Order-theoretic introduction of categories

Ordinary ordered algebraic structures like quantales, residuated lattices, ordered groups, ordered fields, etc add additional algebraic structure to the objects of a preorder. Categories, on the other hand, are different in that they add algebraic structure to the edges of a preorder instead. In a category, the edges of the preorder are mapped to sets of morphisms by the hom set function, and it is these morphisms that are given additional algebraic structure. Categories are therefore a different type of preordered algebraic structure, and by changing the units of ordering, they present a different outlook on order relations and therefore potentially all of mathematics.

The essence of categories:

Input/output ordering:

As categories are such a general concept, we will first comment on the differences between functions and relations. In the theory of relations, there is not necessarily a preferred input for the relation, whereas functions are characterized by specific inputs and outputs. While this distinguishes functions from relations, it is also the source of their utility. Functions can be used as basic units of ordering, to describe any ordered process by which one thing leads to a next thing, as denoted by the arrow notation. \[ f: I \to O \]

Units of ordering and categories

If we consider the general lattice of preorders, then the join irreducible elements are always either single elements or ordered pairs. Starting with these simple components, we can build up any preordering relation. In the case of categories, these same units of ordering are enhanced with additional algebraic structure. This leads to ordered algebraic units which are also called morphisms.

• Ordered units
• Algebraic ordered units

Morphism properties:

Once we have motivated the introduction of morphisms, we can then describe a hierarchy of properties of morphisms using a suborder of the lattice of apartnenss relations. Generally speaking, the identity of a morphism is not determined by its input/output pair unless the category is a preposetal category, as it is different morphisms in hom classes that give categories their algebraic distinctiveness.

Ordered pairs of morphisms:

Once we have got past the basic properties of morphisms like their input/output pair, their input object, and their output object then we can consider ordered pairs of morphisms. This naturally leads to a tree of properties descended from the first and second morphisms, but there are two other properties that mix the two objects that are of relevance in category theory: the inner objects and the outer objects. This produces the following partial order of properties:

The inner objects and the outer objects of an ordered pair of morphisms are relevant to the definition of composition. The other properties follow from the properties of morphisms and ordered pairs.

• Inner objects: the domain of composition of a category consists of all inner-equal ordered pairs of morphisms. We can denote this domain of composition $M_{*}^2$.
• Outer objects: the input-output pair of composition in a category is necessarily equal to the outer objects pair

These two conditions: that inner objects coincide and that the outer objects form a new input-output pair, make it so that composition generalizes transitivity. Indeed, the entire purpose of category theory is to examine algebraic generalizations of transitivity.

An equivalent definition:

We can now define categories using more order-theoretic terminology then done previously, as presented below. Equivalent definition. let $O$ be a class of objects, $M$ be a class of morphisms, $R$ a binary relation on $O$, and $M_{*}^2$ be the class of inner-equal pairs of morphisms, then we can define a pairing function $p$ and an associative partial binary operation $\circ$ of composition: \[ p : M \to R \] \[ \circ: M_{*}^2 \to M \] Then if the following two conditions are satisified (1) generalized reflexivity: for every object $X$ there is an identity endomorphism $1_X$ such that $f \circ 1_x = f$ and $1_x \circ g = g$ for all $f$ that end at $X$ and all $g$ that begin at $X$ and (2) generalized transitivity: the input-output pair of the composition $M_{*}^2$ consists of the outer pair of objects then the structure forms a category. The generalized reflexivity and transitivity conditions clearly make $R$ a preorder, which we can now all call the morphic preorder of the category.

Elementary constructions:

Objects:

As objects are one of the basic elements of any category, we can naturally form a partially ordered set of classes of them. The main elementary classes of objects, used in abstract category are initial and terminal objects. Taking an order-theoretic foundation, these are generalisations of minimal and maximal elements of order-theory with the extra condition that hom classes have only one element.

Morphisms:

Each morphism produces a left-action and a right-action on morphisms by composition (the two are distinguished because categories are non-commutative). A morphism is an epimorphism if its left-action is invertible and its a right action if its right-action is invertible. We can therefore produce preorders on common output morphism systems and common input morphism systems for each object by how epi/mono they are based upon degrees of reversibility. A morphism is split if it not only is invertible, it has an inverse. The other classes of morphisms follow producing the following partial ordered set of classes.

Object systems:

Several classes of collections of morphisms can be introduced based upon the morphism preordering and the symmetric preordering of isomorphism. Morphism lower sets, upper sets, principal ideals, and principal filters can be defined based upon the natural morphic preordering of any category. This is another way to describe the morphic preordering, only instead using a more set-theoretic approach.

Morphism systems:

Morphism systems are the basic objects of most of category-theory. Hom classes appear here as the inverse images of the function that maps any morphism to its ordered pair, and they are non-empty for any ordered pair in the morphic preordering. In other words, there is a mapping $f : R \to Hom$ that maps each ordered pair to a non-empty set of morphisms which are generalizations of the edges of the preorder. Other types of morphism systems include common input and common output families of morphisms, of which parallel morphism systems are an intersection. Common output systems are used in the definition of sites, for example. Hom classes are maximal parallel families of morphisms.

While we demonstrated how objects in a category are naturally preordered, it is not hard to see how the morphisms in a category are preordered as well. Algebraic preorders, introduced naturally in monoid, can equally be applied to categories to get a preordering of morphisms based upon reachability. In the case of single-object categories this of course naturally coincides with the monoid definition. Algebraic lower sets, upper sets, and principal ideals and filters in categories naturally arise as classes of morphism systems from the preordering on morphisms.

The class of composition closed morphisms is directly related to the lattice of subcategories, because the axioms of categories make it so that identity morphisms and objects coincide with one another. To some extent then, the case of subcategories can be reduced to composition closed morphism systems. Another class of morphisms that plays a central role in category-related applications is morphism sequences, as these are used to define exact sequences and chain complexes which are central objects of study in homology.

Object properties:

We previously described properties of morphisms, naturally therefore we would be amiss if we didn't talk about properties of objects. The main property of objects in any category is the isomorphism type, which is a part of the isomorphism identity. Except in the case of skeletal categories in which the two equivalence classes coincide, rather then one being a part of the other. The isomorphism type of an object is merely one part of a lattice of properties, where parts of the isomorphism type in the lattice are typically called invariants.

For example, the isomorphism type of the fundamental group is a a part of the homotopy type, which is a part of the isomorphism type of a topology, which is in turn part of the topological identity. Parts are defined by the lattice of apartness relations in the natural manner. Another example is the algebraic preorder type of a commutative aperiodic semigroup which is an invariant, we can use this to define a preorder on isomorphism types of algebraic preorder isomorphism type-equal classes of semigroups, which is useful in the study of commutative aperiodic semigroups.

Universal algebraic constructions:

Lattice of subcategories:

Let $C$ be a category, then we can naturally form a lattice of subcategories $Sub(C)$. This contains all categories on subsets of objects and morphisms, such that units and composition are preserved. As mentioned previously, $Sub(C)$ is essentially the same as a lattice as to the family of unital composition-closed morphism systems of a category, because objects coincide with initial morphisms. The first thing we can notice about $Sub(C)$ is that the morphic preordering produces a monotone function (a morphism in the category of partial orders) between itself and $Sub(C)$.

Proposition. the mapping $f : Sub(C) \to (F,\subseteq)$ where $(F,\subseteq)$ is the principal ideal in the lattice of preorders of the morphic preordering of $C$ forms a monotone map.

Proof. in any subcategory the collection of morphisms of the subcategory is a subclass of its supercategory. Therefore, by the monotonicity of images, the image of the pairing function on the supercategory, which is the morphic preordering, must be larger then the image of the pairing function on the subcategory. Therefore, the morphic preordering morphism is a monotone morphism of preorders.

Proposition. for a thin category $C$ the lattice of subcategories $Sub(C)$ is isomorphic to the lattice of subpreorders of the morphic preorder on objects of $C$.

Proof. This follows directly from the correspondence between generalized reflexivity and generalized transitivity in thin categories and the reflexivity and transitivity conditions in preorders. This shows that lattices of preorders emerges out of category theory.

Proposition. thin categories are subalgebra-closed.

Proof. Let $C$ be a thin category then for all $a,b \in O$ we have $|Hom_C(a,b)| \le 1$. Let $S$ be a subcategory of $C$ then it has a class $N$ of morphisms which is a subclass of the class $M$ of morphisms of $C$ and any two objects $a,b$ in $S$ are also in $C$ therefore $|Hom_S(a,b)| \le |Hom_C(a,b)| \le 1$.

Corollary. the subalgebra system of all thin subcategories of a category forms a lower class of $Sub(C)$.

Lattice of congruences:

Congruences on categories are defined by refinements of the hom equivalence relation, that satisfies the standard input/output relationship used to define congruences. If a congruence is denoted $E$ then a quotient category can be defined by $\frac{C}{E}$ by reducing class of morphisms to individual morphisms. Based upon this definition, the quotient $\frac{C}{Hom}$ is clearly equal to the morphic preordering. It is in this sense that we can say that categories are merely algebraic generalisations of the transitivity relationship at the basis of order theory.

Higher category theory:

Treated as a branch of universal algebra, categories can be applied to any algebraic structure including themselves. In particular, category theorists have established a total ordering on classes of categories based upon how their height, so that categories themselves can be ordered based upon how nested they are. As a generalization of order theory, there are two types of functors between categories: covariant functors and contravariant functors corresponding to monotone and antitone maps. The opposite category for example is a contravariant functor, which generalizes the dual ordering concept in order theory.

Products and coproducts:

As categories are a preordered algebraic structure, we can define morphic lower bounds and upper bounds in any category. A natural question is how we can generalize joins and meets to an arbitrary category. This requires two preliminaries:

1. If $a,b$ are two objects, then for the meet $a \vee b$ to be the greatest lower bound then for any other lower bound $L$ there must be a weak suborder of the form $[\{L\}, \{a \vee b\}, \{a, b\}$]. The same applies for joins in the other direction.
2. The commutative diagrams in category theory correspond to preposetal subcategories, and therefore commutative diagrams are entirely order-theoretic construction.

The definition of a product in a category is that for any other lower bound $L$ other then the product then there must be a unique weak-ordered preposetal subcategory of the form $[\{L\}, \{a \times b\}, \{a,b\} ]$ (and therefore there must be some unique morphism which makes this subcategory preposetal, that is for it to commute). Coproducts are defined dually. In other words, products and coproducts are generalizatoins of meets/joins:

• Product $\leftrightarrow$ Meet
• Coproduct $\leftrightarrow$ Join

The arithmetical properties of semilattices defined by commutative group-free semigroups are well-understood. Generally speaking, most semilattices are not multiplicative, and semilattices exist with arbitrary arithmetic properties. The one semilattice that describes products is the well known meet operation in the lattice of equivalence relations. It follows that products and coproducts in category theory, are abstract general concepts that describe operations that don't necessarily have anything to do with products.

The origin of the term product is because the category-theoretic product in the category of sets is the direct product. But even in the category of sets, the coproduct is the disjoint union which is an additive operation. Indeed, we define measures by a generalization of the additive property of the disjoint union. Products/coproducts are better thought of as general classes of operations which generalize meet/joins of lattice theory.

Categories with additional structure:

So far we have only dealt with abstract categories with no additional structure on them. Firstly, recall that the concept of having additional structure is an order-theoretic concept explained by lattices of equivalence relations and their dual. We can preorder objects by how much mathematical structure they have, in this sense categories are extensions of preorders as we have described. But there is no reason to stop our exploration of this preorder at ordinary categories.

The first thing worth mentioning is concrete categories, technically concrete categories are categories with additional structure, where that additional structure is a functor to the category of sets. This makes the morphisms in the category behave like functions and the objects behave like sets. These are probably the most important class of categories in most applications.

Another direction is adding additional structure to the hom classes of a category, which leads to preadditive categories, quantaloids, etc. Preadditive categories have a commutative group over each hom class, while quantaloids have a complete lattice subjected to additional axioms. Abelian categories are a special case that are probably the most important in most applications. Clearly, we can build a hierarchy of concepts by how much structure they have.