$ \newcommand\cc{\mathcal{C}} \newcommand\set{\mathsf{Set}} \newcommand\sbcat{\mathit{Category}} \newcommand{\lcat}[1][S]{\mathbb{L}(#1)} \newcommand{\rcat}[1][S]{\mathbb{R}(#1)} \newcommand{\cc}{\mathcal{C}} \newcommand{\cd}{\mathcal{D}} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\la}{\lambda} \newcommand{\La}{\Lambda} \newcommand{\set}{\mathsf{Set}} \newcommand{\sbcat}{\mathcal{S}-\text{Category}} \newcommand{\preord}{\mathcal{P}} \newcommand{\scat}{\mathsf{Cat_{o}}} \newcommand{\vrt}[1]{\boldsymbol{\mathfrak{v}}{#1}} \newcommand{\Grp}{\mathsf{Grp}} \newcommand{\abgrp}{\mathsf{Ab}} \def\inc#1#2{{\jmath}_{{#1}}^{{#2}}} \newcommand{\cod}{\operatorname{cod}} \newcommand{\dom}{\operatorname{dom}} \newcommand{\e}{\epsilon} \newcommand{\im}{\operatorname{im}} \newcommand{\dom}{\operatorname{dom}} \newcommand{\vmap}[1][{}]{{#1}_{\mathfrak{A}}} \def\relax{} $

Introduction

The concept of cross-connection between partially ordered sets was originally introduced by ??? in order to classify the class of fundamental regular semigroups. Grillet defines a cross-connection between two regular partial ordered sets $I$ and $\Lambda $ as a pair of order-preserving mappings \begin{equation} \label{eq:1} \Gamma:I\to\Lambda^{*}\quad\text{and}\quad\Delta:\Lambda\to I^{*} \end{equation} satisfying certain axioms, where $I^{*}$ denote the dual of $I$. See ? for definition of regular partially ordered sets, duals and axioms for cross-connections. Every regular semigroup $S$ induces a cross-connection between the partially ordered set $\Lambda S$ of all principal left ideals of $S$ under inclusion and the partially ordered set $IS$ of all principal right ideals. In ?, this was generalized to cross-connections between normal categories. Again, given a regular semigroup $S$, category $\lcat $ of all principal left ideals is normal and similarly, the category $\rcat $ of all principal right ideals is also normal (see Propositions ?? and ?? for definitions of these categorioes). We can see that every regular semigroup $S$ induces a cross-connection between categories $\lcat $ and $\rcat $.

We observe that the configuration that occur in Grillet’s definition as well as its generalization in ? occures in many areas in mathematics. Our aim here is to describe cross-connections of a more general class of categories. We first identify an appropriate class of categories for which cross-section can be defined. These are categories with subobjects in which objects are sets, morhisms are mappings and inclusions are inclusions of sets. We refer to these categories as set-based categories ($\relax \mathcal {S}$-category for short). A cross-connection between to $\relax \mathcal {S}$-categorys $\cc $ and $\cd $ consists of two set-valued bifunctors \[ \Ga :\cc \times \cd \to \set \quad \text {and}\quad \De :\cc \times \cd \to \set \] and a natural isomorphism \[ \chi :\Ga \to \De \] between them. We can show that every cross-connection determines a semigroup. Also for any semigroup $S$ there is a cross-connection between $\lcat $ and $\rcat $ and there is a natural representation of $S$ by the cross-connection semigrpoup determined by the cross-connection between $\lcat $ and $\rcat$.

In the following discussion, we will follow (?) for ideas regarding categories, subobject relations, etc. Moreover, to save repetition, we shall assume that categories under consideration are small, unless otherwise provided, so that they may be treated as partial algebras.

Categories with subobjects

We assume that the reader is familiar with the basic concepts such as categories, functors, natural transformations, etc. (see for example ?). For the more specialized concepts like categories with subobjects and related ideas, we follow ?. We begin by reviewing briefly the definition of categories with subobjects.

Subobjects

Preorders

Recall that two morphisms $f,g\in \preord $ are parallel if $\dom f=\dom g$ and $\cod f=\cod g$. A preorder $\preord $ is a category such that for all $f,g\in \preord $, $f$ is parallel to $g$ if and only if $f=g$. Equivalently, $\preord $ is a preorder if and only if $\preord p,q$ contain atmost one element for all $p,q\in \vrt \preord $. It follows that the relation $\preceq $, defined for all $p,q\in \vrt \preord $, by \begin{equation*} p\preceqq\iff\preordp,q\ne\emptyset \end{equation*} is a quasi-order on the class $\vrt \preord $. In particular, if $\preord $ is small, then $\vrt \preord ,\preceq $ is a quasi-ordered set. Conversely, it is clear that any quasi-ordered set $\La $ uniquely determines a small preorder $\preord $ such that $\La =\vrt \preord $. The preorder $\preord $ is said to be strict if the quasi-order relation $\preceq $ above is a partial order; that is, $\preord $ has the property that for all $p,q\in \vrt \preord $, \[ \preord p,q\ne \emptyset \quad \text {and}\quad \preord q,p\ne \emptyset \implies p=q. \] The concepts of a small strict preorder and a partially ordered set are equivalent and will be used interchangeably in the sequel.

Choice of subobjects:

Recall ? that a choice of subobject in a category $\cc $ is sub-preorder $\preord \subseteq \cc $ satisfying the conditions:

1. $\preord $ is a strict preorder with $\vrt \preord =\vrt \cc $.
2. Every $f\in \preord $ is a monomorphism in $\cc $.
3. If $f,g\in \preord $ and if $f=hg$ for some $h\in \cc $, then $h\in

    \preord $.
     

Categories with subobjects

If $\preord $ is a choice of subobjects in $\cc $, the pair $\cc ,\preord $ is called a category with subobjects. For brevity, we shall say that a category $\cc $ has subobjects if a choice of subobjects in $\cc $ has been specified. Also, we shall use the notation $\vrt \cc $ to denote the preorder of subobjects as well as the partially ordered set of vertexes in $\cc $. No ambiguity will arise since both these are equivalent. We may then use the usual notation $\subseteq $ (see ?) to denote subobject relation in $\cc $. If $c\subseteq d$, the unique morphism in $\vrt \cc $, the inclusion of $c$ in $d$, is denoted by $\inc {c}{d}$. If $f\in \cc c,d$ and $c'\subseteq c$, then we write \[ f|c'=\inc {c'}{c}\circ f. \] As usual, $f|c'$ is called the restriction of $f$ to $c'$.

In the following $\cc , \cd $, etc. stands for catagories with subobjects in which $\vrt \cc , \vrt \cd $, etc. denote the corresponding preorder of subobjects. A functor $F:\cc \to \cd $ is said to be inclusion preserving if its vertex map $\vrt F$ is an order-preserving map of $\vrt \cc $ to $\vrt \cd $; that is, $\vrt F:\vrt \cc \to \vrt \cd $ is a functor of preorders. $F$ is an embedding if $F$ is faithful and $\vrt F$ is an order-embedding of $\vrt \cc $ into $\vrt \cd $. In particular, $\cc $ is a subcategory (with subobjects) of $\cd $ if $\cc \subseteq \cd $ as partial algebras. In this case, the inclusion $\subseteq $ is a category embedding of $\cc $ in $\cd $ whose vertex map is $\vrt \cc \subseteq \vrt \cd $.

The category $\set $ is clearly a category with subobjects in which the subobject relation coincides with usual set-theoretic inclusion. Similarly categories of groups $\Grp $, abelian groups $\abgrp $, etc., are categories with natural subobject relations.

It is clear that there is a category $\scat $ whose objects are small categories with subobjects and morphisms are inclusion preserving functors. Further, the assignments

\[ \vrt {}:\cc \mapsto \vrt \cc ,\quad \text {and}\quad F\mapsto \vrt F \] is a functor of the category $\scat $ to the category of preorders (or the category of partially ordered sets).