A Tale of Two Categories: Inductive groupoids and Cross-connections
P. A. Azeef Muhammed, Mikhail V. Volkov
TL;DR
This paper establishes a categorical equivalence between inductive groupoids and cross-connections, two models used to represent regular semigroups, thereby unifying different categorical approaches in semigroup theory.
Contribution
It proves a direct category equivalence between the category of inductive groupoids and the category of cross-connections, clarifying their relationship in the theory of regular semigroups.
Findings
Proves a categorical equivalence between inductive groupoids and cross-connections.
Unifies two categorical models of regular semigroups.
Enhances understanding of the structure of regular semigroups.
Abstract
A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category whose object set is a strict preorder and the morphisms admit a factorisation property. A pair of `related' normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad \cite{mem,cross} as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.
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A Tale of Two Categories:
Inductive groupoids and Cross-connections
P. A. Azeef Muhammed
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, Al Khobar 31952, Kingdom of Saudi Arabia.
Institute of Natural Sciences and Mathematics, Ural Federal University, Lenina 51, 620000 Ekaterinburg, Russia
and
Mikhail V. Volkov
Institute of Natural Sciences and Mathematics, Ural Federal University, Lenina 51, 620000 Ekaterinburg, Russia
Abstract.
A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category whose object set is a strict preorder and the morphisms admit a factorisation property. A pair of ‘related’ normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.
Key words and phrases:
Biordered set, inductive groupoid, normal category, cross-connection, regular semigroup
2010 Mathematics Subject Classification:
18B40, 20M17, 18B35, 18A32, 20M50, 20M10
1. Introduction
In the 1880s, Sophus Lie introduced pseudogroups111Lie termed them ‘infinite continuous groups’ as opposed to ‘finite continuous groups’, i.e., Lie groups in modern terminology. as generalisations of Lie groups appropriate in the context of his work in geometries of infinite dimension; see [17, Chapter 1]. After algebra had managed to exempt the idea of a group from its geometric cradle and developed the abstract concept of a group, the quest for the abstract structure represented by pseudogroups began. This quest led to two main solutions in the 1950s. The first solution proposed independently by Wagner [34] and Preston [24] involved the introduction of inverse semigroups. The second solution provided by Ehresmann [10, 11] used the categorical idea of a groupoid. Later, Schein [29, 30] connected Ehresmann’s work on differential geometry with Wagner’s ideas on inverse semigroups to provide a structure theorem for inverse semigroups using groupoids.
Incidentally, inverse semigroups, born around the 1950s, were a special instance of a much more general concept invented earlier and for a completely different purpose. In the 1930s, von Neumann [32, 33] introduced and made a deep study of regular rings in his ground-breaking work on continuous geometry. A ring is said to be (von Neumann) regular if for every , there exists such that . Observe that the regularity of a ring is in fact a property of its multiplicative semigroup so that it is fair to say that von Neumann introduced regular semigroups as well, even though he did not use the latter term. Beyond the class of multiplicative semigroups of regular rings, examples of regular semigroups are plentiful and include many natural and important objects such as the semigroup of all transformations of a set. Certain species of regular semigroups were studied as early as 1940; see [8, 27], and by the 1970s, studying regular semigroups became a hot topic in the blossoming area of the algebraic theory of semigroups.
Regular semigroups arguably form the most general class of semigroups which admits a notion of inverse elements that naturally extends the corresponding notion for groups. Namely, two elements of a semigroup are said to be inverses of each other if and . It is well known and easy to see that a semigroup is regular if and only if each of its elements has an inverse. Inverse semigroups can be defined as semigroups in which every element has a unique inverse. In a given inverse semigroup , the map that assigns to each element its unique inverse can be seen to constitute an involutary anti-automorphism of , similarly to the case of groups. Hence, the usual right-left duality reduces to a mere isomorphism: if denotes the dual222Recall that is defined on the same set as but the multiplication on is defined by , where stands for the multiplication in . of , then is isomorphic to under the map , whenever is inverse. Thus, an inverse semigroup seen from the left looks the same as from the right. As we move to general regular semigroups, this inbuilt symmetry is lost, and the right-left duality may become highly nontrivial. In particular, providing structure theorems for regular semigroups using categories requires inventing categorical structures that would be less symmetric than groupoids on the one hand but still possess some intrinsic duality on the other hand, quite a challenging task!
There are two successful approaches to this task, both due to Nambooripad. They rely on the idea of replacing a ‘too symmetric’ object by a couple of interconnected ‘unilateral’ objects. We observe in passing that this idea has got other interesting incarnations in categorical algebra. As examples, consider Loday’s approach to Leibniz algebras [18] or the recent notion of a constellation studied by Gould, Hollings, and Stokes [12, 13, 31]333Leibniz algebras are non-anticommutative versions of Lie algebras. A standard way to produce a Lie algebra is to define Lie bracket on an associative algebra ; the bracket operation is clearly anticommutative, that is, . Loday ‘splits’ the multiplication in into two by considering two associative operations: the ‘right’ product and the ‘left’ product which are interconnected by certain axioms. The axioms ensure that the bracket satisfies the Leibniz law \bigl{[}[x,y],z\bigr{]}=\bigl{[}[x,z],y\bigr{]}+\bigl{[}x,[y,z]\bigr{]} while the anticommutativity may fail. In a similar way, constellations are partial algebras that are one-sided generalisations of categories..
Nambooripad’s first approach [22] takes as its starting point the structure of idempotents in a semigroup. Recall that an element of a semigroup is called an idempotent if . On the set of all idempotents of , one can define the relation letting for if and only if . It is easy to see that is a partial order on . In inverse semigroups idempotents commute (in fact, it is this property that specifies inverse semigroups within the class of regular semigroups) whence is a semilattice. This semilattice plays a crucial role in the structure theory of inverse semigroups. Nambooripad [22] considered general semigroups, for which he ‘split’ the partial order into two interconnected preorders. This led him to the notion of a biordered set as the abstract model of the idempotents of an arbitrary semigroup. Nambooripad isolated a property that characterises biordered sets of regular semigroups and developed the notion of an inductive groupoid on the base of this characterisation. This way, the category of inductive groupoids arose, the first of the two categories being the objects of the present paper. Using this category, Nambooripad generalised Schein’s work to regular semigroups and proved a category equivalence between the category of regular semigroups and the category .
Later, Nambooripad [23], building on an alternative approach initiated by Hall [15] and Grillet [14], introduced the notion of a normal category as the abstract categorical model of principal one-sided ideals of a regular semigroup. Each regular semigroup gives rise to two normal categories: one that models the principal left ideals of and another one that corresponds to the principal right ideals. In the treatise [23], which was — quoting Meakin and Rajan [20] — “somewhat reminiscent of von Neumann’s foundational work on regular rings”, Nambooripad devised a structure called a cross-connection which captured the non-trivial interrelation between these two normal categories. Cross-connections also form a category, denoted by , which constitutes the second main object of the present paper. Nambooripad mastered the category as an alternate technique to describe regular semigroups; namely, he proved that the category of regular semigroups is equivalent to .
The two approaches of Nambooripad seemed unrelated if not orthogonal to each other. However, the present authors [7] established an equivalence between inductive groupoids and cross-connections that ‘bypasses’ regular semigroups in the sense that the equivalence between and exposited in [7] was not a mere composition of the aforementioned categorical equivalences found in [22, 23]. Still, the equivalence from [7] remained in the realm of regular semigroups: what we did is that we explored the inter-relationship between the idempotent structure and ideal structure in an arbitrary regular semigroup to establish how one can be retrieved from the other. In the present paper, we make one further step. Namely, we discuss inductive groupoids and cross-connections in a purely categorical setting and build upon this a direct category equivalence between the categories and , thus divorcing ourselves completely from a semigroup setting.
The rest of the paper is divided into five sections. In Section 2, we briefly discuss some preliminaries needed for the sequel; in particular, we introduce inductive groupoids and cross-connections. In the next section, we construct the inductive groupoid associated with a given cross-connection and build a functor . In Section 4, we construct a cross-connection from an inductive groupoid and the corresponding functor . In Section 5, we verify that the functor is naturally isomorphic to the functor and the functor is naturally isomorphic to the functor , thus establishing the category equivalence between and . The final section re-discusses the results and outlines some possible developments.
2. Preliminaries
We assume the reader’s acquaintance with basic notions of category theory [19]. As mentioned, the ideas discussed in the paper arose in the realm of regular semigroups; however, here we deal with them in the realm of categories only. So, although a semigroup background may be helpful, it is not a prerequisite for understanding the constructions in the paper. The reader interested in a more detailed presentation of the genesis of the concepts of an inductive groupoid and a cross-connection and their role within semigroup theory may find rather a self-contained account of this material in [7, Sections 1 and 2].
Our basic notational conventions are the following. For a small category , its set of objects is denoted by and its set of morphisms is denoted by itself. For , the set all morphisms from to is denoted by . We compose functions and morphisms from left to right so that in expressions like or etc., the left factor applies first.
2.1. Biordered sets
Let be a set with a partial binary operation denoted by juxtaposition. Let stand for the domain of the partial operation. Define two binary relations
and
on the set as follows:
e$$f$$\iff(e,f)\in D_{E} and ;e$$f$$\iff(f,e)\in D_{E} and .
We use the notation
and
for the ‘symmetric versions’ of respectively and
;
that is,
:= (
and
:= (.
Also, we let
:=
.
Recall that a preorder is a reflexive and transitive binary relation. The partial algebra as above is said to be a biordered set if the following axioms hold for all :
- (B1)
both and are preorders, and D_{E}=$$\cup$$\cup(\cup$$)^{-1}; 2. (B2)
if e$$f;fe$$e$$f, then
if e$$f;ef$$e$$f, then 3. (B3)
if e$$f$$g,
then fe$$ge and
;
if
e$$f$$g,
then ef$$eg and
; 4. (B4)
if e$$f$$g,then ;
if e$$f$$g,then ; 5. (B5)
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f^{\prime}$$g$$eand ;
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{44.93379pt}{-1.18056pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{ge,}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} { {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{ {}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }{}{}{}{}{{}}\pgfsys@moveto{28.9854pt}{0.0pt}\pgfsys@lineto{41.40079pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{28.9854pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{array} there exists such that
f^{\prime}$$g$$eand .
Further, for any two elements , their sandwich set is defined as follows:
e$$h$$f
and for every such that e$$g$$f,
one has
gf$$hfand eg$$eh.
A biordered set is regular if for every , the sandwich set is non-empty.
Given two biordered sets and with the domains of partial operations and respectively, we define a bimorphism as a mapping satisfying:
- (BM1)
; 2. (BM2)
.
Observe that a bimorphism necessarily preserves arrows: for example, if e$$f,
then e\theta$$f\theta, and so on. If is a regular biordered set, then a bimorphism is called a regular bimorphism if it satisfies:
- (RBM)
The above used arrow notation for the preorders in biordered sets was introduced by Easdown [9]. It allows one to present the axioms of a biordered set in a concise way and may be helpful for fresh readers. In the sequel, we shall refer to the partial binary operation of a biordered set as the basic product and we mostly use the following alternative notation to denote the preorders (for ease of writing and to save the arrow for maps and morphisms):
e\mathrel{\leqslant_{\ell}}f\iff e$$f\iff e\>f=eand e\mathrel{\leqslant_{r}}f\iff e$$f\iff f\>e=e.
In a given biordered set with preorders and , we can easily see that the relations
and
are equivalence relations while the relation
is a partial order.
Although the axioms of biordered sets are complicated and may appear slightly artificial, biordered sets arise quite naturally in several mathematical contexts. If a semigroup has idempotents, the set of all idempotents of can be seen to form a biordered set whose partial operation is a certain restriction of the multiplication of . In [22], the biordered sets of the form where is a regular semigroup were characterised by Nambooripad as regular biordered sets. Later, Easdown [9] showed that given a biordered set , we can always construct a semigroup such that and are isomorphic as biordered sets. Beyond semigroups, Putcha [25] showed that pairs of opposite parabolic subgroups of a finite group of Lie type form a biordered set.
In this paper, we do not need the explicit use of the biorder axioms except in a few proofs; nevertheless we have included the full set of axioms for the sake of completeness.
2.2. Ordered groupoids
The notion of an ordered groupoid was introduced by Ehresmann [10, 11] in the context of his work on pseudogroups. Ordered groupoids are essentially groupoids such that their morphisms admit a partial order compatible with the composition. Recall that our convention is to compose the morphisms from left to right.
Definition 2.1**.**
Let be a groupoid and denote by and its domain and codomain maps, respectively. Let be a partial order on . Then is called an ordered groupoid if the following hold for all and all .
- (OG1)
If , and , , then . 2. (OG2)
If , then . 3. (OG3)
If , then there exists a unique morphism (called the restriction of to ) such that and . 4. (OG3∗)
If , then there exists a unique morphism (called the corestriction of to ) such that and .
Observe that in an ordered groupoid , the restriction of to the identity morphisms in induces a partial order on the set of the objects of the groupoid. An order preserving functor between two ordered groupoids is said to be a -isomorphism if its object map is an order isomorphism.
2.3. Inductive groupoids
We are approaching the definition of the category of inductive groupoids, the first of the two main objects of this paper. Roughly speaking, an inductive groupoid is an ordered groupoid whose object set is a regular biordered set and containing certain distinguished morphisms which are induced by alternating sequences of - and -related elements of the biordered set. To formalise this rough idea, we start with associating an ordered groupoid with any given (not necessarily regular) biordered set .
An -path is a sequence of elements of such that for all . An element in such an -path is called inessential if or . Two -paths that share the same first and last elements are said to be essentially the same if each of them can be obtained from the other by a sequence of adding or removing inessential elements. Clearly, this defines an equivalence relation on the set of all -paths. The equivalence class of an -path relative to this relation is referred to as an -chain. In the sequel, we take the liberty of using expressions like ‘let be an -chain’, meaning, of course, the equivalence class of all -paths that are essentially the same as .
The set of all -chains can be thought of as a groupoid if we consider each -chain as a morphism with domain and codomain . The composition of two -chains, say, as above and , is defined if and only if , and if so, then the product is set to be equal to the -chain containing the -path
[TABLE]
The inverse of is the -chain .
Further, we introduce a binary relation on the set . Let and be two -chains. Suppose that in and define the sequence inductively by letting and for each . Using the assumption and the axioms of a biordered set, it is easy to check that all are indeed well-defined elements of , and moreover, forms an -path. Now we let if and only if and the -paths and are essentially the same. Alternatively, if and only if for any -path in the -chain , there exists an -path in the -chain such that for each . It can be shown (see [22, Proposition 3.3]) that is a partial order on and the pair constitutes an ordered groupoid. The partial order may be seen as an extension of the natural partial order of the biordered set to the set .
Given a biordered set , a matrix \bigl{[}\begin{smallmatrix}e&f\\ g&h\end{smallmatrix}\bigr{]} of elements of such that
[TABLE]
or, in Easdown’s arrow notation,
e$$f$$g$$h
stands for the -path . We refer to the -chain corresponding to this -path as an -square and allow ourselves expressions like ‘\bigl{[}\begin{smallmatrix}e&f\\ g&h\end{smallmatrix}\bigr{]} forms an -square’.
Given in a biordered set such that and , one can easily deduce from axioms (B2) and (B3) that \bigl{[}\begin{smallmatrix}g&h\\ eg&eh\end{smallmatrix}\bigr{]} forms an -square. Such an -square is called row-singular. Dually, we define a column-singular -square, and an -square is said to be singular if it is either row-singular or column-singular.
Given an ordered groupoid and a -isomorphism , an -square \bigl{[}\begin{smallmatrix}e&f\\ g&h\end{smallmatrix}\bigr{]} is said to be -commutative if
[TABLE]
Observe that to simplify notation in (1), we denoted the image of the -chain under by rather than and did so also for the other -chains that occur in (1). This convention, of leaving out unnecessary braces when there is no ambiguity regarding the expression, shall be followed in the sequel.
Definition 2.2**.**
Let be a regular biordered set, let be an ordered groupoid, and let be a -isomorphism, called an evaluation functor. We say that forms an inductive groupoid if the following axioms and their duals hold.
- (IG1)
Let and for , let , be such that , and . Then , and
[TABLE] 2. (IG2)
All singular -squares are -commutative.
Let and be two inductive groupoids with biordered sets and respectively. Suppose that is an order preserving functor such that its object map is a regular bimorphism of biordered sets. Then induces a unique order preserving functor between the ordered groupoids of -chains and defined as follows; see [22, Proposition 3.3]. For every -chain in ,
[TABLE]
The order preserving functor is said to be inductive if the following diagram commutes.
[TABLE]
Proposition 2.1** ([22, Remark 3.1]).**
Inductive groupoids with inductive functors as morphisms form a category.
We denote the category of inductive groupoids with inductive functors by .
2.4. Normal categories
Now, we proceed to introduce the second main object of this paper: the category of cross-connections. Again, the construction requires several steps. We begin by discussing normal categories.
Recall that a preorder is said to be strict if identity morphisms are the only isomorphisms in . Observe that a small preorder is strict if and only if it is induced by a partially ordered set.
Let be a small category and a subcategory of such that is a strict preorder category with . The pair is called a category with subobjects if, first, every is a monomorphism in and, second, if and are such that , then .
In a category with subobjects, the morphisms in are called inclusions. We write if there is an inclusion , and we denote this inclusion by . An inclusion splits if there exists a morphism such that . In this situation, the morphism is called a retraction.
A normal factorization of a morphism in is a factorization of the form where is a retraction, is an isomorphism, and is an inclusion, where are such that , . Figure 1 represents the normal factorisation property.
The morphism is called the epimorphic component of the morphism and is denoted by . It can be seen that is uniquely determined by . The codomain of is called the image of the morphism and is denoted by . The following properties of epimorphic components may prove crucial in the sequel:
Proposition 2.2** ([23, Corollary II.4]).**
Let be a category with normal factorisation property where inclusions split.
- (1)
If and are composable such that the inclusion of is , then
[TABLE] 2. (2)
If is an epimorphism, then .
Definition 2.3**.**
Let be a category with subobjects and . A map is called a cone444Notice that what we call a cone here was called a normal cone in [23, 7]. from the base to the apex if:
- (Ncone1)
is a morphism from to for each ;
- (Ncone2)
if , then ;
- (Ncone3)
there exists such that is an isomorphism.
The apex of the cone shall be denoted by in the sequel.
A cone is said to be idempotent if . It is easy to verify that for any cone and any epimorphism , the map defined by is a cone with apex . This cone is denoted by .
Definition 2.4**.**
A category with subobjects is called a normal category if the following holds.
- (NC1)
Any morphism in has a normal factorization. 2. (NC2)
Every inclusion in splits. 3. (NC3)
For each , there is an idempotent cone with apex .
Natural examples of a normal category include the powerset category (subsets of a set with functions as morphisms) [5], the subspace category (subspaces of a vector space with linear transformations as morphisms) [3], etc.
2.5. Normal dual
The normal dual of a normal category is a full subcategory of the category of all functors from to the category . The objects of are certain functors and the morphisms are natural transformations between them. Namely, for each cone in , we define a functor (called an H-functor and denoted by ) from to as follows. For each and for each ,
[TABLE]
We define the M-set of a cone as
[TABLE]
It can be shown that if two -functors and are equal, so are the -sets of the cones and . Hence we can define the -set of an -functor as .
It can be seen that -functors are representable functors such that for a cone , there is a natural isomorphism . Here is the hom-functor determined by .
It can be shown that the normal dual of a normal category is, in fact, normal (see [23, Section III.4.2]). The proof is quite non-trivial since it involves characterising the morphisms in the normal dual, which are natural transformations. In some special cases, the normal dual can be transparently described; for instance, the normal dual of the subspace category has been described via the annihilator category [3], wherein the algebraic duality coincides with the cross-connection duality. In general, however, such simple descriptions do not appear to be possible.
2.6. Cross-connections
An ideal of a normal category generated by its object is the full subcategory of , denoted , whose objects are given by
[TABLE]
A functor between two normal categories and is said to be a local isomorphism if is inclusion preserving, fully faithful and for each , the restriction of to the ideal is an isomorphism of onto the ideal .
Definition 2.5**.**
Let and be normal categories. A cross-connection from to is a triplet (often denoted by just ), where is a local isomorphism such that for every , there is some such that .
Observe that in the above definition, is the -set of the -functor . Given a cross-connection between two normal categories and , we define the set as:
[TABLE]
For a cross-connection from to , it can be shown that there is a unique dual cross-connection from to such that if and only . Then, for , there is a unique idempotent cone in such that and ; this cone is denoted by , in the sequel. Similarly denotes a unique idempotent cone in such that .
Given a cross-connection with dual , , and , the morphism is called the transpose of if and make the following diagram commute:
[TABLE]
It is worth noting that cross-connection transposes enjoy several properties of usual matrix transposes. For instance, and .
Definition 2.6**.**
Let and be two cross-connections. A morphism of cross-connections is a pair of inclusion preserving functors and satisfying the following axioms:
- (M1)
if , then and for all ,
[TABLE] 2. (M2)
if and is the transpose of , then is the transpose of .
Proposition 2.3** ([23, Section V.2.1]).**
The cross-connections with cross-connection morphisms form a category.
We denote by the category of cross-connections with cross-connection morphisms.
3. Inductive groupoid of a cross-connection
Recall that the aim of the present paper is to establish a category equivalence between the categories and . In this section, given a cross-connection , we identify the inductive groupoid associated with the cross-connection . Further, we prove that this correspondence is also functorial.
3.1. Biordered set of a cross-connection
First, observe that for an element , we can uniquely associate with it the pair of idempotent cones . By suitably defining the basic products and preorders [23], the set can be realised as the regular biordered set associated with the cross-connection .
Define two preorders and on as follows. For any two elements and in ,
[TABLE]
Also, we define basic products in as:
[TABLE]
Then as defined in (2) forms a regular biordered set with preorders and basic products as defined in (3) and (4) respectively. This biordered set shall serve as the set of objects of our required inductive groupoid .
3.2. Ordered groupoids of a cross-connection
Let . Consider an isomorphism . Then it has an inverse . Now let be the transpose of relative to , i.e., . Then by [23, Corollary IV.23] and properties of transposes, the morphism will be an isomorphism so that will be a pair of isomorphisms in . So, a morphism in the inductive groupoid from to is defined as a pair . This is well-defined by the uniqueness of inverses and transposes for a fixed pair of objects in .
Lemma 3.1**.**
* is a groupoid.*
Proof.
First, observe that given two morphisms from to and from to , then by the composition in and , we have and . So and by [23, Corollary IV.22],
[TABLE]
So, the composition is well-defined and associative. The morphism is the identity morphism at . Since is a pair of isomorphisms in , in . Hence is a groupoid. ∎
Now given a morphism from to and a morphism from to , define a relation on as follows:
[TABLE]
where is the inclusion from to and stands for the epimorphic component of a morphism in the normal category .
Lemma 3.2**.**
The relation is a partial order on .
Proof.
Clearly is reflexive.
Let and . Then and . So . Also since is an isomorphism, . Hence we have
[TABLE]
So, is is anti-symmetric.
Let and where is a morphism from to . Then clearly . Also,
[TABLE]
So , and thus is transitive. Hence is a partial order on . ∎
Observe that the partial order restricted to the identities of reduces to the natural partial order on the biordered set , and may be written as follows.
[TABLE]
Now define restrictions and corestrictions on as follows. Take a morphism from to . If , then is an inclusion in from to , and is an inclusion in from to . So, is a pair of monomorphisms in from to . Consider their epimorphic components, say
[TABLE]
Observe that and are isomorphisms, and . So, the restriction of to in is defined as the pair .
Similarly, if , then is an inclusion in from to , and is an inclusion in from to . So, is a pair of monomorphisms in from to . Consider their epimorphic components . These are isomorphisms with domain . Now take their inverses, say
[TABLE]
Then, the corestriction of to in is defined as the pair of isomorphisms with codomain .
Theorem 3.3**.**
* is an ordered groupoid with restrictions and corestrictions defined as above.*
Proof.
First, let be a morphism in from to , a morphism from to , a morphism from to and a morphism from to such that and .
Then and are morphisms in from to and to respectively, such that and .
We have
[TABLE]
Hence , and (OG1) is satisfied.
Now if is a morphism in from to and a morphism from to such that , then we need to show . Clearly, and .
Observe that using Proposition 2.2,
[TABLE]
and
[TABLE]
Hence , and similarly we show that . So,
[TABLE]
Hence , and (OG2) is satisfied.
Finally, if is a morphism in from to , and , then we define the restriction of to as . So and , and (OG3) is satisfied. Similarly, we can verify (OG3*).
Hence is an ordered groupoid. ∎
Now, since is a regular biordered set, we can build an ordered groupoid of the -chains of . But to that end, we need to discuss how we can compose two cones in a normal category.
Recall that for a cone in the category and an epimorphism , the map from to is a cone with apex . Hence, given two cones and , we can compose them as follows:
[TABLE]
where is the epimorphic component of the morphism .
Now, we define a partial order on the set of the -chains of . First, for an -chain and for with , let
[TABLE]
where and for , the pairs are such that
[TABLE]
and the cones in the right hand sides of (6a) and (6b) are composed as in (5).
Then for -chains as above and , define
[TABLE]
Now, it may be verified that forms an ordered groupoid with restriction .
Further, we define an evaluation map as follows. In the sequel, for convenience, we often denote the idempotent cones and by and , respectively. The object function of the evaluation functor is , and for an arbitrary -chain in , we let
[TABLE]
Before proceeding, we need to verify the following lemma.
Lemma 3.4**.**
If , then .
Proof.
First, observe that since and are idempotent cones, we have and . Since is an -chain, either or . If , then by (3), and so . Otherwise, if , then by [23, Proposition III.7], is an isomorphism. In either case, we see that is an isomorphism. Similarly, we can verify that is an isomorphism. So,
[TABLE]
Since each is an isomorphism, we conclude that the morphism is an isomorphism in . Similarly
[TABLE]
and so is an isomorphism in .
Now to show that , we need to show that
[TABLE]
But since , and , it suffices to show that
[TABLE]
Using [23, Page 94, Equation (25) version of Lemma IV.18], we have
[TABLE]
So it remains to verify that
[TABLE]
Observe that since is an -chain, and are idempotent cones such that or . If , then , so that
[TABLE]
Otherwise, if , then and
[TABLE]
That is,
[TABLE]
Equating the first equation at the apex and the second equation at the apex , we get
[TABLE]
That is, ; hence the lemma. ∎
Proposition 3.5**.**
* is an evaluation functor.*
Proof.
To prove the proposition, we need to show that is a -isomorphism.
First, we show that is a functor. Suppose are such that exists. Then
[TABLE]
Now, to show that is order preserving, it suffices to show that for an arbitrary in and ,
[TABLE]
In the sequel, we shall denote the idempotent cones , and by , and respectively. So according to our notations, and .
First observe that since or , either or . So,
[TABLE]
Similarly .
Also since , we have . Now,
[TABLE]
Similarly .
So,
[TABLE]
Hence is an evaluation functor. ∎
Remark 3.1*.*
Observe that in the proof of Theorem 3.3 that is an ordered groupoid, we have not used any ‘cross-connection’ properties. Hence using the same proof, one could show that the isomorphisms in the normal categories and , denoted by and respectively, form ordered groupoids. The one-sided subgroupoids of , i.e., the groupoids and (which are subgroupoids of and respectively) obtained by restricting to the categories and respectively also form ordered groupoids. Observe that restricting all the above described groupoids to the image of the evaluation functor also give rise to ordered groupoids. Hence given a cross-connection , we can associate with it several interesting ordered groupoids, which are all subgroupoids of .
3.3. The inductive groupoid
Before we proceed to prove that is an inductive groupoid, we need the following important lemma concerning the ordered groupoid which need not necessarily hold in its ordered subgroupoids. It describes the relationship between the retractions in the normal categories and the restrictions in the inductive groupoid.
Lemma 3.6**.**
Let be a morphism in from to and let be such that . If is the restriction with codomain , then
[TABLE]
Proof.
First, recall that, using [23, Proposition IV.24],
[TABLE]
Also, all the morphisms on the right hand side are retractions.
Now, since , we have
[TABLE]
Taking transposes we get
[TABLE]
Similarly we can prove that . Hence the lemma. ∎
Theorem 3.7**.**
* is an inductive groupoid.*
Proof.
We need to verify that satisfies the axioms of Definition 2.2. First, we need to verify (IG1) of Definition 2.2. To this end, consider an ordered groupoid and an evaluation functor . Let and for , let , be such that , and . Then we need to verify that , and
[TABLE]
The above condition can be illustrated by the commutativity of the solid arrows in Figure 2. Observe that Lemma 3.6 concerns with the commutativity of the square consisting of the elements , , and in Figure 2.
First, let be a morphism in from to such that for . Then the codomain is .
By Remark 3.1, the groupoid is an ordered groupoid; so if , then clearly . Also, by the partial binary composition of the biordered set,
[TABLE]
As the reader sees, here we relabel as . This is done for the sake of notational convenience as we want the indices of s and s to match the indices of in the subsequent proof. Similarly,
[TABLE]
Now,
[TABLE]
Also,
[TABLE]
Since , the right hand sides coincide and thus we have verified (IG1). Similarly, we can verify its dual.
Now we need to verify (IG2), that is, every singular -square is -commutative. Let \bigl{[}\begin{smallmatrix}(h_{1},k_{1})&(h_{1},k_{1})(c,d)\\ (h_{2},k_{2})&(h_{2},k_{2})(c,d)\end{smallmatrix}\bigr{]} be a column-singular -square such that and . Then,
[TABLE]
So,
[TABLE]
Also,
[TABLE]
So, the column-singular -square \bigl{[}\begin{smallmatrix}(h_{1},k_{1})&(h_{1},k_{1})(c,d)\\ (h_{2},k_{2})&(h_{2},k_{2})(c,d)\end{smallmatrix}\bigr{]} is -commutative. Dually, we can show that every row-singular -square is also -commutative. So (IG2) also holds.
Hence is an inductive groupoid. ∎
3.4. The functor
We have seen that given a cross-connection , it has a corresponding inductive groupoid . Now, we extend this correspondence to morphisms and also show that it is, in fact, functorial.
Given two cross-connections and , a morphism of cross-connections is a pair of inclusion preserving functors and satisfying the axioms of Definition 2.6.
Clearly is a functor from to and . So is also a functor.
Proposition 3.8**.**
* is an inductive functor from to .*
Proof.
First, by [23, Lemma V.4], is a regular bimorphism from to .
Recall from [23, Section V.2] that any inclusion preserving functor between two normal categories preserves normal factorisation and in particular it preserves epimorphic components. Now, since and are inclusion preserving,
[TABLE]
So, is order preserving.
Given , as earlier, we denote the idempotent cones and by and , respectively, and the cones and by and , respectively. Then,
[TABLE]
Hence the following diagram commutes:
[TABLE]
Thus is an inductive functor. ∎
Verification of the next result is routine.
Theorem 3.9**.**
The assignments
[TABLE]
is a functor .
4. Cross-connection from an inductive groupoid
Having constructed the inductive groupoid of a cross-connection, now we attempt the converse. Given the inductive groupoid with biordered set having preorders and , we construct a cross-connection . Unlike the previous case where it sufficed to identify the category sitting inside the cross-connection, here we have to ‘split’ the inductive groupoid and then ‘extend’ each part to the required normal category.
4.1. The normal category .
We begin with building the ‘left’ normal category associated to the inductive groupoid . The crucial property of a normal category that will guide us in this construction is that every morphism has a normal factorisation into a retraction, an isomorphism and an inclusion. So, we shall build three separate categories: one category ‘responsible’ for inclusions, the other one ‘responsible’ for isomorphisms, and the last one ‘responsible’ for retractions. Then we combine these categories to build our required category by extending the composition of the isomorphisms (which is inherited from the given inductive groupoid).
The major obstacle in this procedure arises from the fact that normal factorisation of a morphism is not unique. But fortunately, the epimorphic component (retraction + isomorphism) of a morphism is indeed unique. Exploiting this fact, we first build an intermediate category from the categories and , and then finally realise as a suitable product of and .
Given the inductive groupoid with regular biordered set , let the object set be , where . This gives a partially ordered set with respect to the order . In fact, forms a regular partially ordered set, in the sense of Grillet [14]. The proof of this statement may be found in [21]. Given , in the sequel, shall denote the canonical image of in . This set shall act as the object set of all our three categories: , and . That is, .
Let us begin by completing our first category . Recall that a partially ordered set corresponds naturally to a strict preorder category. In fact, our first required category is the preorder category associated with the partially ordered set . Hence, in , we introduce the formal symbol for a morphism from to whenever . So, given two morphisms and in , they are equal if and only if and . Given and , we compose them using the composition induced by the partial binary composition of the biordered set as follows:
[TABLE]
Observe that since , we have in . Now, the verification of the following proposition is routine.
Proposition 4.1**.**
* is a strict preorder category with the object set and the morphisms in as defined above.*
Now, we move onto our second required category, namely which shall be responsible for the isomorphisms in . Recall from Definition 2.2 that an inductive groupoid comes equipped with an evaluation functor , which helps you to ‘evaluate’ -chains of the ordered groupoid in the groupoid . Also, recall that the object set . Then, to define morphisms in the category , given any two morphisms in the inductive groupoid , we first define a relation as follows:
[TABLE]
Lemma 4.2**.**
* is an equivalence relation.*
Proof.
Clearly is reflexive.
Now if , then
[TABLE]
Observe that
[TABLE]
Hence multiplying the equation (8) by on the right and by on the left, we have
[TABLE]
That is, and so the relation is symmetric.
If and , then
[TABLE]
Observe that since and , we have
[TABLE]
So,
[TABLE]
So and the relation is transitive. Hence the lemma. ∎
Remark 4.1*.*
Observe that the relation reduces to the -relation on the identities of . Informally speaking, the relation (7) may be seen as a ‘left-sided’ version of the crucial -relation of [22, Section 4], which is defined later in this article: see equation (11).
We shall require the following observation in the sequel.
Proposition 4.3**.**
Let be a cross-connection with its inductive groupoid . For ,
[TABLE]
Proof.
Suppose that for . Let so that . Then, by (3), we have . Similarly we have . Further the last condition in (7) implies that:
[TABLE]
Since and , we have and . Thus, equating the first components gives us .
Conversely, suppose that . Since and , by (3), we have that and . Also, since and , we have that . Now observe that in the inductive groupoid , the morphisms and are transposes of and , respectively. But since we know that , and also that both the morphisms and are from to , by the uniqueness of transposes (for a given domain and codomain), we have that . Hence we have . ∎
Given a morphism in from to , we shall denote the -class of containing the morphism by . We shall define as a morphism in from to . Further, for such that , we define a composition in as
[TABLE]
Proposition 4.4**.**
* is a groupoid.*
Proof.
First, we verify that the composition is well-defined. Suppose and are such that and exist. Let and ; so we need to show that .
Since , we have that and since , we have . Also since , we get
[TABLE]
So,
[TABLE]
Hence and so the composition is well-defined. The associativity of the composition follows from the associativity of the composition in .
Now given from to , since
[TABLE]
is the identity morphism at the apex . Also,
[TABLE]
So and hence is a groupoid. ∎
Remark 4.2*.*
It can be shown that, in fact, is an ordered groupoid with respect to the order induced from the inductive groupoid.
Having built our required two categories, we move onto our third category . Recall again that . Now, if , for each in the biordered set such that and , we define a morphism in from to . Then two such morphisms and are equal if and only if and in . In that case, since , observe that is a basic product. In particular, if , then and so .
Further, if we have two morphisms and in such that , then we define a composition on as
[TABLE]
Since , so is a basic product in and by axiom (B2), . Since , using axiom (B4), we have . That implies . Combining this with the fact that , we have . Further, since , by transitivity, we have . Hence is a morphism in and the above composition is well-defined.
Proposition 4.5**.**
* is a category.*
Proof.
We need to verify associativity and identity. If , and are composable morphisms in , then observe that . Then,
[TABLE]
So it is associative. Also, since
[TABLE]
the identity morphism at in is . Hence is a category. ∎
We have successfully built all our three ingredient categories, namely , , and . Now we need to synthesise their composition to construct our required category from the inductive groupoid composition. As mentioned earlier, in this process, we shall rely heavily on the uniqueness of epimorphic components of morphisms in a normal category. To that end, we first build the category (that would be responsible for epimorphisms) using the categories and . For this, we need the following concept of a quiver.
Definition 4.1**.**
A quiver consists of a set of objects (denoted as ) together with a set of morphisms (denoted by itself) and two functions giving the domain and codomain of each morphism.
Now given categories and , consider an intermediary quiver with object set and morphisms as follows:
[TABLE]
Then, consider the following relation on the morphisms of the quiver : for any and ,
[TABLE]
The following lemma can be easily verified.
Lemma 4.6**.**
* is an equivalence relation.*
Now, we define the category whose object set is and whose morphisms are -classes of the quiver . We take the liberty of referring to as a category even though we have not yet defined any composition in ; if fact, heading to such a definition, we first need to specify a ‘good’ representative in each -class.
It is easy to see that for an arbitrary morphism , the morphism from the inductive groupoid satisfies and . From the latter property, we conclude that lies in the -class of . This allows us to represent each -class by a morphism of the form which we refer to as a right epi in the category . In the sequel, unless otherwise stated, a morphism in the category shall always be represented by its right epi. For brevity, whenever there is no scope of confusion, we shall denote the above right epi by just .
Observe that a right epi represents the following -class of morphisms in the quiver :
[TABLE]
We allow ourselves the notation , meaning, of course, the -class just shown.
So, given two right epis and in the category , they are related if and only if , and . So, if are such that and , then if and only if . Also if and only if and . In particular, if , we have .
As in [22, Section 4], consider the following relation on : for ,
[TABLE]
In contrast to (7), the definition of is ‘bilateral’; in [22, Section 4], it is verified that is an equivalence relation. As in [22], we shall denote the -class of containing by .
The following lemma which is crucial in the further considerations gives the relationship between the morphisms of the required normal category and the given inductive groupoid .
Lemma 4.7**.**
Let , be right epis in the category . Then if and only if and .
Proof.
Let , then , and . That is and . So .
Conversely, if and , then , and . Now since , we have . So, . Hence the lemma. ∎
Now we proceed to define a partial composition in the category as follows. Let and be right epis in the category such that and . If , then
[TABLE]
where for ,
[TABLE]
Observe that the sandwich set is well-defined as it depends only on the -class of and -class of . To verify that the composition is well-defined, we need to prove the following lemmas. The first lemma shows that the composition in (12) is independent of the representing element of the morphism in .
Lemma 4.8**.**
Let and be morphisms in the category such that and . Then for any fixed ,
[TABLE]
Proof.
Clearly and using [22, Lemma 4.7], we have . Hence the lemma follows from Lemma 4.7. ∎
We also need to show that (12) is independent of the choice of the sandwich element in the sandwich set.
Lemma 4.9**.**
Let such that . For ,
[TABLE]
Proof.
The lemma follows from [22, Lemma 4.8] and Lemma 4.7. ∎
The partial composition defined in the category may be illustrated using Figure 3 in the inductive groupoid . The solid arrows correspond to the relevant morphisms in the category .
Remark 4.3*.*
Observe that the partial composition of and in the category does not depend on the condition that . Hence this composition may be defined between any two morphisms in .
Proposition 4.10**.**
* is a category.*
Proof.
Here, given two morphisms and in the category , we shall prove well-definedness and associativity of the composition when . Clearly, the result also holds, in particular, when , i.e., when .
Let and be morphisms in the category such that and . Then for , by Lemma 4.8 and Lemma 4.9,
[TABLE]
Hence the composition is well-defined.
Now we need to verify identity and associativity. Given a morphism in from to ,
[TABLE]
So is the identity element at .
Also, if , and are composable morphisms in the category , then for and , by [22, Lemma 4.4 ], there exist and such that
[TABLE]
So,
[TABLE]
Thus associativity also holds. Hence is a category. ∎
Having prepared all the necessary ingredients, now we are in a position to define our category . Recall that and we define the morphisms in as follows:
[TABLE]
In the sequel, a morphism in shall be denoted as where and . Hence, by Lemma 4.7, the morphisms if and only if , and . So, if and in the biordered set , we have that if and only if and . In particular, in if and only if .
Further, given two morphisms in the category such that , we define a partial composition in as follows. For ,
[TABLE]
Proposition 4.11**.**
* is a category.*
Proof.
Well-definedness and associativity of the composition in easily follow from well-definedness and associativity of the composition in the category . (See Proposition 4.10 and Figure 3.)
Also, given a morphism in from to , we can see that
[TABLE]
and
[TABLE]
So is the identity element at . Hence is a category. ∎
Remark 4.4*.*
We know that . We can see that in fact all these categories are subcategories of by the following identification.
[TABLE]
Lemma 4.12**.**
* is a category with subobjects.*
Proof.
Clearly, is a small category and by Proposition 4.1, the category is a strict preorder subcategory of such that . Let be a morphism in . If , then . That is, . So by Lemma 4.7, we have and where is the -class of in (see (11)). Hence and so is a monomorphism.
Now if for and , then since , we have . Then , , and . So , i.e., . Hence is a category with subobjects. ∎
So, if , a morphism shall be an inclusion in the category . So, two inclusions and are equal if and only if and if and only if .
Lemma 4.13**.**
Every inclusion in splits.
Proof.
Let be an inclusion in . Then since , we have a retraction in such that
[TABLE]
Hence the lemma. ∎
Remark 4.5*.*
If , a morphism is a right inverse to the inclusion . Hence it will be a retraction in the category . Observe that two retractions and are equal if and only if and if and only if .
For , we can easily verify that the morphism is an isomorphism in . Observe that two isomorphisms and are equal if and only if and if and only if .
Also, given an arbitrary morphism in , we can easily see that the morphism is the epimorphic component of the morphism . The epimorphic component shall be denoted as just in the sequel.
Now we proceed to construct certain distinguished cones in the category . Given a morphism in the inductive groupoid , recall that is an isomorphism in from to . Also, for an arbitrary , observe that is the identity morphism at .
Recall from Remark 4.3 that we can compose any two epimorphisms in and similarly in the category . In particular, we can compose with , using the rule (12) for the corresponding right epis and , respectively. We get the following morphism in from to :
[TABLE]
where .
Fixing and making run over , we define a map as
[TABLE]
where and .
Remark 4.6*.*
Observe that for such that , we have by [22, Lemma 4.7]. So using Lemma 4.7, we have . That is, .
Lemma 4.14**.**
The map is a cone in .
Proof.
Since is independent of and , we can see that is a well-defined map. Now if , then and so we have an inclusion in . Then,
[TABLE]
Also since is an isomorphism in the category , we see that is a cone in . ∎
Figure 4 illustrates the cone in the category . The dashed arrow represents the morphism . The solid arrows give rise to the relevant morphisms in arising from the inductive groupoid. If , then is a retraction in from to . The morphism is an isomorphism in from to . So, is an isomorphism in from to . Also, is an inclusion from to . Composing these morphisms, we get a morphism from to in the category .
Similarly, since , we can build cones by taking and .
We shall need the following lemma in the sequel.
Lemma 4.15**.**
Let and be cones in the category as defined above. Then
[TABLE]
Proof.
Let be such that and , then for an arbitrary , using the composition (5) and for and , we have
[TABLE]
Then, using [22, Lemma 4.4] with , , , and , we have
[TABLE]
The last equality holds as and .
Combining the above arguments, for an arbitrary , we have
[TABLE]
Hence, . ∎
The above proof essentially shows how the role played by sandwich sets in inductive groupoid theory is captured by the cone multiplication (5) in cross-connection theory.
Remark 4.7*.*
Observe that if is a basic product in , then in the inductive groupoid , where . Hence using Remark 4.6 and Lemma 4.15, we have
[TABLE]
Hence for an element , the cone is an idempotent cone in .
Theorem 4.16**.**
* forms a normal category.*
Proof.
By Lemma 4.12, is a category with subobjects. Given an arbitrary morphism in ,
[TABLE]
is a normal factorisation such that is a retraction, is an isomorphism and is an inclusion. By Lemma 4.13, every inclusion in splits. Also, given an object , the map is an idempotent cone with apex . Hence the theorem. ∎
Remark 4.8*.*
By Remarks 3.1 and 4.5, we have several associated ‘one-sided’ ordered groupoids with a given cross-connection. The above theorem describes the normal category (a ‘one-sided’ category with a regular partially ordered set [14] as its object set) constructed from an inductive groupoid. So, it may be worthwhile to investigate if we can associate a normal category from a suitable ordered groupoid by assuming its object set to form a regular partially ordered set.
4.2. The normal category
Dually, given an inductive groupoid , we proceed to build a ‘right-hand side’ normal category via intermediary categories , and , as follows. One major difference from the construction of the category in the preceding subsection is that the morphisms in the ordered groupoid are induced from the inductive groupoid in the opposite direction (see below). We shall omit most of the details as the dual arguments of the construction of will suffice.
First, given the inductive groupoid with regular biordered set , consider the quotient set where . This gives a regular partially ordered set with respect to . We shall denote the -class of in by .
Now we aim to build three categories — , and — that all have as the object set, that is, . It remains to define the morphisms in each of these three categories.
We begin with which has a unique morphism, denoted , from to for each pair such that . Clearly, is a strict preorder category under the following composition:
[TABLE]
We proceed with defining the morphisms in . Given any two morphisms in the inductive groupoid , we first define an equivalence relation as follows:
[TABLE]
Given a morphism , we denote the -class of containing by and we treat as a morphism in from to ; observe the direction change! By the definition of , we have so that for every .
Further for such that , we define a composition in as
[TABLE]
so that will be a morphism in from to . It can be shown that is a groupoid under the above composition.
Finally, for the morphisms in , if , then for each such that and , we define a morphism in from to . If we have two morphisms and in such that , then we compose them as follows:
[TABLE]
so that forms a category.
Now, to build the category using the categories and , we consider an intermediary quiver with object set and morphisms as follows:
[TABLE]
Then, we define an equivalence relation on the set as follows: for , ,
[TABLE]
Now define the category with the object set and with morphisms being -classes of morphisms in . A morphism in such that is called a left epi in the category and shall be denoted by in the sequel. We use left epis as ‘good’ representatives of -classes: the left epi represents the -class
[TABLE]
Then we define a composition in as follows. Let and be left epis in the category . If , then we define
[TABLE]
where for ,
[TABLE]
Finally, we define the morphisms in as follows:
[TABLE]
We shall denote a morphism in by where and . Hence if and only if , and . So, if and in the biordered set , we have that if and only if and . Recall that here represents the -class of in (see (11)). In particular, in if and only if .
Further, given two morphisms in such that , we define a partial composition in as follows. For ,
[TABLE]
We can easily verify that is a category such that , , and are all subcategories of by the following identification.
[TABLE]
As in the case of , we can easily see that given a morphism , it is an inclusion, a morphism is an isomorphism and a morphism is a retraction in the category . Also, we can verify that forms a normal category with distinguished principal cones defined as follows. For a morphism in the inductive groupoid and for every ,
[TABLE]
where and .
4.3. The cross-connection of an inductive groupoid
Now, we proceed to construct the required cross-connection . Recall that the principal cone is defined as for each . The cone determines an -functor so that . Also recall that is the natural isomorphism between the -functor and the covariant hom-functor determined by the object . Now, define a functor as follows:
[TABLE]
for each and for each morphism . The natural transformation above may also be described by the following commutative diagram:
[TABLE]
Now, we proceed to show that is a cross-connection and for that, first we need to prove that is a local isomorphism. One can prove directly that the functor is a local isomorphism by working with the morphisms in the functor category , but it would involve a rather cumbersome argument and use of several undefined ideas. So, we take an alternate easier route. We shall realise the functor as a composition of two functors, namely a local isomorphism and an isomorphism ; thereby proving that is a local isomorphism. It is worth noting that a similar technique has been employed in [23, Section IV.1]. To this end, we shall require the following discussion about the structure arising from a given normal category. As the reader shall see, this is the only place where regular semigroups surface in our discussion.
Given a normal category , it can be seen that [23] the set of all the cones in with the special composition as defined in (5) forms a regular semigroup known as the semigroup of cones in and denoted by .
Also, given a regular semigroup , we can naturally associate with it two normal categories, each arising from the principal left and right ideals respectively (denoted as and respectively). An object of the category is a principal left ideal for , and a morphism from to is a partial right translation . Two morphisms and are equal in if and only if , , , , , and, consequently, . Dually, we can define .
Given an abstractly defined normal category with an associated regular semigroup , the relationship of the normal categories arising from the semigroup is described in the following theorem.
Theorem 4.17** ([23, Theorem III.19, Theorem III.25]).**
Let be a normal category with normal dual . The category is normal category isomorphic to the category and the category is normal category isomorphic to the normal dual .
So now, we first proceed to prove that defined as follows is a local isomorphism. For each and for each morphism ,
[TABLE]
To show that is a local isomorphism, we begin by verifying the following lemma.
Lemma 4.18**.**
* is a covariant functor.*
Proof.
We first need to verify that the is well-defined. If , then . Then for an arbitrary ,
[TABLE]
Hence, in the semigroup , we have . Similarly, we can show that . So, and so, the right ideal . Hence is well-defined.
Now, suppose that . That is, , and . Then, as shown above and . Also, for an arbitrary ,
[TABLE]
That is, in the semigroup , we have and so . Hence is well-defined. Also, if and are composable morphisms in the category , then
[TABLE]
Also,
[TABLE]
Hence,
[TABLE]
Moreover,
[TABLE]
Hence is a well-defined covariant functor. ∎
Before we proceed further, we prove the following lemma which relates the biordered set of an inductive groupoid with the biordered set of the semigroup of cones in the associated normal category .
Lemma 4.19**.**
The map given by is a regular bimorphism of biordered sets. Further, if with , then there exists such that and .
Proof.
First, suppose that . Suppose that , i.e., . Then, . So and . Similarly we can verify for the preorders , and . Hence (BM1) is satisfied.
Also, using Remark 4.7, we can see that
[TABLE]
Hence the condition (BM2) is also satisfied and is a bimorphism.
Now, if , then as above, we can easily verify that ; thus (RBM) is satisfied and so is a regular bimorphism from to .
The rest of the statement of the lemma directly follows from [22, Proposition 2.14]. ∎
Proposition 4.20**.**
The functor is a local isomorphism.
Proof.
Lemma 4.18 shows that is a well-defined functor. To show that is a local isomorphism, we need to show that is inclusion preserving, fully faithful and for each , is an isomorphism of the ideal onto . First observe that for an inclusion , we have . Then by dual of [23, Proposition IV.13(d)], the morphism is an inclusion in and so is inclusion preserving.
Now we proceed to show that is fully-faithful. Suppose that and are two morphisms in from to such that . Then, , that is, . In particular, for , we have . That is, . But since , we have , i.e., . Also since , we have . Thus we see that . This implies that, in particular, . So, we have and hence is a faithful functor.
To show that is full, let be an arbitrary morphism in such that . Then and so, and . Observe that is an epimorphism in and so for some such that . So,
[TABLE]
So, and is full. Hence is fully-faithful.
Now, to complete the proposition, we need to show that for any , the functor is an isomorphism. Since is already shown to be fully-faithful, it suffices to show that the vertex map is an order isomorphism. To this end, suppose that such that . So, we have , i.e., , in the biordered set . Then in the biordered set , we have such that . So by Lemma 4.19, there exists such that and . Then for and using Remark 4.7, we have
[TABLE]
Evaluating the apices of the cones, we get . Also since , we have that , i.e., . But since also, we have . Then,
[TABLE]
So by Lemma 4.7, we see that . That is, and since , we have . Thus, and so is injective on .
Now, to prove is surjective on , let us suppose that such that . Then and so . Without any loss in generality, we can assume that and then we have . If , then and so . Since , we can assume that . Now since implies , using [23, Lemma III.3], there is a unique epimorphism such that . Then,
[TABLE]
Observe that is an inclusion and so is a retraction in of the form for some , i.e., . Then, since is an epimorphism and using (5),
[TABLE]
Thus with and so is surjective on . This also implies that if . Hence the object map is an order isomorphism on . Thus, is a local isomorphism. ∎
Theorem 4.21**.**
The triple is a cross-connection.
Proof.
First, recall from Theorem 4.17 that for a normal category , the normal category is isomorphic to the normal dual . For the case when , an isomorphism can be defined as follows. For each and for each morphism , we set
[TABLE]
Here , as defined in [23, Lemma III.22].
Now we make use of the local isomorphism as defined in (19). For a morphism , we have by definition. Since , we have
[TABLE]
We see that the functor defined in (18) is equal to the composition of the local isomorphism and the isomorphism . Hence the functor is a local isomorphism from to .
Further, for every , we have . Then the component is an isomorphism, so that . Thus is a cross-connection from to . ∎
Dually, we can show that defined by the functor as follows
[TABLE]
is a cross-connection.
We can now see that the biordered set of the cross-connection is given by the set
[TABLE]
Here the element corresponds to the pair of cones . Further if we define the preorders and on as follows:
[TABLE]
then forms a regular biordered set and it is biorder isomorphic to the biordered set of the inductive groupoid (also see [21]). Moreover, it can be easily verified that given , the morphism will be the transpose of the morphism , relative to the cross-connection .
4.4. The functor
Having built the cross-connection associated with an inductive groupoid , now we proceed to show that this correspondence is, in fact, functorial.
Suppose that and are inductive groupoids with biordered sets and , respectively. Let their associated cross-connections be and , respectively. If is an inductive functor between the inductive groupoids and , then define two functors and as follows:
[TABLE]
[TABLE]
Proposition 4.22**.**
The pair is a cross-connection morphism from to .
Proof.
Since is an inductive functor, we can easily verify that and are well-defined functors. Since is order preserving, and are inclusion preserving.
For , if , then since is a bimorphism, we have and so . Then, for an arbitrary ,
[TABLE]
Hence the pair of functors satisfies the condition (M1).
Suppose that such that the morphism is the transpose of the morphism . Then,
[TABLE]
So, the pair of functors satisfies the condition (M2) also and hence is a morphism of cross-connections. ∎
The proof of the following theorem is routine with all the preparations we have made so far.
Theorem 4.23**.**
The assignments
[TABLE]
is a functor .
5. The category equivalence
Having built the functors and the functor in the previous sections, now we proceed to prove the category eqivalence between the categories and . To this end, we need to show that the functor is naturally isomorphic to the functor and the functor is naturally isomorphic to the functor .
We first show that . Suppose that is a cross-connection with an associated inductive groupoid . Recall that for any two elements and in the biordered set ,
[TABLE]
That is, and so for an arbitrary , the canonical image . Similarly, we have .
Also, for a morphism in the groupoid , by Proposition 4.3, we have . So, the corresponding morphism in the category can be represented by just . Similarly, we have .
So, if is a a cross-connection with an associated inductive groupoid and the cross-connection associated with the inductive groupoid is given by , then the functors defined as follows constitute a cross-connection isomorphism. The functors and are given by
[TABLE]
[TABLE]
Hence, for each cross-connection , we can easily see that the assignment
[TABLE]
will be a natural isomorphism between the functor and the functor . That is, for an arbitrary cross-connection morphism , the following diagram commutes:
[TABLE]
Now, conversely suppose is an inductive groupoid such that , then define the functor as follows:
[TABLE]
We can easily verify that is an inductive isomorphism. Further, for an inductive functor , the assignment
[TABLE]
makes the following diagram commute:
[TABLE]
Hence, the functor is naturally isomorphic with the functor .
Summarising the discussion in this section, we have the following theorem which is the main result of this paper.
Theorem 5.1**.**
The category is equivalent to the category .
6. Conclusion
We have described the inductive groupoid associated with a cross-connection, and conversely, we have built the cross-connection associated with an inductive groupoid. We emphasise once again that both constructions have been accomplished within a purely category theoretic framework, free of any semigroup theoretic assumptions, under which the concept of an inductive groupoid and that of a cross-connection were initially conceived almost half a century ago.
Because of certain historical circumstances555See [7, Section 1] for a discussion of these circumstances., the category was much less known than the category . The recent studies by the first author and Rajan [5, 3, 2, 4] have shown that admits worthwhile applications within semigroup theory. We anticipate that our present result may serve as a starting point for looking for a wider spectrum of applications. As a concrete (though, most probably, difficult) task, one can consider developing an abelian version of the theory of cross-connections, aiming at a new categorical framework for the class of von Neumann regular rings. In a sense, this would bring the theory back to its initial origin — see our discussion in the introduction.
Another possible development consists in involving categories enhanced with an appropriate topology. Many natural groupoid-based structures like pseudogroups of transformations and étale groupoids [28] come with inbuilt topology by the very definition, and a topological version of the theory of inductive groupoids has already been considered by Rajan [26]. This suggests that topological variants of the theory of cross-connections should be possible and might be relevant.
A further direction is associated with the generalisations of the categorical structures involved. Several generalisations of inductive groupoids have been proposed and studied in the recent years (see [1, 16, 36, 35], for instance). On the other side, the cross-connections of consistent categories (which are generalisations of normal categories) have been considered in [6]. It may be worthwhile to explore the inter-connections between the categorical structures arising from these generalisations. In particular, our results can guide in the characterisation of the cross-connection constructions corresponding to the aforementioned generalisations of inductive groupoids. In this regard, as the anonymous referee has pointed out, “the cross-connections might prove a useful tool for non-regular semigroups and rings, where inductive groupoids and their generalisations fail, since the latter rely on idempotents whereas the former (in theory at least) do not”.
Acknowledgements
The authors thank the referee for their very meticulous reading of the manuscript, generous comments and several excellent suggestions which have helped improve the readability of the paper.
We acknowledge the financial support by the Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2021-1387).
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