For compact Lie group actions on metrizable spaces, equivariant A-category is invariant under Morita equivalence, yielding Morita-invariant equivariant LS-category and invariant topological complexity, and hence an orbifold topological complexity.
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Morita Invariance of Equivariant Lusternik-Schnirelmann Category and Invariant Topological Complexity
For compact Lie group actions on metrizable spaces, equivariant A-category is invariant under Morita equivalence, yielding Morita-invariant equivariant LS-category and invariant topological complexity, and hence an orbifold topological complexity.