Approximation properties of torsion classes

We strengthen a result of Bagaria and Magidor (Trans. Amer. Math. Soc. 366 (2014), no. 4, 1857-1877) about the relationship between large cardinals and torsion classes of abelian groups, and prove that the Maximum Deconstructibility principle introduced in Cox (J. Pure Appl. Algebra 226 (2022), no. 5) requires large cardinals; it sits, implication-wise, between Vop & ecaron;nka's Principle and the existence of an omega 1$\omega _1$-strongly compact cardinal. While deconstructibility of a class of modules always implies the precovering property by Saor & iacute;n and & Scaron;& tcaron;ov & iacute;& ccaron;ek (Adv. Math. 228 (2011), no. 2, 968-1007), the concepts are (consistently) nonequivalent, even for classes of abelian groups closed under extensions, homomorphic images, and colimits.