Category Theory/Categories

< Category Theory

This is the Categories chapter of Category Theory.

Contents

DefinitionEdit

A category   consists of four kinds of data subject to three axioms, as listed below:

DataEdit

Objects
  has objects denoted by  ,  ,  ,…
Morphisms
For each ordered pair of objects  ,   in  , there is a class of morphisms or arrows from   to  . The notation   means that   is a morphism from   to  . The class of all morphisms from   to   is denoted by   or sometimes simply  .
Composition
For each ordered triple of objects  ,  ,   in  , there is a law of composition: If   and  , then the composite of   and   is a morphism  
Identity
For each object   there is a designated identity morphism on  , notated as  , from   to  .

AxiomsEdit

These data satisfy the following three axioms, of which the first is in the nature of a convention, while the remaining two are more substantial:

Unique typing
  and   are disjoint unless  ,  .
Associative Law
  if the composites are defined. Note that if one composite is defined, the other is necessarily defined.
Identity is a “neutral element”
For the identity morphism   associated to each object  , two equations must hold for each pair of objects   and   and each pair of arrows  ,  :
  •  
  •  

Terminology and fine pointsEdit

  • If   in a category,   is called the domain or source of  , and   is called the codomain or target of  .
  •   is called a hom class or a hom set if it is indeed a set. In general a hom set may be empty, but for any object  ,   is not empty because it contains the identity morphism.
  • The hom class   may be denoted by   or   if it is necessary to specify which category is referred to.
  • An object in a category need not be a set; the object need not have anything called elements.
  • Morphisms may also be called maps. This does not mean that every morphism in any category is a set function (see #Baby examples and #Preorders). Arrow is a less misleading name.
  • The composite   may be written  .
  • It might be more natural to write the composite of   and   as   instead of   but the usage given here is by far the most common. This stems from the fact that if the arrows are set functions and  , then  . Thus   is best read as "do   after  ".

Large and smallEdit

The definition says that a category 'has' objects and 'has' morphisms. This means that for any category   and   is any mathematical object, the statement '  is an object of  ' is either true or false, and similarly for the statement '  is a morphism of  '. The objects (or arrows) of a category need not constitute a set. If they do, the category is said to be small. If they don't, the category is large.

The requirement that the collection of morphisms from   to   be a set makes a category locally small. In this book, all categories are locally small.

DiscreteEdit

A category is discrete if every morphism is an identity.

PreorderEdit

A category   is a preorder if for every pair of objects  , there exists at most one morphism  .

Examples of categoriesEdit

Baby examplesEdit

These examples are trivial and maybe uninteresting. But do not underestimate the power of baby examples. For one thing, they are sometimes counterexamples to possible theorems.

0 (the empty category)
This category has no objects and no morphisms.
1
The category 1 has one object and one morphism, which must necessarily be the object's identity arrow.
1+1
This category has two objects and two morphisms: the identities on each object.
2
This category has two objects and three morphisms. The third morphism goes from one object to the other.
Remarks
  • The objects of these baby categories are nodes in a graph (not sets) and the morphisms are arrows in the graph (not functions).
  • For these baby categories we don't have to say what the composition operation does: it is always forced.
  • It is impolite to say that categorists think that 1 + 1 is not equal to 2.

The category of setsEdit

The category of sets, denoted by Set, is this category:

  • The objects are all sets
  • A morphism from a set   to a set   is a function with domain   and codomain  .
  • The composition is the usual composition: If   and   then   is defined by   for all  .
  • The identity morphism   on a set   is the identity function defined by   for  .
Terminology and fine points
  • In order to preserve the unique typing in a function definition, it is necessary to include its codomain. For example,   is a different function from the inclusion function   to some set   properly including  .
  • In most approaches to the foundations of math, the collection of all sets is not a set. This makes Set a large category. However, it is still locally small since the class of all functions between two sets,   and  , is a subclass of their Cartesian product   which is a set.

Mathematical structures as categoriesEdit

PreordersEdit

A preorder   on a set   is a reflexive and transitive relation on  , which means that for all  ,   and for all  ,  ,   in  , if   and  , then  .

A preorder "is" a category in the following sense: Given a preorder ( ,  ) the category structure is this:

  • The objects of the category are the elements of  .
  • There is exactly one morphism from   to   if and only if  .

The existence of identities is forced by reflexivity and the composition law is forced by transitivity. It follows that the category structure has the property that there is at most one morphism from any object   to any object  .

Conversely, suppose you have a category with set   of objects, with the property that there is at most one morphism between any two objects. Define a relation   on   by requiring that   if and only if there is a morphism from   to  . Then ( ,  ) is a preorder.

The statements in the two preceding paragraphs describe an equivalence of categories between the category of small categories with at most one morphism between any two objects and all functors between such categories, and the category of preorders and order-preserving maps.

Remark: Given a preorder, the morphisms of the corresponding category exist by definition. There is exactly one morphism from   to   if and only if  . This is an axiomatic definition; in a model a morphism from   to   could be anything, for example the pair ( ,  ). In no sense is the morphism required to be a function.

GroupsEdit

Every group   can be viewed as a category   as follows:   has one single object; call it  . Therefore it has only one homset  , which is defined to be the underlying set of the group   (in other words, the arrows are the group elements.) We take as composition the group multiplication. It follows that the identity element of   is  . Notice that in the category  , every morphism is an isomorphism (invertible under composition). Conversely, any one-object category in which all arrows are isomorphisms can be viewed as a group; the elements of the group are the arrows and the multiplication is the composition of the category. This describes an equivalence between the category of groups and homomorphisms and the category of small categories with a single object in which every morphism is an isomorphism.

This can be generalized in two ways.

A category   is called a groupoid if every morphism is an isomorphism. Thus a groupoid can be called "a group with many objects."

A monoid is a set with an associative binary operation that has an identity element. By the same technique as for groups, any monoid "is" a category with exactly one object and any category with exactly one object "is" a monoid.

MatricesEdit

This is a good example of a category whose objects are not sets and whose arrows are not functions.   is the category whose objects are the positive integers and whose arrows   are   matrices where composition is matrix multiplication, for any commutative ring  . For any object  ,  , the   identity matrix.

Categories of sets with structureEdit

finite sets and functions; denoted FinSet.
monoids and morphisms; denoted Mon.
groups and homomorphisms; denoted Grp.
abelian groups and homomorphisms; denoted Ab.
rings and unit-preserving homomorphisms; denoted Rng.
commutative rings and unit-preserving homomorphisms; denoted CRng.
left modules over a ring   and linear maps; denoted  -Mod.
right modules over a ring   and linear maps; denoted Mod- .
modules over a commutative ring   and linear maps; denoted  -Mod.
subsets of Euclidean space of 3 dimensions and Euclidean movements
subsets of Euclidean space of n dimensions and continuous functions
topological spaces and continuous functions; denoted Top.
topological spaces and homotopy classes of functions; denoted Toph.
Remarks

The law of composition is not specified explicitly in describing these categories. This is the custom when the objects have underlying set-structure, the morphisms are functions of the underlying sets (transporting the additional structure), and the law of composition is merely ordinary function-composition. Indeed, sometimes even the specification of the morphisms is suppressed if no confusion would arise—thus one speaks of the category of groups.

The examples of sets with structure suggest a conceptual framework. For example, the concept of group may be regarded as constituting a first-order abstraction or generalization from various concrete, familiar realizations such as the additive group of integers, the multiplicative group of nonzero rationals, groups of permutations, symmetry groups, groups of Euclidean motions, and so on. Then, again, the notion of a category constitutes a second-order abstraction, the concrete realizations of which consist of such first-order abstractions as the category of groups, the category of rings, the category of topological spaces, and so on.

Properties of objects and morphismsEdit

IsomorphismsEdit

A morphism   in a category is said to be an isomorphism if there is a morphism   in the category with  ,  . It is easy to prove that   is then uniquely determined by  . The morphism   is called the inverse of f, written  . It follows that  . If there is an isomorphism from   to  , we say   is isomorphic to  , and it is easy to prove that "isomorphism" is an equivalence relation on the objects of the category.

ExamplesEdit

  • A function from   to   in the category of sets is an isomorphism if and only if it is bijective.
  • A homomorphism of groups is an isomorphism if and only if it is bijective.
  • The isomorphisms of the category of topological spaces and continuous maps are the homeomorphisms. In contrast to the preceding example, a bijective continuous map from one topological space to another need not be a homeomorphism because its inverse (as a set function) may not be continuous. An example is the identity map on the set of real numbers, with the domain having the discrete topology and the codomain having the usual topology.

Monomorphisms and EpimorphismsEdit

A morphism   in a category   is a monomorphism if, for any morphisms   and  , if   and   are defined and   then  .


A morphism   in a category   is an epimorphism if, for any morphisms   and  , if   and   are defined and   then  .

Initial and terminal objectsEdit

  is said to be a terminal (or final) object when   is a unique morphism for any   in  . The law of composition ensures that if   and   are terminal objects in  , they are isomorphic, i.e.,   is unique up to isomorphism. In the categories of sets, groups, and topological spaces, the terminal objects are singletons, trivial groups, and one-point spaces, respectively. "b" is the terminal object in 2 as depicted above.

Constructions on categoriesEdit

SubcategoriesEdit

A subcategory   of a category   is a category in which:

  • The class of objects of   is contained in the class of objects of  .
  • The class of arrows of   is contained in the class of arrows of  .
  • For every arrow   in  , the domain and codomain of   are in  .
  • For every object   in  , the identity arrow   is in  .
  • For every pair of arrows   in  , the arrow   is in   where it is defined.

Opposite categoryEdit

Given a category  , the opposite (or dual) category   has the same objects as   and for every arrow   in  , the arrow   is in  . In other words, it has the same objects, and arrows are reversed.

The product of two categoriesEdit

Given two categories   and  , the product category, denoted  , is given by the following data:

  • The objects of   are   where   is an object of   and   is an object of  .
  • The arrows of   are   where   is an arrow of   and   is an arrow of  .
  • Composition is given by  .
  • The product   is called the cylinder category, denoted  

Arrow categoriesEdit

  • A functor category   is a category whose objects are functors   and arrows are natural transformations.
  • A functor category   is called an arrow category, denoted  . Its objects are arrows   in   and its arrows are pairs of arrows   such that  .

Comma categoriesEdit

  • Given categories and functors   and  , the comma category   has objects   where   is an object in  ,   is an object in  , and   is an arrow in  . Its arrows are   where   is an arrow in   and   is an arrow in   such that  . Composition is given by  . The identity is  .

Special types of comma categoriesEdit

Using the definition of comma category above, assume that we have  ,  , and  . Let   denote the object in  . Then   for some object   in  . In this case, we write the comma category as   and call it the slice category of   (or the category of objects over  ).

Now assume, instead, that we have  ,  , and  . Let   denote the object in  . Then   for some object   in  . In this case, we write the comma category as   and call it the coslice category of   (or the category of objects under  ).

Finally, if we have   and  , then the comma category  .