On some categories of structured sets

Abstract

Given an arbitrary set $$\Omega $$ Ω , we consider the collections $$\textrm{SS}\hspace{0.55542pt}(\Omega )$$ SS ( Ω ) , $$\textrm{SR}\hspace{0.55542pt}(\Omega )$$ SR ( Ω ) and $$\textrm{SO}\hspace{0.55542pt}(\Omega )$$ SO ( Ω ) of all the set systems, the binary set relations and the set operators on $$\Omega $$ Ω . We introduce the notion of linking map on $$\Omega $$ Ω as any map whose domain and codomain may be chosen between the above collections. After providing a descriptive overview useful for framing the notion of linking map in a broad non-specialized context, we explain how linking maps occur in a very natural way in two specific results. The first of these results concerns the classic identification between the subfamily $$\textrm{EQ}\hspace{0.55542pt}(\Omega )$$ EQ ( Ω ) of all the equivalence relations on $$\Omega $$ Ω and the subfamily $$\textrm{SP}\hspace{0.55542pt}(\Omega )$$ SP ( Ω ) of all the set partitions on $$\Omega $$ Ω . Starting from it, we introduce a new subfamily $$\textrm{ESO}\hspace{0.55542pt}(\Omega )$$ ESO ( Ω ) of closure operators on $$\Omega $$ Ω and four linking maps whose restrictions to the subfamilies $$\textrm{EQ}\hspace{0.55542pt}(\Omega )$$ EQ ( Ω ) , $$\textrm{SP}\hspace{0.55542pt}(\Omega )$$ SP ( Ω ) and $$\textrm{ESO}\hspace{0.55542pt}(\Omega )$$ ESO ( Ω ) are bijections. The second result concerns the identification between the subfamily $$\textrm{CSO}\hspace{0.55542pt}(\Omega )$$ CSO ( Ω ) of all closure set operators on $$\Omega $$ Ω and the subfamily $$\textrm{CSS}\hspace{0.55542pt}(\Omega )$$ CSS ( Ω ) of all closure set systems on $$\Omega $$ Ω . Starting from it, we introduce a new subfamily $$\textrm{DSR}\hspace{0.55542pt}(\Omega )$$ DSR ( Ω ) of binary set relations and four linking maps whose restrictions to the subfamilies $$\textrm{CSO}\hspace{0.55542pt}(\Omega )$$ CSO ( Ω ) , $$\textrm{CSS}\hspace{0.55542pt}(\Omega )$$ CSS ( Ω ) and $$\textrm{DSR}\hspace{0.55542pt}(\Omega )$$ DSR ( Ω ) are again bijections. In an attempt to extend in a natural way the above linking maps to categorical isomorphisms, after fixing a nonnegative integer k, we introduce three categories $$\mathbf{SS^k}$$ SS k , $$\mathbf{SR^k}$$ SR k and $$\mathbf{SO^k}$$ SO k , whose detailed study mainly occupies the first part of the present work. Objects and arrows of these three categories are obtained by means of k-iterations of the powerset functor "Equation missing", and they generalize the notions of set systems, set relations and set operators, respectively. In the second part of the paper, we extend the linking maps previously described at a categorical level in terms of isomorphisms between specific categories of set systems, binary set relations and set operators generalizing the occurring collections introduced before, and also prove numerous other results concerning the main properties of all these categories, such as completeness, cocompleteness and Cartesian closedness.