Categorical Interpretations of Cryptomorphisms and Maps arising from Matroidal and Combinatorial Contexts

We outline a formal framework, based on functorial submaps (briefly {\em fsm}) and mappings (called {\em fsm-transfers}) between them, to translate at a categorical level the study of maps between collections of set systems, binary set relations and set operators defined on a fixed ground set. In this context we formalize as {\em fsm-cryptomorphism} the vague notion of cryptomorphism and give the concept of {\em displacement} as functorial extension of fsm-transfers and of fsm-cryptomorphisms. For any subcategory $\mfC$ of $\SET$, we introduce a quasicategory ${\bf fsM(\mfC)}$ of $\mfC$-functorial submaps and fsm-transfers between them. In ${\bf fsM(\mfC)}$ we can describe all the cases in which the axiomatizations of two set-theoretical structures are related by some fsm-transfers and, in addition, to get an ambient within which to formalize fsm-cryptomorphisms as suitable pairs of morphisms of ${\bf fsM(\mfC)}$. Next, by means of two new collections of categories $\SRkroh$ and $\OPkroh$ of $(k,h)$-set relations and $(k,h)$-set operators, with morphisms defined by a condition involving a fixed pre-order $\rho$ on $\txnObj(\SET)$, we frame in an appropriate categorical context the mathematical structures we work with and also translate fsm-transfers as functors between $\SET$-concrete categories. We determine an fsm-cryptomorphism between closable finitary abstract simplicial complexes and algebraic symmetrizing closure operators, and exhibit a partial displacement for it. The fsm-transfers involved in the previous result will be also used to determine fsm-cryptomorphisms and displacements for greedoids and matroids. Through similar techniques, we construct displacements of fsm-cryptomorphisms involving dependence relations, closure operators and Moore set systems and, next, introduce a category ${\bf DR}$ of dependence relations that provides a non-trivial model of proper $\SET$-Moore subcategory of ${\bf SR^{1,\subseteq}_2}$ and that allows us to undertake different categorical constructions in corresponding categories of Moore set systems and closure operators. Always in this context, we analyze further fsm-transfers leading to non-trivial commutative diagrams involving dependence relations. Finally, we determine several fsm-cryptomorphisms for matroids on a finite set and use dependence relations to derive a new matroid axiom system which, jointly with an extra condition, yields a new fsm-cryptomorphism between such an axiom system and algebraic symmetrizing closure operators.