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Bijective correspondences are established between (1) silting objects, (2)simple-minded collections, (3) bounded $t$-structures with length heart and (4)bounded co-$t$-structures. These correspondences are shown to commute withmutations. The results are valid for finite-dimensional algebras. A concreteexample is given to illustrate how these correspondences help to compute thespace of Bridgeland's stability conditions.

Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras