Weight structures (also known as co-t-structures) and t-structures on triangulated categories are closely related by orthogonality. I will explain how this relation can be characterized in terms of simple-minded collections and silting collections, using derived projective covers which are an analog of projective covers in triangulated categories. If time permits, I will also discuss a result about Koszul duality between simple-minded and silting collections. The talk is based on arXiv:2309.00554.