Is the face poset of a simplicial or nice enough cellular complex a Heyting algebra in some natural way?
Edited to add: For the benefit of illustration, here's a few face posets:
the boundary of a square: Elements $a,b,c,d$ of dimension 0 and $x,y,z,w$ of dimension 1; poset relations $a,b < x$; $b,c < y$; $c,d < z$; $a,d < w$.
Two 2-simplices joined along a common edge: Elements $a,b,c,d$ of dimension 0, $j,k,l,m,n$ of dimension 1 and $x,y$ of dimension 2. Poset relations are: $a,b < j$; $a,c < k$; $b,c < l$; $b, d < m$; $c,d < n$ and $j,k,l < x$ and $l,m,n < y$.
