The five-sequence of adjoints for combinatorial simplicial complexes

Abstract: For a set $A$ let ${\mathbf {SC}_A}$ be the poset of simplicial complexes whose vertices are in $A$. For a function $f : A \rightarrow B$ there are functors $ f^{! !}, f^{**}, f^{ii}: {\mathbf {SC}_A} \rightarrow {\mathbf {SC}_B}, \quad f^{!*}, f^{i*} : {\mathbf {SC}_B} \rightarrow {\mathbf {SC}_A}, $ forming a five sequence of adjoints $f^{ !!} \dashv f^{* !} \dashv f^{* *} \dashv f^{*i} \dashv f^{ii}$. 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.