The five-sequence of adjoints for combinatorial simplicial complexes

For a set $A$ let ${\mathbf{S}\mathbf{C}}_A$ be the poset of simplicial complexes whose vertices are in $A$. For a function $f:A\to B$ there are functors $f^{!!},f^{\ast \ast },f^{ii}:{\mathbf{S}\mathbf{C}}_A\to {\mathbf{S}\mathbf{C}}_B,f^{!\ast },f^{i\ast }:{\mathbf{S}\mathbf{C}}_B\to {\mathbf{S}\mathbf{C}}_A,$ forming a five sequence of adjoints $f^{!!}⊣f^{\ast !}⊣f^{\ast \ast }⊣f^{\ast i}⊣f^{ii}$. We investigate in detail these functors, and use this to give three categorical structures on simplicial complexes on finite sets such that the Stanley-Reisner correspondence to commutative monomial rings gives dualities.