Abstract: A common occurrence, in commutative algebra and elsewhere, is a family of ideals $I_n \subseteq k[x_1,\ldots,x_n]$ in a family of polynomial rings, satisfying that $f(I_n) \subseteq I_{n+1}$ for any $f$ in a certain family of ring homomorphisms. In the context where $f$ is any order-preserving function of the indices of the variables, the theory of OI-algbras gives a categorical re-framing of this situation. In this framework one can study OI-ideals using resolutions by free OI-modules. After introducing and motivating the subject, we will exhibit a family of OI-ideals (coming from strongly-stable ideals) that have explicit free resolutions supported on OI-simplicial complexes. This is based on work in progress with Uwe Nagel.