The main purpose of this talk will be to describe a link between posets and operads. The homology of the lattice of partitions of the sets {1, ..., n} is a classical subject studied by many authors. Since the symmetric groups Sn act on these posets, its homology groups are Sn-modules. The only non-vanishing homology groups are in top dimension (Cohen-Macaulay poset) and they are related to the free Lie algebra, as an Sn-module. Such a phenomenon can be explained by the Koszul duality of operads. An operad is a simple algebraic object that models multilinear operations acting on algebraic structures. In this talk, we will make a short review of the notion of operad and Koszul duality. To any operad in the category of sets, we associate a family of partition type posets. We prove that an operad is Koszul if and only if the related posets are Cohen-Macaulay. In this case, the top homology groups of the posets are easy to compute as Sn-modules : they are equal to the Koszul dual (co)operad. On the other hand, this result gives a simple method to prove that an operad is Koszul. |