Operadic Kazhdan-Lusztig-Stanley theory

We introduce a new type of operad-like structure called a P-operad, which depends on the choice of some collection of posets P, and which is governed by chains in posets of P. We introduce several examples of such structures which are related to classical poset theoretic notions such as poset homology, Cohen--Macaulayness and lexicographic shellability. We then show that P-operads form a satisfactory framework to categorify Kazhdan--Lusztig polynomials of geometric lattices and their kernel. In particular, this leads to a new proof of the positivity of the coefficients of Kazhdan--Lusztig polynomials of geometric lattices.