M and σ∈M alone .