Simple duality for finite simplicial complexes
(An informal talk by Mike Barr)

ABSTRACT:
A finite simplicial complex is a downclosed set of subsets of a finite
set.  If S is such a set, then define S* to be the complement of the set
of complements of S (= the set of complements of the complement of S).
Then S** = S (using the commutativity indicated in the previous sentence).
Moreover, if the original set has N + 1 elements (is an N-simplex), then
the homology of S* in dimension n is isomorphic to the cohomology of S in
dimension N - n - 2.  I cannot say it is natural because I do not see how
(-)* is a contravariant functor.  Peter Freyd thinks this may be a
simplicial version of Spanier-Whitehead duality and Alex Heller agrees
with that conjecture, but neither of them have any idea how to prove
that.  In any case, that duality is quite difficult and this one is
trivial.