Derived equivalences between diagram categories of finite posets

This thesis studies universal derived equivalences between categories of diagrams over finite posets. Given a finite poset X and an abelian category A, one considers the category A^X of diagrams over X with values in A, and asks when two posets X and Y give rise to derived equivalent diagram categories for every abelian category A. After recalling the necessary background on triangulated categories, complexes, and derived categories, the thesis presents a detailed proof of a criterion, due to Ladkani, under which suitable combinatorial data on finite posets produce pairs of universally derived equivalent posets. The proof is based on the construction of explicit combinatorial data, called formulas, which give rise to functors on complexes of diagrams and induce triangulated functors on the corresponding derived categories. The final chapter develops original results aimed at producing posets that are universally derived equivalent to a given finite poset. More precisely, it studies when the order structure of a finite poset can be described in terms of a partition into two subsets and an order-preserving map between them, proves that this map is uniquely determined whenever such a description exists, and formulates algorithmic procedures for detecting these partitions and constructing the corresponding universally derived equivalent posets, together with some results on when different constructions may lead to isomorphic outputs.