Inspired by prior results by Stanley and Leroux showing what information can be recovered from an isomorphism of incidence algebras, we investigate the very same idea applied to decomposition spaces. We review the work of Stanley and Leroux and provide sufficient background on the homotopy theory of groupoids to be able to define decomposition spaces, equivalences of them as linear functors and solve the isomorphism problem for both the groupoid-level coalgebra and for the numerical incidence algebra.