2. Categories with subobjects

We assume that the reader is familiar with the basic concepts such as categories, functors, natural transformations, etc. [see for example MacLane, 1971]. For the more specialized concepts like categories with subobjects and related ideas, we follow Nambooripad [1994]. We begin by reviewing briefly the definition of categories with subobjects.

2.1. Subobjects

Recall that two morphisms $f,g\in \mathcal{𝒫}$ are parallel if $domf=domg$ and $codf=codg$. A preorder $\mathcal{𝒫}$ is a category such that for all $f,g\in \mathcal{𝒫}$, $f$ is parallel to $g$ if and only if $f=g$. Equivalently, $\mathcal{𝒫}$ is a preorder if and only if $\mathcal{𝒫}\left(p,q\right)$ contain atmost one element for all $p,q\in \mathfrak{𝖛}\mathcal{𝒫}$. It follows that the relation $\preccurlyeq$, defined for all $p,q\in \mathfrak{𝖛}\mathcal{𝒫}$, by

is a quasi-order on the class $\mathfrak{𝖛}\mathcal{𝒫}$. In particular, if $\mathcal{𝒫}$ is small, then $\left(\mathfrak{𝖛}\mathcal{𝒫},\preccurlyeq \right)$ is a quasi-ordered set. Conversely, it is clear that any quasi-ordered set $\Lambda$ uniquely determines a small preorder $\mathcal{𝒫}$ such that $\Lambda =\mathfrak{𝖛}\mathcal{𝒫}$. The preorder $\mathcal{𝒫}$ is said to be strict if the quasi-order relation $\preccurlyeq$ above is a partial order; that is, $\mathcal{𝒫}$ has the property that for all $p,q\in \mathfrak{𝖛}\mathcal{𝒫}$,

The concepts of a small strict preorder and a partially ordered set are equivalent and will be used interchangeably in the sequel.

Recall E. Krishnan [2000] that a choice of subobject in a category $\mathcal{𝒞}$ is sub-preorder $\mathcal{𝒫}\subseteq \mathcal{𝒞}$ satisfying the conditions:

(1)

$\mathcal{𝒫}$ is a strict preorder with $\mathfrak{𝖛}\mathcal{𝒫}=\mathfrak{𝖛}\mathcal{𝒞}$.

(2)

Every $f\in \mathcal{𝒫}$ is a monomorphism in $\mathcal{𝒞}$.

(3)

If $f,g\in \mathcal{𝒫}$ and if $f=hg$ for some $h\in \mathcal{𝒞}$, then $h\in \mathcal{𝒫}$.

If $\mathcal{𝒫}$ is a choice of subobjects in $\mathcal{𝒞}$, the pair $\left(\mathcal{𝒞},\mathcal{𝒫}\right)$ is called a category with subobjects. For brevity, we shall say that a category $\mathcal{𝒞}$ has subobjects if a choice of subobjects in $\mathcal{𝒞}$ has been specified. Also, we shall use the notation $\mathfrak{𝖛}\mathcal{𝒞}$ to denote the preorder of subobjects as well as the partially ordered set of vertexes in $\mathcal{𝒞}$. No ambiguity will arise since both these are equivalent. We may then use the usual notation $\subseteq$ (see E. Krishnan [2000]) to denote subobject relation in $\mathcal{𝒞}$. If $c\subseteq d$, the unique morphism in $\mathfrak{𝖛}\mathcal{𝒞}$, the inclusion of $c$ in $d$, is denoted by $ȷdc$. If $f\in \mathcal{𝒞}\left(c,d\right)$ and $c\prime \subseteq c$, then we write

As usual, $f\mid c\prime$ is called the restriction of $f$ to $c\prime$.

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

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 $Ab$, etc., are categories with natural subobject relations.

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

is a functor of the category $Cato$ to the category of preorders (or the category of partially ordered sets).

2.2. Categories with factorization

Recall that two monomorphisms $f,g\in \mathcal{𝒞}$ are equivalent if there exists $h,k\in \mathcal{𝒞}$ with $f=hg$ and $g=kf$ (see Nambooripad [1994], Equation I.4(15)). The fact that $f$ and $g$ are monomorphisms imply that the morphism $h$ is an isomorphism and $k=h^{-1}$. A monomorphism $f\in \mathcal{𝒞}\left(c,d\right)$ is called an embedding if $f$ is equivalent to an inclusion. Clearly, every inclusion is an embedding. It is important to notice that no two inclusions can be equivalent as mopnomorphisms [E. Krishnan, 2000, see Lemma 1.10].

Eqivalence of epimorphisms are defined dually. As above, one can see that two epimorphisms $f,g\in \mathcal{𝒞}$ are equivalent if and only if there is an isomorphism $h\in \mathcal{𝒞}$ with $g=fh$.

Recall also that a monomorphism $f:c\to d$ [epimorphism $h:c\to d$] splits if there exists $g:d\to c$ [$k:d\to c$] such that $fg=1_c$ [$kh=1_d$]. Notice that when $f$ is a split monomorphism, then $g$ is a split epimorphism and similarly if $h$ is a split epi, $k$ is a split mono. If an inclusion $ȷdc$ is said to be split if it is split as a monomorphism; that is, there exist $𝜖\in \mathcal{𝒞}$ with $ȷdc𝜖=1_c$. In this case, $𝜖$ is called a retraction which is clearly a split epimorphism.

A category $\mathcal{𝒞}$ with subobjects is said to have factorization property if every morphism $f\in \mathcal{𝒞}$ can be factorized as $f=kh$ where $k$ is an epimorphism and $h$ is an embedding. In this case we can choose $k$ and $h$ so that $h$ is an inclusion (see E. Krishnan [2000], § 3.2.2). A factorization of the form $f=qj$ where $q$ is an epimorphism and $j$ is an inclusion, is called a canonical factorization. Thus $\mathcal{𝒞}$ has factorization property if and only if every morphism has canonical factorization. $\mathcal{𝒞}$ is said to have unique factorization if every morphism in $\mathcal{𝒞}$ has unique canonical factorization.

Let $f$ be a morphism in the categopry $\mathcal{𝒞}$ with factorization. A canonical factorization $f=qj$ of $f$ is called an image factorization if $f=qj$ has the following universal property: if $f=q\prime j\prime$ is any canonical factorization of $f$, there is an inclusion $j\prime \prime$ with $q\prime =qj\prime \prime$ (see E. Krishnan [2000], § 3.2.3). The factorization $f=qj$ with the universal property above, if it exists, is unique. The epimorphism $q=f^{\circ }$ is called the epimorphic component of $f$. If $j$ is the inclusion such that $f=f^{\circ }j$ then the vertex

is called the image of $f$. Also the image facorizatrion of $f$ is the unique canonical factorzation

A morphism $f$ such that $f^{\circ }=f$ is said to be surjective. Note that any surjection is an epimorphism but not conversely. Also, the universal property of the image-factorization implies that, the epimorphic component $f^{\circ }$ of a morphism $f$ is surjective; that is, ${\left(f^{\circ }\right)}^{\circ }=f^{\circ }$.

The category $\mathcal{𝒞}$ is said to have images if every morphism in $\mathcal{𝒞}$ has image.

If $\mathcal{𝒞}$ has images, and $f\in \mathcal{𝒞}$ then, as usual, we may write

This defines a mapping

of the principal order ideal of the partially ordered set $P=\mathfrak{𝖛}\mathcal{𝒞}$ of subobjects of $domf$ to the partially ordered set of subobjects of $codf$. It is easy to see that the map $f\mathfrak{𝔄}$ is order-preserving. Moreover, for $f\in \mathcal{𝒞}$ and $x\subseteq codf$ we write

If the vertex $f^{-1}\left(x\right)$ exists it is called the inverse image or preimage of $x$. Observe that, for all $x\subseteq codf$, the set

is an order ideal in the partially ordered set $P=\mathfrak{𝖛}\mathcal{𝒞}$ and the preimage of $x$ exists if and only if this ideal is non-empty and principal.