To all experts in algebraic topology: Is the duality I describe below known? It does not appear to be Poincare duality since there is no separate treatment of the torsion and torsion free parts. Let S be a simplicial complex, that is a set of faces of an N simplex closed under face operations. For what I am about to say, it is necessary that the empty face in dimension -1 be included so that the (co-)homology will be reduced. This means, for example, that there is a distinction between the empty simplicial complex (which has no homology) and the simplicial complex consisting of only the empty set (which has one cyclic homology group in degree -1). In fact, the former is dual to the N simplex and the latter to the N sphere. Let S' be defined as the set of complements of the complement of S. That is, for each simplex sigma not in S, the complementary simplex, whose vertices are those not in sigma, belongs to S' (and nothing else). Then for i = -1,...,N, the ith homology of S is the N-i-1st cohomology of S'. The algebraic topology books I have looked at do not mention Poincare duality, except for Lefschetz, written in 1940, and he uses a complex different from S'; in fact just opposite poset to S, since he is doing things in the generality of an arbitrary poset equipped with "incidence numbers" that are used in defining the boundary operator. In particular, his dual has the same number of elements as S, rather than 2^{N+1} - that number. One thing I find curious is that although this gives, essentially, the cohomology of S, there is no obvious way of making that into a contravariant functor.